Question

A ball with a mass of 3.54 kg and a radius of 14.5 cm starts from rest at the top of a ramp that has a height of 1.08 m. What is the speed of the ball when it reaches the bottom of the ramp?

Assume 3 significant figures in your answer.

Answer 1

Given

Ball mass m = 3.54 kg , radius r = 14.5 cm , at a height on a ramp is h = 1.08 m

The ball starts from rest so that it will reach the bottom, means its initial velocity is zero ,By conservation of energy of the ball at two points , position 1 is at top of the ramp and position2 is at bottom of the ramp

so at position1 the total energy is only Gravitational potential energy  

   E1 = E2

E1 =m*g*h

and E2 = linea kinetic energy + rotational kinetic energy

treating the ball as solid sphere whose moment of inertia is I = (2/5)*m*r^2

   I = (2/5)*3.54*0.145^2 kg.m^2

   I = 0.0298 kg.m^2

rotational kinetic energy is R.k.e = 0.5*I*w^2 = 0.5*I*v^2/r^2

we know that v = r*W ==> W = v/r

E2 = 0.5*m*v^2 + 0.5*I*v^2/r^2

by conservation of energy

   m*g*h = 0.5*m*v^2 + 0.5*I*v^2/r^2

substituting the values

   3.54*9.8*1.08 = 0.5*3.54*v^2 + 0.5*0.0298*v^2/(0.145)^2

solving for v

       v = 3.89 m/s