Question

A wheel is rotating freely at angular speed 530 rev/min on a shaft whose rotational inertia is negligible. A second wheel, initially at rest and with 5 times the rotational inertia of the first, is suddenly coupled to the same shaft. (a) What is the angular speed of the resultant combination of the shaft and two wheels? (b) What fraction of the original rotational kinetic energy is lost?

Answer 1

Initial angular speed of the first wheel = ₁ = 530 rev/min

Rotational inertia of the first wheel = I₁

Rotational inertia of the second wheel = I₂ = 5I₁

Angular speed after the second wheel is coupled to the same shaft = ₂

By conservation of angular momentum,

I₁₁ = (I₁ + I₂)₂

I₁₁ = 6I₁₂

₂ = ₁/6

₂ = 530/6

₂ = 88.33 rev/min

Original rotational kinetic energy = KE₁

KE₁ = I₁₁²/2

Final rotational kinetic energy = KE₂

KE₂ = (I₁ + I₂)₂²/2

KE₂ = (I₁ + 5I₁)(₁/6)²/2

KE₂ = (6I₁)(₁²/36)/2

KE₂ = I₁₁²/12

Energy lost = E

E = KE₁ - KE₂

Fraction of original rotational kinetic energy lost = f

f = 5/6

f = 0.833

a) Angular speed of the resultant combination = 88.33 rev/min

b) Fraction of original rotational kinetic energy lost = 0.833