In the figure below, a solid brass ball of mass m and radius r will roll without
ID: 1909752 • Letter: I
Question
In the figure below, a solid brass ball of mass m and radius r will roll without slipping along the loop-the-loop track when released from rest along the straight section. From what minimum height h above the bottom of the track must the marble be released to ensure that it does not leave the track at the top of the loop? (The radius of the loop-the-loop is R. Assume R » r. Use any variable stated above along with the following as necessary: g) h = (b) If the marble is released from height 6R above the bottom of the track, what is the magnitude of the horizontal component of the force acting on it at point Q? (Use any variable stated above along with the following as necessary: g)Explanation / Answer
a)
For not leaving contact at highest point of loop, mv^2/R > mg
or v^2 > Rg
KE = 1/2*mv^2 = 1/2*m*(Rg)
PE at heighest point of loop = mg*(2R)
Total energy at heighest point of loop = KE + PE = 1/2*mgR + 2*mgR = 5/2*mgR
Initial total energy = PE = mgh
Energy conservation: mgh = 5/2*mgR
h = 5R/2 = 2.5*R
b)
At Q, PE = mg*R
So, KE = mg*(6R) - mg*R = 5*mgR
1/2*mv^2 = 5*mgR
So, mv^2 = 10*mgR
Horizontal force = centrifugal force = mv^2/R = 10*mgR/R = 10*mg