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) $\frac{1}{\sqrt{2}} \omega$
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Solution

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Solution Steps

Step 1: Understanding the Problem

We need to determine the angular velocity of an object at time t2=2t1 t_2 = 2t_1 given that it starts from rest and has a constant angular acceleration. The object has an angular speed ω1 \omega_1 at time t1 t_1 .

Step 2: Using the Angular Kinematics Equation

The angular velocity ω \omega of an object under constant angular acceleration α \alpha can be described by the equation: ω=ω0+αt \omega = \omega_0 + \alpha t where ω0 \omega_0 is the initial angular velocity. Since the object starts from rest, ω0=0 \omega_0 = 0 .

Step 3: Expressing Angular Velocity at t1 t_1

At time t1 t_1 , the angular velocity ω1 \omega_1 is given by: ω1=αt1 \omega_1 = \alpha t_1

Step 4: Expressing Angular Velocity at t2=2t1 t_2 = 2t_1

At time t2=2t1 t_2 = 2t_1 , the angular velocity ω2 \omega_2 is: ω2=α(2t1) \omega_2 = \alpha (2t_1) Substituting αt1=ω1 \alpha t_1 = \omega_1 from the previous step: ω2=2αt1=2ω1 \omega_2 = 2 \alpha t_1 = 2 \omega_1

Final Answer

The angular velocity at time t2=2t1 t_2 = 2t_1 is: 2ω1 \boxed{2 \omega_1} Thus, the correct answer is not listed among the provided options.

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