[53] A gramophone record takes 4.0 s to reach its angular velocity of 60 rev/min from rest. Find its angular acceleration assuming that it is constant.

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]

Space for your working:



Answer :

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]