A motorcycle, which has an initial linear speed of 7.8 m/s, decelerates to a speed of 1.4 m/s in 5.8 s. Each wheel has a radius of 0.67 m and is rotating in a counterclockwise (positive) directions. What is (a) the constant angular acceleration (in rad/s2) and (b) the angular displacement (in rad) of each wheel

(a) 1.64 rad/s²

(b) 39.94 rad

Explanation:

(a) From the question,

α = a/r Equation 1

Where a = linear acceleration if the motorcycle wheel, r = radius of the wheel, α = angular velocity of the wheel.

But,

a = (v-u)/t Equation 2

Given: v = 1.4 m/s, u = 7.8 m/s, t = 5.8 s

Substitute into equation 2.

a = (1.4-7.8)/5.8

a = -1.1 m/s².

Note: The negative sign shows that the motorcycle is decelerating

Also given: r = 0.67 m.

Substitute into equation 1

α = 1.1/0.67

α  = 1.64 rad/s²

(b)

Θ = s/r Equation 3

Where s = linear displacement, Θ = angular displacement.

But,

s = (v²-u²)/2a Equation 2

s = (1.4²-7.8²)/[2×(-1.1)]

s = -58.88/-2.2

s = 26.76 m.

therefore,

Θ = 26.76/0.67

Θ = 39.94 rad.

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