Using the Work-Energy Theorem to Solve a Textbook Sliding Problem with Friction

[Yosty22]

Homework Statement

A 2.50-kg textbook is forced against a horizontal spring of negligible mass and force constant, k, of 250N/m, compressing the spring a distance of 0.250 m. When released, the textbook slides on a horizontal tabletop with coefficient of kinetic friction μ_k=0.30. Use the work-energy theorem to find how far the text-book moves from its initial position before coming to rest.

Homework Equations

W_tot=K₂-K₁=ΔK (work-energy theorem)
K=0.5mv²

The Attempt at a Solution

I have no idea how to even start. It says that I have to use the work energy theorem, so all I can think of doing is substitution. However, Work-energy theorem has no reference to kinetic friction, just kinetic energy. Any ideas to get me started?

 

[Doc Al]

However, Work-energy theorem has no reference to kinetic friction, just kinetic energy. Any ideas to get me started?

Friction is a force, so consider the work done by it.

 

[Yosty22]

Friction is a force, so consider the work done by it.

Ok, so I used the equation friction=μ*n.

Since the table it is on is horizontal, that means that there is no acceleration or any motion up or down, so W=N. In this case, I use the weight of the box which came out to be 9.8*2.5=24.5N.

I found the force of force of friction to be 7.35N acting in the opposite direction of motion. This is where I am stuck. If I put Work due to friction in the work energy theorem, I have to multiply the force of friction by the distance over which it acts. How do I find the distance that the friction acts over? I thought that the force of friction acted over the same distance as it moved, just in the opposite direction. In this case, I still have two unknowns: the total distance that the block traveled and its velocity, but the problem only asks for the distance. How would I go about finding the velocity if it is not given and I don't know the kenetic energy or how far it traveled or any time interval?

 

[Doc Al]

Hint: What's final speed of the book?

 

[Yosty22]

Hint: What's final speed of the book?

Ahh, ok, it would be zero because it comes to a rest. However, wouldn't it start from rest as well, since it is pushed up against the compressed spring? Or would I designate v₁to be the moment it leaves the spring?

 

[Doc Al]

Ahh, ok, it would be zero because it comes to a rest. However, wouldn't it start from rest as well, since it is pushed up against the compressed spring?

Yes, it starts from rest and ends up at rest.

Or would I designate v₁to be the moment it leaves the spring?

All you need to worry about is the change in KE. What would that be? (No need to worry about intermediate points.)

 

[Yosty22]

So wouldn't that mean the change is 0? In which case, there is no kinetic energy. Does that make sense?

 

[Doc Al]

So wouldn't that mean the change is 0? In which case, there is no kinetic energy. Does that make sense?

The change in kinetic energy is zero, which makes perfect sense. After all, you're trying to find how how far the thing goes before friction brings it to rest. (That doesn't mean that there wasn't kinetic energy while it was moving, but who cares?)

 

[Sydney Austin]

So since friction and the spring are both doing work, would the total work be the (work of the spring) - (the work of friction) since work by friction is against motion?

 

[Doc Al]

So since friction and the spring are both doing work, would the total work be the (work of the spring) - (the work of friction) since work by friction is against motion?

Sure.