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
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 t2=2t1 given that it starts from rest and has a constant angular acceleration. The object has an angular speed ω1 at time t1.
Step 2: Using the Angular Kinematics Equation
The angular velocity ω of an object under constant angular acceleration α can be described by the equation:
ω=ω0+αt
where ω0 is the initial angular velocity. Since the object starts from rest, ω0=0.
Step 3: Expressing Angular Velocity at t1
At time t1, the angular velocity ω1 is given by:
ω1=αt1
Step 4: Expressing Angular Velocity at t2=2t1
At time t2=2t1, the angular velocity ω2 is:
ω2=α(2t1)
Substituting αt1=ω1 from the previous step:
ω2=2αt1=2ω1
Final Answer
The angular velocity at time t2=2t1 is:
2ω1
Thus, the correct answer is not listed among the provided options.