How Far Does the Spring Compress When a 20 kg Mass Slides Down an Incline?

[accc]

A 20 kg mass is released from rest at the top of a plane inclined at an angle of 30 degrees. At the bottom of the plane lies a spring with a spring constant of 200 N/m. The distance between the mass and the spring is 6 m, and the coefficient of sliding friction is .2. I have to answer a series of questions:

a) How far has the spring been compressed when the block is brought to rest?

b) What is the velocity of the mass as it reaches the spring?

c) & d) How far will the spring be compressed when the mass reaches its maximum velocity, and what will its maximum velocity be?

e) What is the velocity of the mass as it is released by the spring back up the plane?

f) How far up the plane does the mass travel?

g) How long does it take for the mass to reach this point?

This was originally assigned a while ago, but I never got around to doing it and I have to hand it in by tomorrow. I came up with an answer for the second question by doing Kinetic Energy + Work done by Friction = Potential Energy, but I'm not sure if that's the right way to do it. I need help on the first question with finding how much force is being applied to the spring, and also for the 3rd and fourth questions on how to find the maximum velocity. After I find these I could probably do the last three questions by myself. Can anyone help?

 

[Doc Al]

Originally posted by accc
I came up with an answer for the second question by doing Kinetic Energy + Work done by Friction = Potential Energy, but I'm not sure if that's the right way to do it. I need help on the first question with finding how much force is being applied to the spring, and also for the 3rd and fourth questions on how to find the maximum velocity. After I find these I could probably do the last three questions by myself.

For a & b, I would use the so-called energy equation: the (negative) work done by friction equals the change in energy of the mass. (Be sure to include all applicable energy forms: gravitational PE, spring PE, and KE.)

For c & d, I would use the same energy equation to express the KE as a function of spring compression. Then I would differentiate, to find the compression for maximum KE. Once you have the spring compression, you can find the KE and velocity.

Hope this helps a bit.

 

[accc]

Originally posted by Doc Al
For a & b, I would use the so-called energy equation: the (negative) work done by friction equals the change in energy of the mass. (Be sure to include all applicable energy forms: gravitational PE, spring PE, and KE.)

I'm pretty sure that's what I did for b. I did 1/2mv^2 + umgcos30*d = mgh. With units, I came up with 10v^2 + (.2)(20)(9.8)(cos30)(6)=(20)(9.8)(6sin30), and came up with v = 6.2. Now I'm not sure how I apply this to question a, for how far the spring is compressed when v = 0?

For c & d, I would use the same energy equation to express the KE as a function of spring compression. Then I would differentiate, to find the compression for maximum KE. Once you have the spring compression, you can find the KE and velocity.

Hope this helps a bit.

Yeah after I find out how far the spring gets compressed in a I'm pretty sure I can figure out the rest. Thanks for your help.

 

[Doc Al]

Originally posted by accc
I did 1/2mv^2 + umgcos30*d = mgh. With units, I came up with 10v^2 + (.2)(20)(9.8)(cos30)(6)=(20)(9.8)(6sin30), and came up with v = 6.2. Now I'm not sure how I apply this to question a, for how far the spring is compressed when v = 0?

I didn't check your arithmetic, but your equation looks good to me. For question a, do the same thing. Only now the KE is zero and you have some compressed spring energy.

 

[accc]

Originally posted by Doc Al
I didn't check your arithmetic, but your equation looks good to me. For question a, do the same thing. Only now the KE is zero and you have some compressed spring energy.

Okay, I got it now. Thanks!