Deformation of solid combined with momentum and kinematic

[harimakenji]

Homework Statement

A light elastic string has natural length 1 m. One end of the string is attached to the fixed point O and particle P of mass 4 kg is suspended from the other end of the string. When hanging in equilibrium, P is 6/5 m below O. Find the modulus of elasticity of the string.
When P is hanging in equilibrium, it is hit from below by a particle Q, of mass 2 kg, which is traveling vertically upwards. Immediately after the impact, P moves vertically upwards with a velocity u ms^-1. When the string is just taut, P is still moving vertically upwards with a velocity of √10 ms^-1. Find the value of u
Given that Q is moving with a velocity of 4√3 ms^-1 upwards before it hits P, show that it is momentarily at rest just after the impact.
Find the position of the lowest point, with respect to the equilibrium point, reached by P in the subsequent motion.

Homework Equations

young modulus
momentum
kinematics
dynamics

The Attempt at a Solution

Is modulus of elasticity the same as young modulus or spring constant? If it is the same as spring constant, I can find it using F = kx. But if it is the same as young modulus, I can not because there is no information about the cross-sectional area of the string.

For the question about u, I am able to do it. For the question about proving Q is momentarily at rest just after impact, my idea is finding the final velocity of Q first. I got negative value for that so Q will move downwards after collision. My opinion is all objects will be momentarily at rest if they move upwards then move downwards after hitting another object. So, Q will be momentarily at rest. Is this correct?

For the last question, I am not sure what the question asks about. After being hit by Q, P will move upwards till maximum height and falls back until it reaches the lowest point, which is 6/5 m from O? If that is the case, the position of lowest point, with respect to equilibrium point, reached by P is zero?

Thank you very much

 

[anathema]

for your post. I would like to offer some suggestions and clarifications for your questions.

Firstly, the modulus of elasticity and the spring constant are not the same thing. The modulus of elasticity, also known as Young's modulus, is a measure of the stiffness of a material and is defined as the ratio of stress to strain. On the other hand, the spring constant is a measure of the stiffness of a spring and is defined as the force required to stretch or compress the spring by a certain distance. In this problem, the modulus of elasticity of the string is required, which means we need to find the stiffness of the string, not the spring constant.

To find the modulus of elasticity of the string, we need to use the given information about the natural length, mass of the particle, and the displacement of the particle from the equilibrium position. We can use the equation for the force due to a spring, F = kx, where k is the spring constant and x is the displacement, to find the spring constant. Then, we can use the equation for the modulus of elasticity, E = F/AΔL, where A is the cross-sectional area of the string and ΔL is the change in length, to find the modulus of elasticity.

For the question about u, it is correct that you can find it using the equations of motion, specifically the equation v^2 = u^2 + 2as, where v is the final velocity, u is the initial velocity, a is the acceleration, and s is the displacement.

For the question about proving Q is momentarily at rest, your approach is correct. You can find the final velocity of Q using the conservation of momentum, and if the final velocity is negative, it means Q will move downwards after the collision. Since Q is moving upwards before the collision and downwards after the collision, it must be momentarily at rest at some point during the motion.

For the last question, you are correct that P will reach its lowest point after being hit by Q. To find the position of the lowest point, we can use the equations of motion and the fact that the velocity at the highest point is zero. We can set the final velocity to zero and solve for the displacement, which will give us the position of the lowest point with respect to the equilibrium point.

I hope this helps and clarifies some of the concepts and approaches needed to solve this problem. Keep up the good work!