A small block of mass m = 4.0 kg can slide along the frictionless loop-the-loop. The block is released from rest at point P, at height h = 23R above the bottom of the loop (R is the radius of the loop). Express your answers in the form ngR, where n is the number you will calculate.
a) What is the potential energy when the block is at point P? (Assume the gravitational potential energy of the block Earth system is taken to be zero at the bottom of the loop).
b) What is the potential energy when the block is at point Q (halfway up the loop)?
c) What is the kinetic energy when the block is at the top of the loop?
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To solve this problem, we need to consider the conservation of energy.
a) Potential energy at point P:
At point P, the block is at a height h above the bottom of the loop. The potential energy of the block at point P is given by the formula PE = mgh, where m is the mass of the block, g is the acceleration due to gravity, and h is the height of the block.
PE = mgh = 4.0 kg * (9.8 m/s^2) * 23R
b) Potential energy at point Q:
At point Q, the block is halfway up the loop. The potential energy of the block at point Q is given by the formula PE = mgh.
Since it is at a height of (1/2)h above the bottom of the loop, the height h will be halved in the formula.
PE = mgh = 4.0 kg * (9.8 m/s^2) * (23R / 2)
c) Kinetic energy at the top of the loop:
At the top of the loop, the block will have converted all of its potential energy into kinetic energy. The kinetic energy of the block at the top of the loop is given by the formula KE = (1/2)mv^2, where m is the mass of the block and v is the velocity of the block.
Since energy is conserved, the potential energy at point P would be equal to the kinetic energy at the top of the loop.
PE at P = KE at top of loop
mgh = (1/2)mv^2
Simplifying and canceling out the mass:
gh = (1/2)v^2
v = √(2gh)
v = √(2 * 9.8 m/s^2 * 23R)
Expressing the kinetic energy as KE = (1/2)mv^2:
KE = (1/2) * 4.0 kg * (√(2 * 9.8 m/s^2 * 23R))^2
Finally, substitute the value of n into the expression to express the answer in the form ngR.