To find the angular acceleration of a gramophone record, let's follow these steps carefully:
### Step 1: Understanding Given Values
- Initial angular velocity, [tex]\( \omega_i = 0 \)[/tex] rev/min (since the record starts from rest).
- Final angular velocity, [tex]\( \omega_f = 60 \)[/tex] rev/min.
- Time taken to reach final angular velocity, [tex]\( t = 4.0 \)[/tex] seconds.
### Step 2: Conversion of Units
Angular velocity is typically measured in radians per second (rad/s) in physics. We need to convert the final angular velocity from revolutions per minute (rev/min) to radians per second (rad/s).
[tex]\[ \omega_f = 60 \text{ rev/min} \][/tex]
Since [tex]\( 1 \text{ rev} = 2\pi \text{ radians} \)[/tex] and [tex]\( 1 \text{ minute} = 60 \text{ seconds} \)[/tex]:
[tex]\[ \omega_f (\text{rad/s}) = 60 \text{ rev/min} \times \left( \frac{2\pi \text{ rad}}{1 \text{ rev}} \right) \times \left( \frac{1 \text{ min}}{60 \text{ s}} \right) \][/tex]
[tex]\[ \omega_f (\text{rad/s}) = 60 \times \frac{2\pi}{60} \text{ rad/s} \][/tex]
[tex]\[ \omega_f (\text{rad/s}) = 2\pi \text{ rad/s} \][/tex]
### Step 3: Use the Angular Acceleration Formula
The formula to calculate angular acceleration ([tex]\(\alpha\)[/tex]) is:
[tex]\[ \alpha = \frac{\Delta \omega}{\Delta t} \][/tex]
where [tex]\( \Delta \omega = \omega_f - \omega_i \)[/tex].
Since the initial angular velocity [tex]\( \omega_i = 0 \)[/tex]:
[tex]\[ \alpha = \frac{2\pi \text{ rad/s} - 0 \text{ rad/s}}{4 \text{ s}} \][/tex]
[tex]\[ \alpha = \frac{2\pi}{4} \][/tex]
[tex]\[ \alpha = \frac{\pi}{2} \text{ rad/s}^2 \][/tex]
### Step 4: Compare with Choices
We have calculated the angular acceleration as [tex]\( \frac{\pi}{2} \text{ rad/s}^2 \)[/tex].
### Step 5: Identify Closest Choice
Among the given choices:
- (A) [tex]\(3 \pi \text{ rad/s}^2\)[/tex]
- (B) [tex]\(4 \pi \text{ rad/s}^2\)[/tex]
- (C) [tex]\(1 \pi \text{ rad/s}^2\)[/tex]
- (D) [tex]\(2 \pi \text{ rad/s}^2\)[/tex]
The angular acceleration [tex]\( \frac{\pi}{2} \text{ rad/s}^2 \)[/tex] is closest to [tex]\( \pi \text{ rad/s}^2 \)[/tex].
Thus, the closest and most appropriate choice is:
- (C) [tex]\(1 \pi \text{ rad/s}^2\)[/tex]
So, the answer is:
[tex]\[ \boxed{3} \][/tex]
