Finding spring constant and friction

[uzman1243]

Homework Statement

Josh pushes a box of mass m which then travels on a horizontal surface. There is a coefficient of
kinetic friction μ between the box and the surface. The box has speed v when it reaches x = 0 and
encounters a spring. The box compresses the spring, stops, and then recoils and travels in the opposite direction. When the box reaches x = 0 on its return trip, it stops.
Determine k, the spring constant, in terms of μ, m, g, and v.

Homework Equations

The Attempt at a Solution

is the diagram I have drawn correct?

If it is, I can use F=ma in x & y direction.
However, how can I get V into this equation?

 

[gneill]

Your diagram is okay as far as the forces involved. Have you considered a conservation of energy approach?

 

[uzman1243]

Your diagram is okay as far as the forces involved. Have you considered a conservation of energy approach?

Would it go like this:

Intial Kinetic energy + Potential energy (0.5*k*x^2) - work (µ*nc*x) = final kinetic energy (0 as it is in rest in final position)

is this correct? I don't understand the value for x though?

 

[gneill]

You have two unknowns here. One is the spring constant and the other is the compression distance, which is your "x". Having two unknowns implies you'll need two equations, right?

You know what the total energy is to begin with (it's all kinetic and you have the mass and velocity as givens). You should also know what the final energy is and where it's "located". Write energy balance equations for the times when the KE is known to be zero...

 

[uzman1243]

You have two unknowns here. One is the spring constant and the other is the compression distance, which is your "x". Having two unknowns implies you'll need two equations, right?

You know what the total energy is to begin with (it's all kinetic and you have the mass and velocity as givens). You should also know what the final energy is and where it's "located". Write energy balance equations for the times when the KE is known to be zero...

Is the equation correct though?

(0.5mv^2) + (0.5kx^2) - ((coefficient of frict) * (normal contact) * x) =0

What will the second equation be?

 

[gneill]

Is the equation correct though?

(0.5mv^2) + (0.5kx^2) - ((coefficient of frict) * (normal contact) * x) =0

What will the second equation be?

No, that equation doesn't look correct when the spring compression is x (fully compressed), although it's close. The system starts out with all the energy as KE. That's your 0.5mv^2 term. When the spring is fully compressed the new KE is zero but the spring has stored PE and some energy has been lost due to friction. Write the energy balance accordingly: OLD = NEW. So something like

KE_i = PE_s + W_f

Where KE_i is the initial KE, PE_s is the spring PE, W_f is the work done by friction. Use the appropriate expressions for the terms.

Write the second equation for when the block has returned to the starting point from the fully compressed position. For that portion of the trip the initial energy was all stored in the spring as PE, so that's your "OLD" part of the balance. Where's all the "NEW" energy at the end of the trip?

 

[uzman1243]

No, that equation doesn't look correct when the spring compression is x (fully compressed), although it's close. The system starts out with all the energy as KE. That's your 0.5mv^2 term. When the spring is fully compressed the new KE is zero but the spring has stored PE and some energy has been lost due to friction. Write the energy balance accordingly: OLD = NEW. So something like

KE_i = PE_s + W_f

Where KE_i is the initial KE, PE_s is the spring PE, W_f is the work done by friction. Use the appropriate expressions for the terms.

Write the second equation for when the block has returned to the starting point from the fully compressed position. For that portion of the trip the initial energy was all stored in the spring as PE, so that's your "OLD" part of the balance. Where's all the "NEW" energy at the end of the trip?

So the first equation would be:
0.5mv^2 = 0.5kx^2 - (µ*nc*x)
(work has to be negative as energy goes out of the system right?)

the second equation would be:
0.5kx^2 = (µ*nc*x)
There is no kinetic energy in the final position as object comes to rest.

Is this correct?

 

[gneill]

So the first equation would be:
0.5mv^2 = 0.5kx^2 - (µ*nc*x)
(work has to be negative as energy goes out of the system right?)

the second equation would be:
0.5kx^2 = (µ*nc*x)
There is no kinetic energy in the final position as object comes to rest.

Is this correct?

Yup, looks good.

[edit: see my correction below]

 

[uzman1243]

Yup, looks good.

Thank you so much. You're a good man

 

[gneill]

So the first equation would be:
0.5mv^2 = 0.5kx^2 - (µ*nc*x)
(work has to be negative as energy goes out of the system right?)

Oops. Let's me correct what I said! Interchange the positions of the KE and spring PE in your equation (or equivalently, change the sign of the friction energy). When the block comes to rest at the instant the spring is fully compressed, all the remaining energy will be in the spring. So the sum of the spring energy and the friction energy should total to the original energy.

(Clearly I need another coffee this morning!)

 

[uzman1243]

Oops. Let's me correct what I said! Interchange the positions of the KE and spring PE in your equation (or equivalently, change the sign of the friction energy). When the block comes to rest at the instant the spring is fully compressed, all the remaining energy will be in the spring. So the sum of the spring energy and the friction energy should total to the original energy.

(Clearly I need another coffee this morning!)

0.5mv^2 - (µ*nc*x) = 0.5kx^2

Is that correct?

 

[gneill]

0.5mv^2 - (µ*nc*x) = 0.5kx^2

Is that correct?

Yup. That's good.