Question

Big Ben in London is the most accurate mechanical clock of its size. The 300-kg hour hand is 2.7 m long, and the 100-kg minute hand is 4.2 m long. Treat each hand as if it were a thin rod. Calculate the rotational kinetic energy of the two-hand system.

Answer 1

Mass of the hour hand = M₁ = 300 kg

Length of the hour hand = L₁ = 2.7 m

Moment of inertia of the hour hand = I₁

I₁ = M₁L₁²/3

I₁ = (300)(2.7)²/3

I₁ = 729 kg.m²

Time period of one full revolution of the hour hand = T₁ = 12 hours = 43200 sec

Angular speed of the hour hand = ₁

₁ = 1.454 x 10^-4 rad/s

Mass of the minute hand = M₂ = 100 kg

Length of the minute hand = L₂ = 4.2 m

Moment of inertia of the minute hand = I₂

I₂ = M₂L₂²/3

I₂ = (100)(4.2)²/3

I₂ = 588 kg.m²

Time period of one full revolution of the minute hand = T₂ = 60 min = 3600 sec

Angular speed of the minute hand = ₂

₂ = 1.745 x 10^-3 rad/s

Rotational energy of the two-hand system = E

E = I₁₁²/2 + I₂₂²/2

E = (729)(1.454x10^-4)²/2 + (588)(1.745x10^-3)²/2

E = 9.03 x 10^-4 J

Rotational kinetic energy of the two-hand system = 9.03 x 10^-4 J