A wheel initially at rest, begins to rotate about its axis with constant angular acceleration. If it rotates through an angle θ 1 in first 2 s and a further angle θ 2 in the next 2 s , the ratio θ

Question

A wheel initially at rest, begins to rotate about its axis with constant angular acceleration. If it rotates through an angle θ 1 ​ in first 2 s and a further angle θ 2 ​ in the next 2 s , the ratio θ 1 ​ : θ 2 ​ is

TestHub · Accepted Answer

Using the kinematic equation for rotational motion, the angle rotated is given by $θ = ω_{0} t + \frac{1}{2} α t^{2}$ .

Since the wheel starts from rest, $ω_{0} = 0$ .

Angle rotated in the first 2 seconds is $θ_{1} = \frac{1}{2} α (2)^{2} = 2 α$ .

Angle rotated in the first 4 seconds is $θ_{t o t a l} = \frac{1}{2} α (4)^{2} = 8 α$ .

Angle rotated in the next 2 seconds is $θ_{2} = θ_{t o t a l} - θ_{1} = 8 α - 2 α = 6 α$ .

The ratio is $\frac{θ _{1}}{θ _{2}} = \frac{2 α}{6 α} = \frac{1}{3}$ .

Answer: $1 : 3$

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