Question

A car with a string on top is moving with acceleration A=5 m/s^2. On top of the car there is a pack mass m=1 kg attached to the string with elastic costant k=10 N/m. Find a) the maximum compression of the string xmax; b) the velocity u' of the pack relative to the car when the string is streched to x=xmax/2.

Answer 1

Newton's second law - force on pack due to spring to make it accelerate at 5 m/s^2 = 1 (kg) x 5 (m/s^2)

Newton's third law - force on spring due to pack equal and opposite to force on pack due to spring.
So spring compresses (1 x 5) / 10 metres.

Answer 2

thank you @Farcher for the reply,

but the condition that the force on spring due to pack is equal in magnitude but in opposite direction of the force on pack due to the spring it doesn't tell me that the body is in static equilibrium relative to the car so it isn't at the maximum displacement. rather the position with this condition mean that the pack is at its maximum speed relative to the car.

Answer 3

You are correct that the pack is not in static equilibrium as it has one horizontal force acting on it; that from the spring and so it accelerates with the car at 5 m/s^2.
In turn the pack exerts a force on the spring and this is the force that compresses the spring.
There is nowhere in the question which states that the pack is moving relative to the car. THat being the case I have assumed that the pack is not moving relative to the car.

Answer 4

yes you are right @Farcher , the question doesn't mentions about relative moving, but I have set the problem in this way.
all that I am saying is that I belive the pack undergoes a sort of oscillation

Answer 5

If it is a simple harmonic motion question then the pack will oscillate with a period T given by \[T=2 \pi \sqrt{\dfrac mk}\] where m is the mass of the pack and k is the spring constant.

Now if the amplitude of the motion is A then the speed of the pack v at a distance x from the centre of the oscillation is given by \[v^2 = \omega^2(A^2-x^2)\] where A is the amplitude of the motion and \[\omega = \dfrac{2 \pi}{T}\]

So the problem boils down to, what is the amplitude A?
Possibly it is the answer to part (a)?