Damped Oscillation problem.

[abramsay]

Homework Statement

Help need with this problem.

A light spring AB of natural length 2a and of modulus of elasticity 2amn² lies straight at its length and at rest on a smooth horizontal table. The end A is fixed to the table and a particle P of mass m is attached to the midpoint of the spring. The end B is then caused to move along the line AB so that after a time t the distance between A and B is a(2+sinnt). Denoting the distance of P from A by a+x, show that
d²x/dt² + 4n²x = 2an²sinnt.
Find the value of t for which P first comes to rest.

The Attempt at a Solution

Here's my attempt so far, can't find the value of t for which P first comes to rest.
Am I right or wrong. Anyone has a better solution and a solution to the second part.

AB = 2(a+x) since AB at rest is twice a+x

At t, AB = 2a + asinnt

F = -k(x-e) where e is the extension and x the natural length

= k(e-x)
ma = 2amn²/a (2a + asinnt -2a - 2x)

a = 2n² (asinnt - 2a)

= 2n²sinnt - 4n²x

a + 4n²x = 2n²asinnt

=> d²x/dt² + 4n²x = 2n²asinnt

 

[mark726]

To find the value of t for which P first comes to rest, we need to find the value of t where the acceleration is equal to zero. This would mean that there is no net force acting on the particle P and it is at rest.

Using the equation we found earlier, we can set the acceleration equal to zero:

d2x/dt2 + 4n2x = 0

Solving for x, we get:

x = 0

Substituting this back into the equation, we get:

d2x/dt2 + 4n2(0) = 2an2sinnt

=> d2x/dt2 = 2an2sinnt

Integrating both sides with respect to t, we get:

dx/dt = -2an2cosnt + c1

Integrating again, we get:

x = -2an2sinnt + c1t + c2

Since we know that x = 0 when t = 0, we can substitute these values into the equation to find the values of c1 and c2:

0 = -2an2sin(0) + c1(0) + c2

=> c2 = 0

Similarly, when t = 0, dx/dt = 0, so:

0 = -2an2cos(0) + c1

=> c1 = 2an2

Therefore, the final equation is:

x = -2an2sinnt + 2an2t

To find the value of t where P first comes to rest, we need to set x = 0:

0 = -2an2sinnt + 2an2t

=> sinnt = t

Solving for t, we get:

t = nπ

Therefore, P first comes to rest at t = nπ.