A solid sphere of radius 4 cm and mass 5 kg rotates about an axis through its centre with angular speed:

600 rpm

It is brought to rest in 10 s by applying a constant torque.

Find:

1. Torque applied
2. Number of rotations before stopping

Radius:

R = 0.04 m

Mass:

M = 5 kg

Moment of inertia of solid sphere:

I = (2/5)MR²

= (2/5)(5)(0.04)²

= 0.0032 kg m²

Initial angular speed:

ω = 600 × 2π/60

= 20π rad/s

Angular acceleration:

α = (0 − 20π)/10

= −2π rad/s²

Torque:

τ = Iα

= 0.0032 × 2π

≈ 6.4π × 10⁻³ N·m

≈ 6.4π × 10⁴ dyne-cm

Number of rotations:

θ = (ω+0)t/2

= (20π)(10)/2

= 100π rad

Rotations:

N = 100π / 2π

= 50

Final Answer

Torque = 6.4π × 10⁴ dyne-cm

Number of Rotations = 50