A small mass m slides without friction along the looped apparatus shown in the f

Question

A small mass m slides without friction along the looped apparatus shown in the figure below. If the object is to remain on the track, even at the top of the circle (whose radius is r), from what minimum height h must it be released? (Use the following as necessary: r.)

h = ________________________________

A small mass m slides without friction along the looped apparatus shown in the figure below. If the object is to remain on the track, even at the top of the circle (whose radius is r), from what minimum height h must it be released? (Use the following as necessary: r.) h = ________________________________

Explanation / Answer

At the top of the loop, it is at a height of 2r, where r is the radius of the loop: We know that at the top of the loop, the velocity must be such that the centripetal acceleration is equal to "g", so g = v2/r, so v2 = gr. Since the formula for kinetic energy uses v2, I will not take the square root. We also know that at the top of the ramp, it has potential energy, and at the top of the loop, it has both potential and kinetic: mgh = 1/2mv2 + mgh2 mgh = 1/2mv2 + mg2r now v2 = gr, so mgh = 1/2mgr + mg2r Divide by mg: h = 1/2r + 2r = 2.5r So the answer is 2.5r Hope that helps, please let me know if you have any other questions. Good luck!

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