In the figure below, a small block of mass m = 0.034 kg can slide along the frictionless loop-the-loop, with loop radius 8 cm. The block is released from rest at point P, at height h = 7R above the bottom of the loop. (For all parts, answer using g for the acceleration due to gravity, and R and m as appropriate.)
(a) How much work does the gravitational force do on the block as the block travels from point P to point Q?
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(b) How much work does the gravitational force do on the block as the block travels from point P to the top of the loop?
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(c) If the gravitational potential energy of the block-Earth system is taken to be zero at the bottom of the loop, what is the potential energy when the block is at point P?
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(d) What is the potential energy when the block is at point Q?
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(e) What is the potential energy when the block is at the top of the loop?
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(f) If, instead of being released, the block is given some initial speed downward along the track, do the answers to (a) through (e) increase, decrease, or remain the same?
increase
decrease
remain the same
To calculate the answers to these questions, we need to apply the principles of conservation of energy. The total mechanical energy of the system remains constant as long as there are no external non-conservative forces acting on the block.
Let's go through each part of the question and explain how to calculate the answers:
(a) To calculate the work done by the gravitational force as the block travels from point P to point Q, we first need to calculate the change in gravitational potential energy. This can be done using the formula:
ΔPE = m * g * Δh
where ΔPE is the change in potential energy, m is the mass of the block, g is the acceleration due to gravity, and Δh is the change in height. In this case, the block is moving from a height of h = 7R to a height of h = R, so Δh = -6R.
Using the given values of m, g, and R, we can calculate the change in potential energy and convert it to work done by multiplying it by -1 (since the work done is negative):
ΔPE = -m * g * Δh = -0.034 kg * g * (-6R)
The answer to part (a) is the negative of this value.
(b) To calculate the work done by the gravitational force as the block travels from point P to the top of the loop, we need to calculate the change in potential energy from P to the top of the loop. This is the same as the answer to part (a), since the height change is still Δh = -6R.
(c) To find the potential energy when the block is at point P, we can use the formula for gravitational potential energy:
PE = m * g * h
where PE is the potential energy, m is the mass of the block, g is the acceleration due to gravity, and h is the height. In this case, the height is 7R.
(d) To find the potential energy when the block is at point Q, we can use the same formula as in part (c), but with the height at Q, which is R.
(e) To find the potential energy when the block is at the top of the loop, we can use the same formula as in part (c), but with the height at the top of the loop, which is 2R.
(f) If the block is given some initial speed downward along the track instead of being released from rest, the potential energy at points P, Q, and the top of the loop will decrease because some of the initial potential energy will be converted into kinetic energy. The work done by the gravitational force between points P and Q, and between point P and the top of the loop, will remain the same because gravity is a conservative force and doesn't depend on the path taken.