Compressed spring with frictional forces problem

[offbeatjumi]

Homework Statement

A book (mass 2.50 kg) is forced against horizontal spring (negligible mass) with force constant 250 N/m, the spring is compressed 0.250m. When it's released the textbook slides horizontally across a surface with kinetic friction coeff = 0.30. How far does the book move from its original position before stopping?

Homework Equations

W = 1/2kX^2... F(s) = kx. U(s) = 1/2kx^2

The Attempt at a Solution

I know I need to use the work energy theorem here.
I was trying to use E2 + W(non-cons) = E1 since there is a nonconservative force in this problem. But isn't the initial and final kinetic energy both zero? I tried using Newton's 2nd law as well... F(net,x) = F(s) - f(k). But at the time when its released, it is not static equillibrium and I can't solve that equation. Unless... does f(k) = (mu)k*n (normal force?) where n = mg.
If anyone can help me set a foot in the correct direction I'd really appreciate it. Thanks!

 

[rock.freak667]

The Attempt at a Solution

I know I need to use the work energy theorem here.
I was trying to use E2 + W(non-cons) = E1 since there is a nonconservative force in this problem. But isn't the initial and final kinetic energy both zero? I tried using Newton's 2nd law as well... F(net,x) = F(s) - f(k). But at the time when its released, it is not static equillibrium and I can't solve that equation. Unless... does f(k) = (mu)k*n (normal force?) where n = mg.
If anyone can help me set a foot in the correct direction I'd really appreciate it. Thanks!

Right good, this is correct. If frictional force F_R= μN = μmg, and the initial energy is 1/2kx², what is this converted into during the motion in general?

 

[kerimek]

I would first start by identifying the key variables and equations relevant to this problem. The key variables here are the mass of the book (2.50 kg), the force constant of the spring (250 N/m), the displacement of the spring (0.250m), and the kinetic friction coefficient (0.30). The equations that can be used to solve this problem are the work-energy theorem, F=ma, and the equations for work and potential energy of a spring.

Next, I would suggest setting up a free-body diagram for the book to visualize the forces acting on it. This would include the force from the compressed spring (F(s)) and the kinetic friction force (f(k)).

To solve for the distance the book moves before stopping, we can use the work-energy theorem. This states that the work done on an object is equal to the change in its kinetic energy. In this case, the work done by the spring is equal to the change in the book's kinetic energy, which is zero at both the starting and stopping points.

We can therefore set up the equation: W(s) + W(f) = 0, where W(s) is the work done by the spring and W(f) is the work done by friction. We can solve for the work done by the spring using the equation W(s) = 1/2kx^2, where k is the force constant and x is the displacement of the spring. We can solve for the work done by friction using the equation W(f) = f(k)d, where d is the distance the book moves before stopping.

Finally, we can set these two equations equal to each other and solve for d to find the distance the book moves before stopping. This will give us the final answer to the problem.