S2m: Coupled mass-spring systems


Example 1

S2m: Coupled mass-spring systems (ver. 1)

Consider a coupled mass-spring system where the inner mass is \(1\) kg, the outer mass is \(1\) kg, the inner spring has constant \(15\) N/m, the outer spring has constant \(4\) N/m. The inner mass is moved \(5\) meters inwards from its natural position, while the outer mass is moved \(\frac{29}{4}\) meters inwards from its natural position. Both masses are then simultaneously released from rest.

Give a system of IVPs that models this scenario, then solve the system. Use your solution to find the position of both masses after \(3\) seconds.

Answer.

The IVP system is given by:

\[1 x_1''=- 19 x_1+ 4 x_2\hspace{2em}x_1(0)= -5 ,x_1'(0)=0\]

\[1 x_2''= 4 x_1- 4 x_2\hspace{2em}x_2(0)= -\frac{29}{4} ,x_2'(0)=0\]

This system solves to:

\[x_1= -3 \, \cos\left(2 \, \sqrt{5} t\right) - 2 \, \cos\left(\sqrt{3} t\right)\]

\[x_2= \frac{3}{4} \, \cos\left(2 \, \sqrt{5} t\right) - 8 \, \cos\left(\sqrt{3} t\right)\]

Thus after \(3\) seconds, the inner mass is located approximately \(-2.91\) meters from its natural position, and the outer mass is located approximately \(-3.23\) meters from its natural position.


Example 2

S2m: Coupled mass-spring systems (ver. 2)

Consider a coupled mass-spring system where the inner mass is \(2\) kg, the outer mass is \(1\) kg, the inner spring has constant \(4\) N/m, the outer spring has constant \(2\) N/m. The inner mass is moved \(6\) meters outwards from its natural position, while the outer mass is moved \(3\) meters outwards from its natural position. Both masses are then simultaneously released from rest.

Give a system of IVPs that models this scenario, then solve the system. Use your solution to find the position of both masses after \(5\) seconds.

Answer.

The IVP system is given by:

\[2 x_1''=- 6 x_1+ 2 x_2\hspace{2em}x_1(0)= 6 ,x_1'(0)=0\]

\[1 x_2''= 2 x_1- 2 x_2\hspace{2em}x_2(0)= 3 ,x_2'(0)=0\]

This system solves to:

\[x_1= 3 \, \cos\left(2 \, t\right) + 3 \, \cos\left(t\right)\]

\[x_2= -3 \, \cos\left(2 \, t\right) + 6 \, \cos\left(t\right)\]

Thus after \(5\) seconds, the inner mass is located approximately \(-1.67\) meters from its natural position, and the outer mass is located approximately \(4.22\) meters from its natural position.


Example 3

S2m: Coupled mass-spring systems (ver. 3)

Consider a coupled mass-spring system where the inner mass is \(1\) kg, the outer mass is \(1\) kg, the inner spring has constant \(15\) N/m, the outer spring has constant \(4\) N/m. The inner mass is moved \(10\) meters outwards from its natural position, while the outer mass is moved \(\frac{75}{4}\) meters outwards from its natural position. Both masses are then simultaneously released from rest.

Give a system of IVPs that models this scenario, then solve the system. Use your solution to find the position of both masses after \(3\) seconds.

Answer.

The IVP system is given by:

\[1 x_1''=- 19 x_1+ 4 x_2\hspace{2em}x_1(0)= 10 ,x_1'(0)=0\]

\[1 x_2''= 4 x_1- 4 x_2\hspace{2em}x_2(0)= \frac{75}{4} ,x_2'(0)=0\]

This system solves to:

\[x_1= 5 \, \cos\left(2 \, \sqrt{5} t\right) + 5 \, \cos\left(\sqrt{3} t\right)\]

\[x_2= -\frac{5}{4} \, \cos\left(2 \, \sqrt{5} t\right) + 20 \, \cos\left(\sqrt{3} t\right)\]

Thus after \(3\) seconds, the inner mass is located approximately \(5.63\) meters from its natural position, and the outer mass is located approximately \(8.48\) meters from its natural position.


Example 4

S2m: Coupled mass-spring systems (ver. 4)

Consider a coupled mass-spring system where the inner mass is \(2\) kg, the outer mass is \(1\) kg, the inner spring has constant \(18\) N/m, the outer spring has constant \(4\) N/m. The inner mass is moved \(2\) meters outwards from its natural position, while the outer mass is moved \(10\) meters inwards from its natural position. Both masses are then simultaneously released from rest.

Give a system of IVPs that models this scenario, then solve the system. Use your solution to find the position of both masses after \(4\) seconds.

Answer.

The IVP system is given by:

\[2 x_1''=- 22 x_1+ 4 x_2\hspace{2em}x_1(0)= 2 ,x_1'(0)=0\]

\[1 x_2''= 4 x_1- 4 x_2\hspace{2em}x_2(0)= -10 ,x_2'(0)=0\]

This system solves to:

\[x_1= 4 \, \cos\left(2 \, \sqrt{3} t\right) - 2 \, \cos\left(\sqrt{3} t\right)\]

\[x_2= -2 \, \cos\left(2 \, \sqrt{3} t\right) - 8 \, \cos\left(\sqrt{3} t\right)\]

Thus after \(4\) seconds, the inner mass is located approximately \(-0.490\) meters from its natural position, and the outer mass is located approximately \(-6.95\) meters from its natural position.


Example 5

S2m: Coupled mass-spring systems (ver. 5)

Consider a coupled mass-spring system where the inner mass is \(2\) kg, the outer mass is \(1\) kg, the inner spring has constant \(18\) N/m, the outer spring has constant \(4\) N/m. The inner mass is moved \(1\) meters inwards from its natural position, while the outer mass is moved \(14\) meters outwards from its natural position. Both masses are then simultaneously released from rest.

Give a system of IVPs that models this scenario, then solve the system. Use your solution to find the position of both masses after \(2\) seconds.

Answer.

The IVP system is given by:

\[2 x_1''=- 22 x_1+ 4 x_2\hspace{2em}x_1(0)= -1 ,x_1'(0)=0\]

\[1 x_2''= 4 x_1- 4 x_2\hspace{2em}x_2(0)= 14 ,x_2'(0)=0\]

This system solves to:

\[x_1= -4 \, \cos\left(2 \, \sqrt{3} t\right) + 3 \, \cos\left(\sqrt{3} t\right)\]

\[x_2= 2 \, \cos\left(2 \, \sqrt{3} t\right) + 12 \, \cos\left(\sqrt{3} t\right)\]

Thus after \(2\) seconds, the inner mass is located approximately \(-6.04\) meters from its natural position, and the outer mass is located approximately \(-9.78\) meters from its natural position.


Example 6

S2m: Coupled mass-spring systems (ver. 6)

Consider a coupled mass-spring system where the inner mass is \(1\) kg, the outer mass is \(1\) kg, the inner spring has constant \(3\) N/m, the outer spring has constant \(2\) N/m. The inner mass is moved \(0\) meters outwards from its natural position, while the outer mass is moved \(5\) meters outwards from its natural position. Both masses are then simultaneously released from rest.

Give a system of IVPs that models this scenario, then solve the system. Use your solution to find the position of both masses after \(4\) seconds.

Answer.

The IVP system is given by:

\[1 x_1''=- 5 x_1+ 2 x_2\hspace{2em}x_1(0)= 0 ,x_1'(0)=0\]

\[1 x_2''= 2 x_1- 2 x_2\hspace{2em}x_2(0)= 5 ,x_2'(0)=0\]

This system solves to:

\[x_1= -2 \, \cos\left(\sqrt{6} t\right) + 2 \, \cos\left(t\right)\]

\[x_2= \cos\left(\sqrt{6} t\right) + 4 \, \cos\left(t\right)\]

Thus after \(4\) seconds, the inner mass is located approximately \(0.555\) meters from its natural position, and the outer mass is located approximately \(-3.55\) meters from its natural position.


Example 7

S2m: Coupled mass-spring systems (ver. 7)

Consider a coupled mass-spring system where the inner mass is \(1\) kg, the outer mass is \(1\) kg, the inner spring has constant \(3\) N/m, the outer spring has constant \(2\) N/m. The inner mass is moved \(8\) meters inwards from its natural position, while the outer mass is moved \(\frac{17}{2}\) meters inwards from its natural position. Both masses are then simultaneously released from rest.

Give a system of IVPs that models this scenario, then solve the system. Use your solution to find the position of both masses after \(5\) seconds.

Answer.

The IVP system is given by:

\[1 x_1''=- 5 x_1+ 2 x_2\hspace{2em}x_1(0)= -8 ,x_1'(0)=0\]

\[1 x_2''= 2 x_1- 2 x_2\hspace{2em}x_2(0)= -\frac{17}{2} ,x_2'(0)=0\]

This system solves to:

\[x_1= -3 \, \cos\left(\sqrt{6} t\right) - 5 \, \cos\left(t\right)\]

\[x_2= \frac{3}{2} \, \cos\left(\sqrt{6} t\right) - 10 \, \cos\left(t\right)\]

Thus after \(5\) seconds, the inner mass is located approximately \(-4.27\) meters from its natural position, and the outer mass is located approximately \(-1.41\) meters from its natural position.


Example 8

S2m: Coupled mass-spring systems (ver. 8)

Consider a coupled mass-spring system where the inner mass is \(1\) kg, the outer mass is \(2\) kg, the inner spring has constant \(14\) N/m, the outer spring has constant \(6\) N/m. The inner mass is moved \(2\) meters inwards from its natural position, while the outer mass is moved \(\frac{20}{3}\) meters outwards from its natural position. Both masses are then simultaneously released from rest.

Give a system of IVPs that models this scenario, then solve the system. Use your solution to find the position of both masses after \(2\) seconds.

Answer.

The IVP system is given by:

\[1 x_1''=- 20 x_1+ 6 x_2\hspace{2em}x_1(0)= -2 ,x_1'(0)=0\]

\[2 x_2''= 6 x_1- 6 x_2\hspace{2em}x_2(0)= \frac{20}{3} ,x_2'(0)=0\]

This system solves to:

\[x_1= -4 \, \cos\left(\sqrt{21} t\right) + 2 \, \cos\left(\sqrt{2} t\right)\]

\[x_2= \frac{2}{3} \, \cos\left(\sqrt{21} t\right) + 6 \, \cos\left(\sqrt{2} t\right)\]

Thus after \(2\) seconds, the inner mass is located approximately \(1.96\) meters from its natural position, and the outer mass is located approximately \(-6.35\) meters from its natural position.


Example 9

S2m: Coupled mass-spring systems (ver. 9)

Consider a coupled mass-spring system where the inner mass is \(1\) kg, the outer mass is \(1\) kg, the inner spring has constant \(3\) N/m, the outer spring has constant \(2\) N/m. The inner mass is moved \(9\) meters outwards from its natural position, while the outer mass is moved \(8\) meters outwards from its natural position. Both masses are then simultaneously released from rest.

Give a system of IVPs that models this scenario, then solve the system. Use your solution to find the position of both masses after \(4\) seconds.

Answer.

The IVP system is given by:

\[1 x_1''=- 5 x_1+ 2 x_2\hspace{2em}x_1(0)= 9 ,x_1'(0)=0\]

\[1 x_2''= 2 x_1- 2 x_2\hspace{2em}x_2(0)= 8 ,x_2'(0)=0\]

This system solves to:

\[x_1= 4 \, \cos\left(\sqrt{6} t\right) + 5 \, \cos\left(t\right)\]

\[x_2= -2 \, \cos\left(\sqrt{6} t\right) + 10 \, \cos\left(t\right)\]

Thus after \(4\) seconds, the inner mass is located approximately \(-6.99\) meters from its natural position, and the outer mass is located approximately \(-4.67\) meters from its natural position.


Example 10

S2m: Coupled mass-spring systems (ver. 10)

Consider a coupled mass-spring system where the inner mass is \(1\) kg, the outer mass is \(1\) kg, the inner spring has constant \(15\) N/m, the outer spring has constant \(4\) N/m. The inner mass is moved \(2\) meters inwards from its natural position, while the outer mass is moved \(\frac{33}{2}\) meters inwards from its natural position. Both masses are then simultaneously released from rest.

Give a system of IVPs that models this scenario, then solve the system. Use your solution to find the position of both masses after \(3\) seconds.

Answer.

The IVP system is given by:

\[1 x_1''=- 19 x_1+ 4 x_2\hspace{2em}x_1(0)= -2 ,x_1'(0)=0\]

\[1 x_2''= 4 x_1- 4 x_2\hspace{2em}x_2(0)= -\frac{33}{2} ,x_2'(0)=0\]

This system solves to:

\[x_1= 2 \, \cos\left(2 \, \sqrt{5} t\right) - 4 \, \cos\left(\sqrt{3} t\right)\]

\[x_2= -\frac{1}{2} \, \cos\left(2 \, \sqrt{5} t\right) - 16 \, \cos\left(\sqrt{3} t\right)\]

Thus after \(3\) seconds, the inner mass is located approximately \(-0.540\) meters from its natural position, and the outer mass is located approximately \(-7.77\) meters from its natural position.


Example 11

S2m: Coupled mass-spring systems (ver. 11)

Consider a coupled mass-spring system where the inner mass is \(2\) kg, the outer mass is \(1\) kg, the inner spring has constant \(10\) N/m, the outer spring has constant \(12\) N/m. The inner mass is moved \(7\) meters inwards from its natural position, while the outer mass is moved \(2\) meters outwards from its natural position. Both masses are then simultaneously released from rest.

Give a system of IVPs that models this scenario, then solve the system. Use your solution to find the position of both masses after \(2\) seconds.

Answer.

The IVP system is given by:

\[2 x_1''=- 22 x_1+ 12 x_2\hspace{2em}x_1(0)= -7 ,x_1'(0)=0\]

\[1 x_2''= 12 x_1- 12 x_2\hspace{2em}x_2(0)= 2 ,x_2'(0)=0\]

This system solves to:

\[x_1= -4 \, \cos\left(2 \, \sqrt{5} t\right) - 3 \, \cos\left(\sqrt{3} t\right)\]

\[x_2= 6 \, \cos\left(2 \, \sqrt{5} t\right) - 4 \, \cos\left(\sqrt{3} t\right)\]

Thus after \(2\) seconds, the inner mass is located approximately \(6.39\) meters from its natural position, and the outer mass is located approximately \(-1.53\) meters from its natural position.


Example 12

S2m: Coupled mass-spring systems (ver. 12)

Consider a coupled mass-spring system where the inner mass is \(1\) kg, the outer mass is \(2\) kg, the inner spring has constant \(14\) N/m, the outer spring has constant \(6\) N/m. The inner mass is moved \(8\) meters outwards from its natural position, while the outer mass is moved \(\frac{34}{3}\) meters outwards from its natural position. Both masses are then simultaneously released from rest.

Give a system of IVPs that models this scenario, then solve the system. Use your solution to find the position of both masses after \(2\) seconds.

Answer.

The IVP system is given by:

\[1 x_1''=- 20 x_1+ 6 x_2\hspace{2em}x_1(0)= 8 ,x_1'(0)=0\]

\[2 x_2''= 6 x_1- 6 x_2\hspace{2em}x_2(0)= \frac{34}{3} ,x_2'(0)=0\]

This system solves to:

\[x_1= 4 \, \cos\left(\sqrt{21} t\right) + 4 \, \cos\left(\sqrt{2} t\right)\]

\[x_2= -\frac{2}{3} \, \cos\left(\sqrt{21} t\right) + 12 \, \cos\left(\sqrt{2} t\right)\]

Thus after \(2\) seconds, the inner mass is located approximately \(-7.67\) meters from its natural position, and the outer mass is located approximately \(-10.8\) meters from its natural position.


Example 13

S2m: Coupled mass-spring systems (ver. 13)

Consider a coupled mass-spring system where the inner mass is \(1\) kg, the outer mass is \(2\) kg, the inner spring has constant \(5\) N/m, the outer spring has constant \(4\) N/m. The inner mass is moved \(6\) meters outwards from its natural position, while the outer mass is moved \(\frac{15}{2}\) meters outwards from its natural position. Both masses are then simultaneously released from rest.

Give a system of IVPs that models this scenario, then solve the system. Use your solution to find the position of both masses after \(2\) seconds.

Answer.

The IVP system is given by:

\[1 x_1''=- 9 x_1+ 4 x_2\hspace{2em}x_1(0)= 6 ,x_1'(0)=0\]

\[2 x_2''= 4 x_1- 4 x_2\hspace{2em}x_2(0)= \frac{15}{2} ,x_2'(0)=0\]

This system solves to:

\[x_1= 2 \, \cos\left(\sqrt{10} t\right) + 4 \, \cos\left(t\right)\]

\[x_2= -\frac{1}{2} \, \cos\left(\sqrt{10} t\right) + 8 \, \cos\left(t\right)\]

Thus after \(2\) seconds, the inner mass is located approximately \(0.334\) meters from its natural position, and the outer mass is located approximately \(-3.83\) meters from its natural position.


Example 14

S2m: Coupled mass-spring systems (ver. 14)

Consider a coupled mass-spring system where the inner mass is \(1\) kg, the outer mass is \(2\) kg, the inner spring has constant \(5\) N/m, the outer spring has constant \(4\) N/m. The inner mass is moved \(0\) meters outwards from its natural position, while the outer mass is moved \(9\) meters inwards from its natural position. Both masses are then simultaneously released from rest.

Give a system of IVPs that models this scenario, then solve the system. Use your solution to find the position of both masses after \(3\) seconds.

Answer.

The IVP system is given by:

\[1 x_1''=- 9 x_1+ 4 x_2\hspace{2em}x_1(0)= 0 ,x_1'(0)=0\]

\[2 x_2''= 4 x_1- 4 x_2\hspace{2em}x_2(0)= -9 ,x_2'(0)=0\]

This system solves to:

\[x_1= 4 \, \cos\left(\sqrt{10} t\right) - 4 \, \cos\left(t\right)\]

\[x_2= -\cos\left(\sqrt{10} t\right) - 8 \, \cos\left(t\right)\]

Thus after \(3\) seconds, the inner mass is located approximately \(-0.0322\) meters from its natural position, and the outer mass is located approximately \(8.92\) meters from its natural position.


Example 15

S2m: Coupled mass-spring systems (ver. 15)

Consider a coupled mass-spring system where the inner mass is \(2\) kg, the outer mass is \(1\) kg, the inner spring has constant \(10\) N/m, the outer spring has constant \(12\) N/m. The inner mass is moved \(2\) meters inwards from its natural position, while the outer mass is moved \(\frac{25}{3}\) meters inwards from its natural position. Both masses are then simultaneously released from rest.

Give a system of IVPs that models this scenario, then solve the system. Use your solution to find the position of both masses after \(3\) seconds.

Answer.

The IVP system is given by:

\[2 x_1''=- 22 x_1+ 12 x_2\hspace{2em}x_1(0)= -2 ,x_1'(0)=0\]

\[1 x_2''= 12 x_1- 12 x_2\hspace{2em}x_2(0)= -\frac{25}{3} ,x_2'(0)=0\]

This system solves to:

\[x_1= 2 \, \cos\left(2 \, \sqrt{5} t\right) - 4 \, \cos\left(\sqrt{3} t\right)\]

\[x_2= -3 \, \cos\left(2 \, \sqrt{5} t\right) - \frac{16}{3} \, \cos\left(\sqrt{3} t\right)\]

Thus after \(3\) seconds, the inner mass is located approximately \(-0.540\) meters from its natural position, and the outer mass is located approximately \(-4.46\) meters from its natural position.


Example 16

S2m: Coupled mass-spring systems (ver. 16)

Consider a coupled mass-spring system where the inner mass is \(1\) kg, the outer mass is \(1\) kg, the inner spring has constant \(15\) N/m, the outer spring has constant \(4\) N/m. The inner mass is moved \(2\) meters inwards from its natural position, while the outer mass is moved \(\frac{33}{2}\) meters inwards from its natural position. Both masses are then simultaneously released from rest.

Give a system of IVPs that models this scenario, then solve the system. Use your solution to find the position of both masses after \(4\) seconds.

Answer.

The IVP system is given by:

\[1 x_1''=- 19 x_1+ 4 x_2\hspace{2em}x_1(0)= -2 ,x_1'(0)=0\]

\[1 x_2''= 4 x_1- 4 x_2\hspace{2em}x_2(0)= -\frac{33}{2} ,x_2'(0)=0\]

This system solves to:

\[x_1= 2 \, \cos\left(2 \, \sqrt{5} t\right) - 4 \, \cos\left(\sqrt{3} t\right)\]

\[x_2= -\frac{1}{2} \, \cos\left(2 \, \sqrt{5} t\right) - 16 \, \cos\left(\sqrt{3} t\right)\]

Thus after \(4\) seconds, the inner mass is located approximately \(-2.05\) meters from its natural position, and the outer mass is located approximately \(-13.1\) meters from its natural position.


Example 17

S2m: Coupled mass-spring systems (ver. 17)

Consider a coupled mass-spring system where the inner mass is \(2\) kg, the outer mass is \(1\) kg, the inner spring has constant \(4\) N/m, the outer spring has constant \(2\) N/m. The inner mass is moved \(3\) meters outwards from its natural position, while the outer mass is moved \(9\) meters inwards from its natural position. Both masses are then simultaneously released from rest.

Give a system of IVPs that models this scenario, then solve the system. Use your solution to find the position of both masses after \(3\) seconds.

Answer.

The IVP system is given by:

\[2 x_1''=- 6 x_1+ 2 x_2\hspace{2em}x_1(0)= 3 ,x_1'(0)=0\]

\[1 x_2''= 2 x_1- 2 x_2\hspace{2em}x_2(0)= -9 ,x_2'(0)=0\]

This system solves to:

\[x_1= 5 \, \cos\left(2 \, t\right) - 2 \, \cos\left(t\right)\]

\[x_2= -5 \, \cos\left(2 \, t\right) - 4 \, \cos\left(t\right)\]

Thus after \(3\) seconds, the inner mass is located approximately \(6.78\) meters from its natural position, and the outer mass is located approximately \(-0.841\) meters from its natural position.


Example 18

S2m: Coupled mass-spring systems (ver. 18)

Consider a coupled mass-spring system where the inner mass is \(1\) kg, the outer mass is \(2\) kg, the inner spring has constant \(5\) N/m, the outer spring has constant \(4\) N/m. The inner mass is moved \(5\) meters outwards from its natural position, while the outer mass is moved \(\frac{13}{4}\) meters outwards from its natural position. Both masses are then simultaneously released from rest.

Give a system of IVPs that models this scenario, then solve the system. Use your solution to find the position of both masses after \(3\) seconds.

Answer.

The IVP system is given by:

\[1 x_1''=- 9 x_1+ 4 x_2\hspace{2em}x_1(0)= 5 ,x_1'(0)=0\]

\[2 x_2''= 4 x_1- 4 x_2\hspace{2em}x_2(0)= \frac{13}{4} ,x_2'(0)=0\]

This system solves to:

\[x_1= 3 \, \cos\left(\sqrt{10} t\right) + 2 \, \cos\left(t\right)\]

\[x_2= -\frac{3}{4} \, \cos\left(\sqrt{10} t\right) + 4 \, \cos\left(t\right)\]

Thus after \(3\) seconds, the inner mass is located approximately \(-4.97\) meters from its natural position, and the outer mass is located approximately \(-3.21\) meters from its natural position.


Example 19

S2m: Coupled mass-spring systems (ver. 19)

Consider a coupled mass-spring system where the inner mass is \(1\) kg, the outer mass is \(1\) kg, the inner spring has constant \(5\) N/m, the outer spring has constant \(6\) N/m. The inner mass is moved \(2\) meters outwards from its natural position, while the outer mass is moved \(\frac{47}{6}\) meters inwards from its natural position. Both masses are then simultaneously released from rest.

Give a system of IVPs that models this scenario, then solve the system. Use your solution to find the position of both masses after \(4\) seconds.

Answer.

The IVP system is given by:

\[1 x_1''=- 11 x_1+ 6 x_2\hspace{2em}x_1(0)= 2 ,x_1'(0)=0\]

\[1 x_2''= 6 x_1- 6 x_2\hspace{2em}x_2(0)= -\frac{47}{6} ,x_2'(0)=0\]

This system solves to:

\[x_1= 5 \, \cos\left(\sqrt{15} t\right) - 3 \, \cos\left(\sqrt{2} t\right)\]

\[x_2= -\frac{10}{3} \, \cos\left(\sqrt{15} t\right) - \frac{9}{2} \, \cos\left(\sqrt{2} t\right)\]

Thus after \(4\) seconds, the inner mass is located approximately \(-7.31\) meters from its natural position, and the outer mass is located approximately \(-0.390\) meters from its natural position.


Example 20

S2m: Coupled mass-spring systems (ver. 20)

Consider a coupled mass-spring system where the inner mass is \(1\) kg, the outer mass is \(1\) kg, the inner spring has constant \(5\) N/m, the outer spring has constant \(6\) N/m. The inner mass is moved \(8\) meters outwards from its natural position, while the outer mass is moved \(\frac{10}{3}\) meters outwards from its natural position. Both masses are then simultaneously released from rest.

Give a system of IVPs that models this scenario, then solve the system. Use your solution to find the position of both masses after \(4\) seconds.

Answer.

The IVP system is given by:

\[1 x_1''=- 11 x_1+ 6 x_2\hspace{2em}x_1(0)= 8 ,x_1'(0)=0\]

\[1 x_2''= 6 x_1- 6 x_2\hspace{2em}x_2(0)= \frac{10}{3} ,x_2'(0)=0\]

This system solves to:

\[x_1= 4 \, \cos\left(\sqrt{15} t\right) + 4 \, \cos\left(\sqrt{2} t\right)\]

\[x_2= -\frac{8}{3} \, \cos\left(\sqrt{15} t\right) + 6 \, \cos\left(\sqrt{2} t\right)\]

Thus after \(4\) seconds, the inner mass is located approximately \(-0.667\) meters from its natural position, and the outer mass is located approximately \(7.47\) meters from its natural position.


Example 21

S2m: Coupled mass-spring systems (ver. 21)

Consider a coupled mass-spring system where the inner mass is \(2\) kg, the outer mass is \(1\) kg, the inner spring has constant \(10\) N/m, the outer spring has constant \(12\) N/m. The inner mass is moved \(1\) meters outwards from its natural position, while the outer mass is moved \(\frac{43}{6}\) meters inwards from its natural position. Both masses are then simultaneously released from rest.

Give a system of IVPs that models this scenario, then solve the system. Use your solution to find the position of both masses after \(5\) seconds.

Answer.

The IVP system is given by:

\[2 x_1''=- 22 x_1+ 12 x_2\hspace{2em}x_1(0)= 1 ,x_1'(0)=0\]

\[1 x_2''= 12 x_1- 12 x_2\hspace{2em}x_2(0)= -\frac{43}{6} ,x_2'(0)=0\]

This system solves to:

\[x_1= 3 \, \cos\left(2 \, \sqrt{5} t\right) - 2 \, \cos\left(\sqrt{3} t\right)\]

\[x_2= -\frac{9}{2} \, \cos\left(2 \, \sqrt{5} t\right) - \frac{8}{3} \, \cos\left(\sqrt{3} t\right)\]

Thus after \(5\) seconds, the inner mass is located approximately \(-1.35\) meters from its natural position, and the outer mass is located approximately \(6.12\) meters from its natural position.


Example 22

S2m: Coupled mass-spring systems (ver. 22)

Consider a coupled mass-spring system where the inner mass is \(1\) kg, the outer mass is \(2\) kg, the inner spring has constant \(14\) N/m, the outer spring has constant \(6\) N/m. The inner mass is moved \(1\) meters outwards from its natural position, while the outer mass is moved \(\frac{25}{2}\) meters outwards from its natural position. Both masses are then simultaneously released from rest.

Give a system of IVPs that models this scenario, then solve the system. Use your solution to find the position of both masses after \(5\) seconds.

Answer.

The IVP system is given by:

\[1 x_1''=- 20 x_1+ 6 x_2\hspace{2em}x_1(0)= 1 ,x_1'(0)=0\]

\[2 x_2''= 6 x_1- 6 x_2\hspace{2em}x_2(0)= \frac{25}{2} ,x_2'(0)=0\]

This system solves to:

\[x_1= -3 \, \cos\left(\sqrt{21} t\right) + 4 \, \cos\left(\sqrt{2} t\right)\]

\[x_2= \frac{1}{2} \, \cos\left(\sqrt{21} t\right) + 12 \, \cos\left(\sqrt{2} t\right)\]

Thus after \(5\) seconds, the inner mass is located approximately \(4.64\) meters from its natural position, and the outer mass is located approximately \(8.16\) meters from its natural position.


Example 23

S2m: Coupled mass-spring systems (ver. 23)

Consider a coupled mass-spring system where the inner mass is \(1\) kg, the outer mass is \(2\) kg, the inner spring has constant \(14\) N/m, the outer spring has constant \(6\) N/m. The inner mass is moved \(1\) meters outwards from its natural position, while the outer mass is moved \(\frac{25}{2}\) meters outwards from its natural position. Both masses are then simultaneously released from rest.

Give a system of IVPs that models this scenario, then solve the system. Use your solution to find the position of both masses after \(3\) seconds.

Answer.

The IVP system is given by:

\[1 x_1''=- 20 x_1+ 6 x_2\hspace{2em}x_1(0)= 1 ,x_1'(0)=0\]

\[2 x_2''= 6 x_1- 6 x_2\hspace{2em}x_2(0)= \frac{25}{2} ,x_2'(0)=0\]

This system solves to:

\[x_1= -3 \, \cos\left(\sqrt{21} t\right) + 4 \, \cos\left(\sqrt{2} t\right)\]

\[x_2= \frac{1}{2} \, \cos\left(\sqrt{21} t\right) + 12 \, \cos\left(\sqrt{2} t\right)\]

Thus after \(3\) seconds, the inner mass is located approximately \(-2.95\) meters from its natural position, and the outer mass is located approximately \(-5.24\) meters from its natural position.


Example 24

S2m: Coupled mass-spring systems (ver. 24)

Consider a coupled mass-spring system where the inner mass is \(1\) kg, the outer mass is \(2\) kg, the inner spring has constant \(14\) N/m, the outer spring has constant \(6\) N/m. The inner mass is moved \(7\) meters outwards from its natural position, while the outer mass is moved \(\frac{31}{6}\) meters outwards from its natural position. Both masses are then simultaneously released from rest.

Give a system of IVPs that models this scenario, then solve the system. Use your solution to find the position of both masses after \(4\) seconds.

Answer.

The IVP system is given by:

\[1 x_1''=- 20 x_1+ 6 x_2\hspace{2em}x_1(0)= 7 ,x_1'(0)=0\]

\[2 x_2''= 6 x_1- 6 x_2\hspace{2em}x_2(0)= \frac{31}{6} ,x_2'(0)=0\]

This system solves to:

\[x_1= 5 \, \cos\left(\sqrt{21} t\right) + 2 \, \cos\left(\sqrt{2} t\right)\]

\[x_2= -\frac{5}{6} \, \cos\left(\sqrt{21} t\right) + 6 \, \cos\left(\sqrt{2} t\right)\]

Thus after \(4\) seconds, the inner mass is located approximately \(5.96\) meters from its natural position, and the outer mass is located approximately \(4.14\) meters from its natural position.


Example 25

S2m: Coupled mass-spring systems (ver. 25)

Consider a coupled mass-spring system where the inner mass is \(1\) kg, the outer mass is \(2\) kg, the inner spring has constant \(14\) N/m, the outer spring has constant \(6\) N/m. The inner mass is moved \(4\) meters inwards from its natural position, while the outer mass is moved \(\frac{17}{3}\) meters inwards from its natural position. Both masses are then simultaneously released from rest.

Give a system of IVPs that models this scenario, then solve the system. Use your solution to find the position of both masses after \(4\) seconds.

Answer.

The IVP system is given by:

\[1 x_1''=- 20 x_1+ 6 x_2\hspace{2em}x_1(0)= -4 ,x_1'(0)=0\]

\[2 x_2''= 6 x_1- 6 x_2\hspace{2em}x_2(0)= -\frac{17}{3} ,x_2'(0)=0\]

This system solves to:

\[x_1= -2 \, \cos\left(\sqrt{21} t\right) - 2 \, \cos\left(\sqrt{2} t\right)\]

\[x_2= \frac{1}{3} \, \cos\left(\sqrt{21} t\right) - 6 \, \cos\left(\sqrt{2} t\right)\]

Thus after \(4\) seconds, the inner mass is located approximately \(-3.36\) meters from its natural position, and the outer mass is located approximately \(-4.57\) meters from its natural position.


Example 26

S2m: Coupled mass-spring systems (ver. 26)

Consider a coupled mass-spring system where the inner mass is \(2\) kg, the outer mass is \(1\) kg, the inner spring has constant \(4\) N/m, the outer spring has constant \(2\) N/m. The inner mass is moved \(5\) meters inwards from its natural position, while the outer mass is moved \(4\) meters inwards from its natural position. Both masses are then simultaneously released from rest.

Give a system of IVPs that models this scenario, then solve the system. Use your solution to find the position of both masses after \(5\) seconds.

Answer.

The IVP system is given by:

\[2 x_1''=- 6 x_1+ 2 x_2\hspace{2em}x_1(0)= -5 ,x_1'(0)=0\]

\[1 x_2''= 2 x_1- 2 x_2\hspace{2em}x_2(0)= -4 ,x_2'(0)=0\]

This system solves to:

\[x_1= -2 \, \cos\left(2 \, t\right) - 3 \, \cos\left(t\right)\]

\[x_2= 2 \, \cos\left(2 \, t\right) - 6 \, \cos\left(t\right)\]

Thus after \(5\) seconds, the inner mass is located approximately \(0.827\) meters from its natural position, and the outer mass is located approximately \(-3.38\) meters from its natural position.


Example 27

S2m: Coupled mass-spring systems (ver. 27)

Consider a coupled mass-spring system where the inner mass is \(1\) kg, the outer mass is \(2\) kg, the inner spring has constant \(14\) N/m, the outer spring has constant \(6\) N/m. The inner mass is moved \(6\) meters inwards from its natural position, while the outer mass is moved \(\frac{35}{3}\) meters inwards from its natural position. Both masses are then simultaneously released from rest.

Give a system of IVPs that models this scenario, then solve the system. Use your solution to find the position of both masses after \(4\) seconds.

Answer.

The IVP system is given by:

\[1 x_1''=- 20 x_1+ 6 x_2\hspace{2em}x_1(0)= -6 ,x_1'(0)=0\]

\[2 x_2''= 6 x_1- 6 x_2\hspace{2em}x_2(0)= -\frac{35}{3} ,x_2'(0)=0\]

This system solves to:

\[x_1= -2 \, \cos\left(\sqrt{21} t\right) - 4 \, \cos\left(\sqrt{2} t\right)\]

\[x_2= \frac{1}{3} \, \cos\left(\sqrt{21} t\right) - 12 \, \cos\left(\sqrt{2} t\right)\]

Thus after \(4\) seconds, the inner mass is located approximately \(-4.98\) meters from its natural position, and the outer mass is located approximately \(-9.43\) meters from its natural position.


Example 28

S2m: Coupled mass-spring systems (ver. 28)

Consider a coupled mass-spring system where the inner mass is \(1\) kg, the outer mass is \(1\) kg, the inner spring has constant \(3\) N/m, the outer spring has constant \(2\) N/m. The inner mass is moved \(8\) meters inwards from its natural position, while the outer mass is moved \(\frac{7}{2}\) meters inwards from its natural position. Both masses are then simultaneously released from rest.

Give a system of IVPs that models this scenario, then solve the system. Use your solution to find the position of both masses after \(2\) seconds.

Answer.

The IVP system is given by:

\[1 x_1''=- 5 x_1+ 2 x_2\hspace{2em}x_1(0)= -8 ,x_1'(0)=0\]

\[1 x_2''= 2 x_1- 2 x_2\hspace{2em}x_2(0)= -\frac{7}{2} ,x_2'(0)=0\]

This system solves to:

\[x_1= -5 \, \cos\left(\sqrt{6} t\right) - 3 \, \cos\left(t\right)\]

\[x_2= \frac{5}{2} \, \cos\left(\sqrt{6} t\right) - 6 \, \cos\left(t\right)\]

Thus after \(2\) seconds, the inner mass is located approximately \(0.321\) meters from its natural position, and the outer mass is located approximately \(2.96\) meters from its natural position.


Example 29

S2m: Coupled mass-spring systems (ver. 29)

Consider a coupled mass-spring system where the inner mass is \(1\) kg, the outer mass is \(2\) kg, the inner spring has constant \(5\) N/m, the outer spring has constant \(4\) N/m. The inner mass is moved \(6\) meters outwards from its natural position, while the outer mass is moved \(3\) meters outwards from its natural position. Both masses are then simultaneously released from rest.

Give a system of IVPs that models this scenario, then solve the system. Use your solution to find the position of both masses after \(2\) seconds.

Answer.

The IVP system is given by:

\[1 x_1''=- 9 x_1+ 4 x_2\hspace{2em}x_1(0)= 6 ,x_1'(0)=0\]

\[2 x_2''= 4 x_1- 4 x_2\hspace{2em}x_2(0)= 3 ,x_2'(0)=0\]

This system solves to:

\[x_1= 4 \, \cos\left(\sqrt{10} t\right) + 2 \, \cos\left(t\right)\]

\[x_2= -\cos\left(\sqrt{10} t\right) + 4 \, \cos\left(t\right)\]

Thus after \(2\) seconds, the inner mass is located approximately \(3.16\) meters from its natural position, and the outer mass is located approximately \(-2.66\) meters from its natural position.


Example 30

S2m: Coupled mass-spring systems (ver. 30)

Consider a coupled mass-spring system where the inner mass is \(1\) kg, the outer mass is \(1\) kg, the inner spring has constant \(15\) N/m, the outer spring has constant \(4\) N/m. The inner mass is moved \(6\) meters inwards from its natural position, while the outer mass is moved \(\frac{31}{2}\) meters inwards from its natural position. Both masses are then simultaneously released from rest.

Give a system of IVPs that models this scenario, then solve the system. Use your solution to find the position of both masses after \(5\) seconds.

Answer.

The IVP system is given by:

\[1 x_1''=- 19 x_1+ 4 x_2\hspace{2em}x_1(0)= -6 ,x_1'(0)=0\]

\[1 x_2''= 4 x_1- 4 x_2\hspace{2em}x_2(0)= -\frac{31}{2} ,x_2'(0)=0\]

This system solves to:

\[x_1= -2 \, \cos\left(2 \, \sqrt{5} t\right) - 4 \, \cos\left(\sqrt{3} t\right)\]

\[x_2= \frac{1}{2} \, \cos\left(2 \, \sqrt{5} t\right) - 16 \, \cos\left(\sqrt{3} t\right)\]

Thus after \(5\) seconds, the inner mass is located approximately \(4.75\) meters from its natural position, and the outer mass is located approximately \(11.1\) meters from its natural position.


Example 31

S2m: Coupled mass-spring systems (ver. 31)

Consider a coupled mass-spring system where the inner mass is \(1\) kg, the outer mass is \(1\) kg, the inner spring has constant \(5\) N/m, the outer spring has constant \(6\) N/m. The inner mass is moved \(2\) meters outwards from its natural position, while the outer mass is moved \(\frac{47}{6}\) meters inwards from its natural position. Both masses are then simultaneously released from rest.

Give a system of IVPs that models this scenario, then solve the system. Use your solution to find the position of both masses after \(4\) seconds.

Answer.

The IVP system is given by:

\[1 x_1''=- 11 x_1+ 6 x_2\hspace{2em}x_1(0)= 2 ,x_1'(0)=0\]

\[1 x_2''= 6 x_1- 6 x_2\hspace{2em}x_2(0)= -\frac{47}{6} ,x_2'(0)=0\]

This system solves to:

\[x_1= 5 \, \cos\left(\sqrt{15} t\right) - 3 \, \cos\left(\sqrt{2} t\right)\]

\[x_2= -\frac{10}{3} \, \cos\left(\sqrt{15} t\right) - \frac{9}{2} \, \cos\left(\sqrt{2} t\right)\]

Thus after \(4\) seconds, the inner mass is located approximately \(-7.31\) meters from its natural position, and the outer mass is located approximately \(-0.390\) meters from its natural position.


Example 32

S2m: Coupled mass-spring systems (ver. 32)

Consider a coupled mass-spring system where the inner mass is \(1\) kg, the outer mass is \(1\) kg, the inner spring has constant \(3\) N/m, the outer spring has constant \(2\) N/m. The inner mass is moved \(2\) meters inwards from its natural position, while the outer mass is moved \(6\) meters outwards from its natural position. Both masses are then simultaneously released from rest.

Give a system of IVPs that models this scenario, then solve the system. Use your solution to find the position of both masses after \(3\) seconds.

Answer.

The IVP system is given by:

\[1 x_1''=- 5 x_1+ 2 x_2\hspace{2em}x_1(0)= -2 ,x_1'(0)=0\]

\[1 x_2''= 2 x_1- 2 x_2\hspace{2em}x_2(0)= 6 ,x_2'(0)=0\]

This system solves to:

\[x_1= -4 \, \cos\left(\sqrt{6} t\right) + 2 \, \cos\left(t\right)\]

\[x_2= 2 \, \cos\left(\sqrt{6} t\right) + 4 \, \cos\left(t\right)\]

Thus after \(3\) seconds, the inner mass is located approximately \(-3.92\) meters from its natural position, and the outer mass is located approximately \(-2.99\) meters from its natural position.


Example 33

S2m: Coupled mass-spring systems (ver. 33)

Consider a coupled mass-spring system where the inner mass is \(2\) kg, the outer mass is \(1\) kg, the inner spring has constant \(4\) N/m, the outer spring has constant \(2\) N/m. The inner mass is moved \(8\) meters inwards from its natural position, while the outer mass is moved \(7\) meters inwards from its natural position. Both masses are then simultaneously released from rest.

Give a system of IVPs that models this scenario, then solve the system. Use your solution to find the position of both masses after \(4\) seconds.

Answer.

The IVP system is given by:

\[2 x_1''=- 6 x_1+ 2 x_2\hspace{2em}x_1(0)= -8 ,x_1'(0)=0\]

\[1 x_2''= 2 x_1- 2 x_2\hspace{2em}x_2(0)= -7 ,x_2'(0)=0\]

This system solves to:

\[x_1= -3 \, \cos\left(2 \, t\right) - 5 \, \cos\left(t\right)\]

\[x_2= 3 \, \cos\left(2 \, t\right) - 10 \, \cos\left(t\right)\]

Thus after \(4\) seconds, the inner mass is located approximately \(3.70\) meters from its natural position, and the outer mass is located approximately \(6.10\) meters from its natural position.


Example 34

S2m: Coupled mass-spring systems (ver. 34)

Consider a coupled mass-spring system where the inner mass is \(1\) kg, the outer mass is \(1\) kg, the inner spring has constant \(3\) N/m, the outer spring has constant \(2\) N/m. The inner mass is moved \(6\) meters inwards from its natural position, while the outer mass is moved \(7\) meters inwards from its natural position. Both masses are then simultaneously released from rest.

Give a system of IVPs that models this scenario, then solve the system. Use your solution to find the position of both masses after \(2\) seconds.

Answer.

The IVP system is given by:

\[1 x_1''=- 5 x_1+ 2 x_2\hspace{2em}x_1(0)= -6 ,x_1'(0)=0\]

\[1 x_2''= 2 x_1- 2 x_2\hspace{2em}x_2(0)= -7 ,x_2'(0)=0\]

This system solves to:

\[x_1= -2 \, \cos\left(\sqrt{6} t\right) - 4 \, \cos\left(t\right)\]

\[x_2= \cos\left(\sqrt{6} t\right) - 8 \, \cos\left(t\right)\]

Thus after \(2\) seconds, the inner mass is located approximately \(1.29\) meters from its natural position, and the outer mass is located approximately \(3.51\) meters from its natural position.


Example 35

S2m: Coupled mass-spring systems (ver. 35)

Consider a coupled mass-spring system where the inner mass is \(1\) kg, the outer mass is \(1\) kg, the inner spring has constant \(3\) N/m, the outer spring has constant \(2\) N/m. The inner mass is moved \(1\) meters outwards from its natural position, while the outer mass is moved \(\frac{19}{2}\) meters outwards from its natural position. Both masses are then simultaneously released from rest.

Give a system of IVPs that models this scenario, then solve the system. Use your solution to find the position of both masses after \(5\) seconds.

Answer.

The IVP system is given by:

\[1 x_1''=- 5 x_1+ 2 x_2\hspace{2em}x_1(0)= 1 ,x_1'(0)=0\]

\[1 x_2''= 2 x_1- 2 x_2\hspace{2em}x_2(0)= \frac{19}{2} ,x_2'(0)=0\]

This system solves to:

\[x_1= -3 \, \cos\left(\sqrt{6} t\right) + 4 \, \cos\left(t\right)\]

\[x_2= \frac{3}{2} \, \cos\left(\sqrt{6} t\right) + 8 \, \cos\left(t\right)\]

Thus after \(5\) seconds, the inner mass is located approximately \(-1.71\) meters from its natural position, and the outer mass is located approximately \(3.69\) meters from its natural position.


Example 36

S2m: Coupled mass-spring systems (ver. 36)

Consider a coupled mass-spring system where the inner mass is \(2\) kg, the outer mass is \(1\) kg, the inner spring has constant \(4\) N/m, the outer spring has constant \(2\) N/m. The inner mass is moved \(6\) meters outwards from its natural position, while the outer mass is moved \(0\) meters outwards from its natural position. Both masses are then simultaneously released from rest.

Give a system of IVPs that models this scenario, then solve the system. Use your solution to find the position of both masses after \(4\) seconds.

Answer.

The IVP system is given by:

\[2 x_1''=- 6 x_1+ 2 x_2\hspace{2em}x_1(0)= 6 ,x_1'(0)=0\]

\[1 x_2''= 2 x_1- 2 x_2\hspace{2em}x_2(0)= 0 ,x_2'(0)=0\]

This system solves to:

\[x_1= 4 \, \cos\left(2 \, t\right) + 2 \, \cos\left(t\right)\]

\[x_2= -4 \, \cos\left(2 \, t\right) + 4 \, \cos\left(t\right)\]

Thus after \(4\) seconds, the inner mass is located approximately \(-1.89\) meters from its natural position, and the outer mass is located approximately \(-2.03\) meters from its natural position.


Example 37

S2m: Coupled mass-spring systems (ver. 37)

Consider a coupled mass-spring system where the inner mass is \(1\) kg, the outer mass is \(2\) kg, the inner spring has constant \(5\) N/m, the outer spring has constant \(4\) N/m. The inner mass is moved \(1\) meters inwards from its natural position, while the outer mass is moved \(7\) meters outwards from its natural position. Both masses are then simultaneously released from rest.

Give a system of IVPs that models this scenario, then solve the system. Use your solution to find the position of both masses after \(4\) seconds.

Answer.

The IVP system is given by:

\[1 x_1''=- 9 x_1+ 4 x_2\hspace{2em}x_1(0)= -1 ,x_1'(0)=0\]

\[2 x_2''= 4 x_1- 4 x_2\hspace{2em}x_2(0)= 7 ,x_2'(0)=0\]

This system solves to:

\[x_1= -4 \, \cos\left(\sqrt{10} t\right) + 3 \, \cos\left(t\right)\]

\[x_2= \cos\left(\sqrt{10} t\right) + 6 \, \cos\left(t\right)\]

Thus after \(4\) seconds, the inner mass is located approximately \(-5.95\) meters from its natural position, and the outer mass is located approximately \(-2.93\) meters from its natural position.


Example 38

S2m: Coupled mass-spring systems (ver. 38)

Consider a coupled mass-spring system where the inner mass is \(2\) kg, the outer mass is \(1\) kg, the inner spring has constant \(18\) N/m, the outer spring has constant \(4\) N/m. The inner mass is moved \(4\) meters outwards from its natural position, while the outer mass is moved \(7\) meters outwards from its natural position. Both masses are then simultaneously released from rest.

Give a system of IVPs that models this scenario, then solve the system. Use your solution to find the position of both masses after \(5\) seconds.

Answer.

The IVP system is given by:

\[2 x_1''=- 22 x_1+ 4 x_2\hspace{2em}x_1(0)= 4 ,x_1'(0)=0\]

\[1 x_2''= 4 x_1- 4 x_2\hspace{2em}x_2(0)= 7 ,x_2'(0)=0\]

This system solves to:

\[x_1= 2 \, \cos\left(2 \, \sqrt{3} t\right) + 2 \, \cos\left(\sqrt{3} t\right)\]

\[x_2= -\cos\left(2 \, \sqrt{3} t\right) + 8 \, \cos\left(\sqrt{3} t\right)\]

Thus after \(5\) seconds, the inner mass is located approximately \(-1.36\) meters from its natural position, and the outer mass is located approximately \(-5.81\) meters from its natural position.


Example 39

S2m: Coupled mass-spring systems (ver. 39)

Consider a coupled mass-spring system where the inner mass is \(2\) kg, the outer mass is \(1\) kg, the inner spring has constant \(10\) N/m, the outer spring has constant \(12\) N/m. The inner mass is moved \(3\) meters outwards from its natural position, while the outer mass is moved \(\frac{61}{6}\) meters inwards from its natural position. Both masses are then simultaneously released from rest.

Give a system of IVPs that models this scenario, then solve the system. Use your solution to find the position of both masses after \(2\) seconds.

Answer.

The IVP system is given by:

\[2 x_1''=- 22 x_1+ 12 x_2\hspace{2em}x_1(0)= 3 ,x_1'(0)=0\]

\[1 x_2''= 12 x_1- 12 x_2\hspace{2em}x_2(0)= -\frac{61}{6} ,x_2'(0)=0\]

This system solves to:

\[x_1= 5 \, \cos\left(2 \, \sqrt{5} t\right) - 2 \, \cos\left(\sqrt{3} t\right)\]

\[x_2= -\frac{15}{2} \, \cos\left(2 \, \sqrt{5} t\right) - \frac{8}{3} \, \cos\left(\sqrt{3} t\right)\]

Thus after \(2\) seconds, the inner mass is located approximately \(-2.54\) meters from its natural position, and the outer mass is located approximately \(9.18\) meters from its natural position.


Example 40

S2m: Coupled mass-spring systems (ver. 40)

Consider a coupled mass-spring system where the inner mass is \(1\) kg, the outer mass is \(1\) kg, the inner spring has constant \(15\) N/m, the outer spring has constant \(4\) N/m. The inner mass is moved \(8\) meters outwards from its natural position, while the outer mass is moved \(15\) meters outwards from its natural position. Both masses are then simultaneously released from rest.

Give a system of IVPs that models this scenario, then solve the system. Use your solution to find the position of both masses after \(2\) seconds.

Answer.

The IVP system is given by:

\[1 x_1''=- 19 x_1+ 4 x_2\hspace{2em}x_1(0)= 8 ,x_1'(0)=0\]

\[1 x_2''= 4 x_1- 4 x_2\hspace{2em}x_2(0)= 15 ,x_2'(0)=0\]

This system solves to:

\[x_1= 4 \, \cos\left(2 \, \sqrt{5} t\right) + 4 \, \cos\left(\sqrt{3} t\right)\]

\[x_2= -\cos\left(2 \, \sqrt{5} t\right) + 16 \, \cos\left(\sqrt{3} t\right)\]

Thus after \(2\) seconds, the inner mass is located approximately \(-7.34\) meters from its natural position, and the outer mass is located approximately \(-14.3\) meters from its natural position.


Example 41

S2m: Coupled mass-spring systems (ver. 41)

Consider a coupled mass-spring system where the inner mass is \(1\) kg, the outer mass is \(1\) kg, the inner spring has constant \(3\) N/m, the outer spring has constant \(2\) N/m. The inner mass is moved \(6\) meters outwards from its natural position, while the outer mass is moved \(7\) meters outwards from its natural position. Both masses are then simultaneously released from rest.

Give a system of IVPs that models this scenario, then solve the system. Use your solution to find the position of both masses after \(4\) seconds.

Answer.

The IVP system is given by:

\[1 x_1''=- 5 x_1+ 2 x_2\hspace{2em}x_1(0)= 6 ,x_1'(0)=0\]

\[1 x_2''= 2 x_1- 2 x_2\hspace{2em}x_2(0)= 7 ,x_2'(0)=0\]

This system solves to:

\[x_1= 2 \, \cos\left(\sqrt{6} t\right) + 4 \, \cos\left(t\right)\]

\[x_2= -\cos\left(\sqrt{6} t\right) + 8 \, \cos\left(t\right)\]

Thus after \(4\) seconds, the inner mass is located approximately \(-4.48\) meters from its natural position, and the outer mass is located approximately \(-4.30\) meters from its natural position.


Example 42

S2m: Coupled mass-spring systems (ver. 42)

Consider a coupled mass-spring system where the inner mass is \(2\) kg, the outer mass is \(1\) kg, the inner spring has constant \(10\) N/m, the outer spring has constant \(12\) N/m. The inner mass is moved \(1\) meters inwards from its natural position, while the outer mass is moved \(10\) meters outwards from its natural position. Both masses are then simultaneously released from rest.

Give a system of IVPs that models this scenario, then solve the system. Use your solution to find the position of both masses after \(5\) seconds.

Answer.

The IVP system is given by:

\[2 x_1''=- 22 x_1+ 12 x_2\hspace{2em}x_1(0)= -1 ,x_1'(0)=0\]

\[1 x_2''= 12 x_1- 12 x_2\hspace{2em}x_2(0)= 10 ,x_2'(0)=0\]

This system solves to:

\[x_1= -4 \, \cos\left(2 \, \sqrt{5} t\right) + 3 \, \cos\left(\sqrt{3} t\right)\]

\[x_2= 6 \, \cos\left(2 \, \sqrt{5} t\right) + 4 \, \cos\left(\sqrt{3} t\right)\]

Thus after \(5\) seconds, the inner mass is located approximately \(1.56\) meters from its natural position, and the outer mass is located approximately \(-8.48\) meters from its natural position.


Example 43

S2m: Coupled mass-spring systems (ver. 43)

Consider a coupled mass-spring system where the inner mass is \(1\) kg, the outer mass is \(1\) kg, the inner spring has constant \(3\) N/m, the outer spring has constant \(2\) N/m. The inner mass is moved \(1\) meters outwards from its natural position, while the outer mass is moved \(\frac{11}{2}\) meters inwards from its natural position. Both masses are then simultaneously released from rest.

Give a system of IVPs that models this scenario, then solve the system. Use your solution to find the position of both masses after \(3\) seconds.

Answer.

The IVP system is given by:

\[1 x_1''=- 5 x_1+ 2 x_2\hspace{2em}x_1(0)= 1 ,x_1'(0)=0\]

\[1 x_2''= 2 x_1- 2 x_2\hspace{2em}x_2(0)= -\frac{11}{2} ,x_2'(0)=0\]

This system solves to:

\[x_1= 3 \, \cos\left(\sqrt{6} t\right) - 2 \, \cos\left(t\right)\]

\[x_2= -\frac{3}{2} \, \cos\left(\sqrt{6} t\right) - 4 \, \cos\left(t\right)\]

Thus after \(3\) seconds, the inner mass is located approximately \(3.43\) meters from its natural position, and the outer mass is located approximately \(3.23\) meters from its natural position.


Example 44

S2m: Coupled mass-spring systems (ver. 44)

Consider a coupled mass-spring system where the inner mass is \(1\) kg, the outer mass is \(2\) kg, the inner spring has constant \(5\) N/m, the outer spring has constant \(4\) N/m. The inner mass is moved \(2\) meters inwards from its natural position, while the outer mass is moved \(5\) meters outwards from its natural position. Both masses are then simultaneously released from rest.

Give a system of IVPs that models this scenario, then solve the system. Use your solution to find the position of both masses after \(3\) seconds.

Answer.

The IVP system is given by:

\[1 x_1''=- 9 x_1+ 4 x_2\hspace{2em}x_1(0)= -2 ,x_1'(0)=0\]

\[2 x_2''= 4 x_1- 4 x_2\hspace{2em}x_2(0)= 5 ,x_2'(0)=0\]

This system solves to:

\[x_1= -4 \, \cos\left(\sqrt{10} t\right) + 2 \, \cos\left(t\right)\]

\[x_2= \cos\left(\sqrt{10} t\right) + 4 \, \cos\left(t\right)\]

Thus after \(3\) seconds, the inner mass is located approximately \(2.01\) meters from its natural position, and the outer mass is located approximately \(-4.96\) meters from its natural position.


Example 45

S2m: Coupled mass-spring systems (ver. 45)

Consider a coupled mass-spring system where the inner mass is \(1\) kg, the outer mass is \(1\) kg, the inner spring has constant \(5\) N/m, the outer spring has constant \(6\) N/m. The inner mass is moved \(2\) meters inwards from its natural position, while the outer mass is moved \(\frac{22}{3}\) meters inwards from its natural position. Both masses are then simultaneously released from rest.

Give a system of IVPs that models this scenario, then solve the system. Use your solution to find the position of both masses after \(3\) seconds.

Answer.

The IVP system is given by:

\[1 x_1''=- 11 x_1+ 6 x_2\hspace{2em}x_1(0)= -2 ,x_1'(0)=0\]

\[1 x_2''= 6 x_1- 6 x_2\hspace{2em}x_2(0)= -\frac{22}{3} ,x_2'(0)=0\]

This system solves to:

\[x_1= 2 \, \cos\left(\sqrt{15} t\right) - 4 \, \cos\left(\sqrt{2} t\right)\]

\[x_2= -\frac{4}{3} \, \cos\left(\sqrt{15} t\right) - 6 \, \cos\left(\sqrt{2} t\right)\]

Thus after \(3\) seconds, the inner mass is located approximately \(2.98\) meters from its natural position, and the outer mass is located approximately \(1.94\) meters from its natural position.


Example 46

S2m: Coupled mass-spring systems (ver. 46)

Consider a coupled mass-spring system where the inner mass is \(1\) kg, the outer mass is \(2\) kg, the inner spring has constant \(14\) N/m, the outer spring has constant \(6\) N/m. The inner mass is moved \(7\) meters outwards from its natural position, while the outer mass is moved \(\frac{25}{3}\) meters outwards from its natural position. Both masses are then simultaneously released from rest.

Give a system of IVPs that models this scenario, then solve the system. Use your solution to find the position of both masses after \(3\) seconds.

Answer.

The IVP system is given by:

\[1 x_1''=- 20 x_1+ 6 x_2\hspace{2em}x_1(0)= 7 ,x_1'(0)=0\]

\[2 x_2''= 6 x_1- 6 x_2\hspace{2em}x_2(0)= \frac{25}{3} ,x_2'(0)=0\]

This system solves to:

\[x_1= 4 \, \cos\left(\sqrt{21} t\right) + 3 \, \cos\left(\sqrt{2} t\right)\]

\[x_2= -\frac{2}{3} \, \cos\left(\sqrt{21} t\right) + 9 \, \cos\left(\sqrt{2} t\right)\]

Thus after \(3\) seconds, the inner mass is located approximately \(0.160\) meters from its natural position, and the outer mass is located approximately \(-4.33\) meters from its natural position.


Example 47

S2m: Coupled mass-spring systems (ver. 47)

Consider a coupled mass-spring system where the inner mass is \(2\) kg, the outer mass is \(1\) kg, the inner spring has constant \(10\) N/m, the outer spring has constant \(12\) N/m. The inner mass is moved \(1\) meters outwards from its natural position, while the outer mass is moved \(10\) meters inwards from its natural position. Both masses are then simultaneously released from rest.

Give a system of IVPs that models this scenario, then solve the system. Use your solution to find the position of both masses after \(2\) seconds.

Answer.

The IVP system is given by:

\[2 x_1''=- 22 x_1+ 12 x_2\hspace{2em}x_1(0)= 1 ,x_1'(0)=0\]

\[1 x_2''= 12 x_1- 12 x_2\hspace{2em}x_2(0)= -10 ,x_2'(0)=0\]

This system solves to:

\[x_1= 4 \, \cos\left(2 \, \sqrt{5} t\right) - 3 \, \cos\left(\sqrt{3} t\right)\]

\[x_2= -6 \, \cos\left(2 \, \sqrt{5} t\right) - 4 \, \cos\left(\sqrt{3} t\right)\]

Thus after \(2\) seconds, the inner mass is located approximately \(-0.702\) meters from its natural position, and the outer mass is located approximately \(9.11\) meters from its natural position.


Example 48

S2m: Coupled mass-spring systems (ver. 48)

Consider a coupled mass-spring system where the inner mass is \(2\) kg, the outer mass is \(1\) kg, the inner spring has constant \(10\) N/m, the outer spring has constant \(12\) N/m. The inner mass is moved \(3\) meters outwards from its natural position, while the outer mass is moved \(\frac{61}{6}\) meters inwards from its natural position. Both masses are then simultaneously released from rest.

Give a system of IVPs that models this scenario, then solve the system. Use your solution to find the position of both masses after \(3\) seconds.

Answer.

The IVP system is given by:

\[2 x_1''=- 22 x_1+ 12 x_2\hspace{2em}x_1(0)= 3 ,x_1'(0)=0\]

\[1 x_2''= 12 x_1- 12 x_2\hspace{2em}x_2(0)= -\frac{61}{6} ,x_2'(0)=0\]

This system solves to:

\[x_1= 5 \, \cos\left(2 \, \sqrt{5} t\right) - 2 \, \cos\left(\sqrt{3} t\right)\]

\[x_2= -\frac{15}{2} \, \cos\left(2 \, \sqrt{5} t\right) - \frac{8}{3} \, \cos\left(\sqrt{3} t\right)\]

Thus after \(3\) seconds, the inner mass is located approximately \(2.37\) meters from its natural position, and the outer mass is located approximately \(-6.19\) meters from its natural position.


Example 49

S2m: Coupled mass-spring systems (ver. 49)

Consider a coupled mass-spring system where the inner mass is \(2\) kg, the outer mass is \(1\) kg, the inner spring has constant \(4\) N/m, the outer spring has constant \(2\) N/m. The inner mass is moved \(3\) meters outwards from its natural position, while the outer mass is moved \(12\) meters outwards from its natural position. Both masses are then simultaneously released from rest.

Give a system of IVPs that models this scenario, then solve the system. Use your solution to find the position of both masses after \(3\) seconds.

Answer.

The IVP system is given by:

\[2 x_1''=- 6 x_1+ 2 x_2\hspace{2em}x_1(0)= 3 ,x_1'(0)=0\]

\[1 x_2''= 2 x_1- 2 x_2\hspace{2em}x_2(0)= 12 ,x_2'(0)=0\]

This system solves to:

\[x_1= -2 \, \cos\left(2 \, t\right) + 5 \, \cos\left(t\right)\]

\[x_2= 2 \, \cos\left(2 \, t\right) + 10 \, \cos\left(t\right)\]

Thus after \(3\) seconds, the inner mass is located approximately \(-6.87\) meters from its natural position, and the outer mass is located approximately \(-7.98\) meters from its natural position.


Example 50

S2m: Coupled mass-spring systems (ver. 50)

Consider a coupled mass-spring system where the inner mass is \(2\) kg, the outer mass is \(1\) kg, the inner spring has constant \(18\) N/m, the outer spring has constant \(4\) N/m. The inner mass is moved \(0\) meters outwards from its natural position, while the outer mass is moved \(\frac{45}{2}\) meters inwards from its natural position. Both masses are then simultaneously released from rest.

Give a system of IVPs that models this scenario, then solve the system. Use your solution to find the position of both masses after \(4\) seconds.

Answer.

The IVP system is given by:

\[2 x_1''=- 22 x_1+ 4 x_2\hspace{2em}x_1(0)= 0 ,x_1'(0)=0\]

\[1 x_2''= 4 x_1- 4 x_2\hspace{2em}x_2(0)= -\frac{45}{2} ,x_2'(0)=0\]

This system solves to:

\[x_1= 5 \, \cos\left(2 \, \sqrt{3} t\right) - 5 \, \cos\left(\sqrt{3} t\right)\]

\[x_2= -\frac{5}{2} \, \cos\left(2 \, \sqrt{3} t\right) - 20 \, \cos\left(\sqrt{3} t\right)\]

Thus after \(4\) seconds, the inner mass is located approximately \(-2.61\) meters from its natural position, and the outer mass is located approximately \(-16.7\) meters from its natural position.