Question

1. A cylindrical wooden log rolls without slipping down a small ramp. The log has a radius of 16 cm, a length of 2.5 m, and a mass of 130 kg. The log starts from rest near the top of the ramp. When the log reaches the bottom of the ramp, it is rolling at a linear speed of 2.3 m/s.

a.) Assuming that the log is a perfect cylinder of uniform density, find the log’s moment of inertia (about its long, central axis). Show your work.

b.) Find the log’s total kinetic energy at the bottom of the ramp. Show your work completely.

c.) Assuming no loss of energy to air resistance or friction, from what height did the log start? Show your work.

Answer 1

a)

m = mass of log = 130 kg

r = radius = 16 cm = 0.16 m

moment of inertia is given as

I = (0.5) m r²

I = (0.5) (130) (0.16)²

I = 1.664 kg/m²

b)

at the bottom :

v = 2.3 m/s

w = v/r = 2.3/0.16 = 14.375 rad/s

Total kinetic energy is given as

KE = (0.5) Iw² + (0.5) m v²

KE = (0.5) (1.664) (14.375)² + (0.5) (130) (2.3)²

KE = 515.775 J

c)

using conservation of energy

Potential energy = Total kinetic energy at bottom

mgh = 515.775

(130) (9.8) h = 515.775

h = 0.405 m