Questions: An object at rest begins to rotate with a constant angular acceleration. If this object has an angular speed ω at time t₁, what will the angular velocity be at time t₂=2t₁? (a) 1/4 ω (b) 4 ω₁ (c) 20 (d) 1/2 ω (e) 1/√2 ω

An object at rest begins to rotate with a constant angular acceleration. If this object has an angular speed ω at time t₁, what will the angular velocity be at time t₂=2t₁?

(a) 1/4 ω
(b) 4 ω₁
(c) 20
(d) 1/2 ω
(e) 1/√2 ω

Transcript text: An object at rest begins to rotate with a constant angular acceleration. If this object has an angular speed $\omega$ at time $t_{1}$. what will the angular velocity be at time $t_{2}=2 t_{1}$ ? (a) $\frac{1}{4} \omega$ (b) $4 \omega_{1}$ (c) 20 (d) $\frac{1}{2} \omega$ (e) $\frac{1}{\sqrt{2}} \omega$

Solution

Solution Steps

Step 1: Understanding the Problem

We need to determine the angular velocity of an object at time $ t_2 = 2t_1 $ given that it starts from rest and has a constant angular acceleration. The object has an angular speed $ \omega_1 $ at time $ t_1 $.

Step 2: Using the Angular Kinematics Equation

The angular velocity $ \omega $ of an object under constant angular acceleration $ \alpha $ can be described by the equation: \[ \omega = \omega_0 + \alpha t \] where $ \omega_0 $ is the initial angular velocity. Since the object starts from rest, $ \omega_0 = 0 $.

Step 3: Expressing Angular Velocity at $ t_1 $

At time $ t_1 $, the angular velocity $ \omega_1 $ is given by: \[ \omega_1 = \alpha t_1 \]

Step 4: Expressing Angular Velocity at $ t_2 = 2t_1 $

At time $ t_2 = 2t_1 $, the angular velocity $ \omega_2 $ is: \[ \omega_2 = \alpha (2t_1) \] Substituting $ \alpha t_1 = \omega_1 $ from the previous step: \[ \omega_2 = 2 \alpha t_1 = 2 \omega_1 \]