Circular Motion Applications

This lesson brings together the theory of circular motion with challenging application problems: the loop-the-loop, particles on the inside of bowls, and combined problems involving energy and forces.

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1 Loop-the-Loop

A particle slides down a slope and enters a circular loop of radius r. The key question: what is the minimum height h from which the particle must start to complete the loop?

Analysis

At the top of the loop (height 2r from the bottom):

For the track to maintain contact: N + mg = mv_top^2/r, where N >= 0.

Critical case: N = 0, v_top^2 = gr.

Energy conservation from start (height h) to top of loop (height 2r):

mgh = (1/2)mv_top^2 + mg(2r)

gh = (1/2)gr + 2gr = (5/2)gr

h = 5r/2

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2 Worked Example 1

A marble rolls (without sliding) from height h down a smooth track into a vertical loop of radius 0.3 m. Find the minimum h for the marble to complete the loop (treating the marble as a particle).

h = 5r/2 = 5(0.3)/2 = 0.75 m

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3 Worked Example 2