Question

A wheel has an initial angular velocity of 1 rad/s and undergoes an angular acceleration of 10 rad/s^2. (a) How many radians does it rotate in 4 sec? (b) What is its angular velocity at this time?

Answer 1

Solution:

Given data: The initial angular velocity of the wheel ω_i = 1 rad/s

Initial angular position θ_i = 0 rad

The angular acceleration of the wheel, α = 10 rad/s²

Part (a) To find; final angular position θ_f.

Since the wheel is spinning about an rotation axis, it rotates through angle Δθ (= θ_f – θ_i, angular displacement) after some time Δt. We have the relation,

Δθ = θ_f – θ_i = ω_i*t+ (1/2)α*t²
Thus angular displacement in t = 4 s is given by,

θ_f – 0 rad = (1 rad/s)*(4 s) + (1/2)(10 rad/s²)(4 s)²

θ_f = 1 rad + 80 rad

θ_f = 81 rad

Thus wheel rotates 81 rad in 4 sec.

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Part (b) To find; final angular velocity ω_f.

The angular velocity at t= 4 s is given by,

ω_f = ω_i + αt

ω_f = 1 rad/s + (10 rad/s²)*(4 s)

ω_f = 41 rad/s

Thus at t = 4 s, wheel’s angular velocity is 41 rad/s