Question

The following objects are released simultaneously from rest at the top of a 1.80 m long ramp inclined at 3.30 degrees to the horizontal: a solid sphere, a solid cylinder, a hollow cylindrical shell, and a hollow ball. Which wins the race? At the moment the winner reaches the bottom, find the positions of the other three objects.

Answer 1

As they roll down the ramp, their initial potential energy is converted into two types of kinetic energy let determine initial potential energy PE = mgh = m times 9.8 times 1.80 = 17.64 m The two types are linear & rotational kinetic energy Linear = 1/2 m V^2 Rotational = 1/2 I w^2 For solid sphere I = 2/5 mr^2 for solid cylindrical I = mr^2 shell for a hollow ball = I = 2/3 mr^2 To convert angular velocity to linear velocity use w = v/r, w^2 = v^2/r^2 for a solid sphere K.E = 1/2 2/5 m r^2 v^2/r^2 = 0.2 mV^2 Total K.E = 0.7 mv^2 Total final velocity Initial PE = 0.7 mv^2 = m times 17.64 V^2 = 25.2 V = 5 m/s Now to determine time when sphere reaches at bottom of the ramp we need to find a