A block of mass 0.32kg starts from rest at point A and slides down a frictionles

Question

A block of mass 0.32kg starts from rest at point A and slides down a frictionless hill of height h. At the bottom of the hill it slides across a horizontal piece of track where the coefficient of kinetic friction is 0.27. This section (from points B to C) is 5.07m in length. The block then enters a frictionless loop of radius r= 2.07m. Point D is the highest point in the loop. The loop has a total height of 2r. Note that the drawing above is not to scale.

1.What is the minimum speed of the block at point D that still allows the block to complete the loop without leaving the track?

2.What is the minimum kinetic energy for the block at point C in order to have enough speed at point D that the block will not leave the track?

3.What is the minimum kinetic energy for the block at point B in order to have enough speed at point D that the block will not leave the track?

4.What is the minimum height from which the block should start in order to have enough speed at point D that the block will not leave the track?

Explanation / Answer

OK so the first two are relatively easy.

1). This can be answered completely conceptually. Any velocity greater than 0 at the top of the loop will allow the block to make it around. This hopefully is intuitave. If the velocity is zero at the top it will fall but anything greater than 0 will aloow it to keep going around and as gravity pulls the block down, the track follows its path precisely.

2). So since the blcok needs a velocity greater than the 0 at the top we can use this as our lower limit of energy. At the top of the loop if its velocity is exactly 0 its only energy is potential energy. PE = mass * g * height. Here the height is 2r. So at point C it needs that at least that amount of engery to make it around the loop. Boom there's the answer. The Kinettic energy at C must be greater than the Potential energy the block has at Point D. Plug in you numbers and solve.

3). First you need the energy it loses from friction. Energy is Force*distance. You are given the distance so to find the energy lost to friction all you need is the force of friction. F(friction) = Normal force * coeficcient of friction. Since from B to C it's level, the normal force equals the force of gravity on the block. Now with these numbers you get the energy lost to friction, just add that to the energy you know you need from 2 to get the answer for this part since the block needs at point B enough energy to make it through the part B-C and then around the loop.

4). Part 3 was the hardest part. You now know at B how much energy the block needs to make it. Since B is at the bottom of the ramp, here the block has 0 potential energy. This means the energy it has is all kinetic. Since at the start the block is at rest at the top it only has potential energy. Value for the kinetic energy you calculated in part 3 equal to the potential energy formula and solve for the height of the ramp.

(KE from part 3) = mass * g * height of ramp.

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