Question

In the figure here, a solid brass ball of mass 0.331 g will roll smoothly along a loop-the-loop track when released from rest along the straight section. The circular loop has radius R = 0.15 m, and the ball has radius r << R. (a) What is h if the ball is on the verge of leaving the track when it reaches the top of the loop? (b) If the ball is released at height h = 4R, what is the magnitude of the horizontal force component acting on the ball at point Q?

Answer 1

(a) using conservation of energy

mgh_i = mgh_f + 1/2mv² + 1/2 (2/5mr²)w²

where w = v/r

mgh_i = mgh_f + 1/2mv² + 1/2 (2/5mr²) (v/r)²

gh_i = gh_f + v² / 2 + v² / 5

gh_i = 2gr + 7v² / 10

at the verge of leaving,

v² = rg

so,

gh_i = 2gr + 7rg / 10

h_i = 2r + 7r / 10

h_i = 2.7r

h_i = 2.7 * 0.15

h_i = 0.405 m or 40.5 cm

------------------------------------------------------------------------------------------

(b) now, if h_i = 4r

mg(4r) = mgr + 1/2mv² + 1/2 (2/5mr²) (v/r)²

g(4r) = gr + 1/2v² + 1/2 (2/5r²) (v/r)²

g(4r) = gr + 7v² / 10

3gr = 7v² / 10

v² = 30gr / 7

so,

F = mv² / r

F = m*30gr / 7r

F = m*30*g / 7

F = 0.013902 N

or

F = 1.39e-2 N