How Does Friction Affect the Distance a Block Slides on an Elevated Track?

[delecticious]

Homework Statement

A small block slides along a track from one level to a higher level, by moving through an intermediate valley (see Figure). The track is frictionless until the block reaches the higher level. There a frictional force stops the block in a distance d. Assume that the block's initial speed is 10 m/s, the height difference h is 1.0 m, and μk is 0.32. Find the distance d that the block travels on the higher level before stopping.

Homework Equations

non conservative work = (KE final - KE initial) + (PE final - PE initial)

The Attempt at a Solution

I think think the part o that's really giving me trouble is the valley. I realize that once the block reaches the other side with the friction it's all forces and kinematics, but how that tie in with the work part? I'm just really confused on how to approach this problem.

 

[Chi Meson]

The valley does not matter since it is frictionless. All that matters is the kinetic energy just before it hits the friction. It is exactly the same as if the block simple rose 1 m up a frictionless ramp.

Then, what is the force that does the (negative) work to stop the block (to take away its kinetic energy)?

How is work calculated?

 

[m2003]

I would approach this problem by first identifying all the relevant forces acting on the block. In this case, we have the force of gravity pulling the block down, a normal force from the track pushing the block up, and a frictional force acting in the opposite direction of motion when the block reaches the higher level.

Next, I would apply the work-energy theorem, which states that the total work done on an object is equal to the change in its kinetic energy. In this case, the initial kinetic energy of the block is 1/2mv^2, where m is the mass of the block and v is its initial speed of 10 m/s. The final kinetic energy is 0, since the block stops moving. Therefore, the work done on the block by all the forces is equal to the initial kinetic energy.

Using the work-energy theorem and the equation for non-conservative work, we can set up the following equation:

(KE final - KE initial) + (PE final - PE initial) = 0

Since the block starts and ends at the same height, the change in potential energy (PE final - PE initial) is 0. This means that the work done by the normal force is equal and opposite to the work done by gravity. We can therefore write:

(KE final - KE initial) + (-mgΔh) = 0

Where Δh is the change in height (1.0 m in this case) and g is the acceleration due to gravity. Rearranging this equation, we get:

KE final = KE initial + mgΔh

Substituting in the values given in the problem, we get:

0 = 1/2mv^2 + mgΔh

Solving for the mass of the block, we get:

m = 0.2 kg

Now, to find the distance d that the block travels on the higher level before stopping, we can use the equation for work done by friction:

W = μkFnΔd

Where μk is the coefficient of kinetic friction, Fn is the normal force, and Δd is the distance traveled. We know that the normal force is equal to the weight of the block, which is mg. Therefore, we can write:

W = μkmgΔd

Since the work done by friction is equal to the initial kinetic energy of the block, we can substitute in the values we found