The speed of a motor increases from 1200 rpm to 1800 rpm in 20 S. How many revolutions does it make in this period of time?

Options

A. 400
B. 200
C. 500
D. 800

Answer

To solve the problem of how many revolutions a motor makes as its speed increases from 1200 rpm to 1800 rpm in 20 seconds, we can follow these steps:

### Step 1: Convert RPM to Radians per Second
First, we need to convert the initial and final speeds from revolutions per minute (rpm) to radians per second (rad/s).

- Initial speed (ω₀) = 1200 rpm
- Final speed (ω) = 1800 rpm

Using the conversion factor:
1 revolution = 2π radians
1 minute = 60 seconds

We can convert:
\[ \omega_0 = 1200 \, \text{rpm} = \frac{1200 \times 2\pi \, \text{radians}}{60 \, \text{seconds}} = 40\pi \, \text{rad/s} \]
\[ \omega = 1800 \, \text{rpm} = \frac{1800 \times 2\pi \, \text{radians}}{60 \, \text{seconds}} = 60\pi \, \text{rad/s} \]

### Step 2: Calculate Angular Acceleration
Next, we calculate the angular acceleration (α) using the formula:
\[ \alpha = \frac{\Delta \omega}{\Delta t} = \frac{\omega - \omega_0}{t} \]
Where:
- Δω = ω - ω₀
- t = 20 seconds

Substituting the values:
\[ \alpha = \frac{60\pi - 40\pi}{20} = \frac{20\pi}{20} = \pi \, \text{rad/s}^2 \]

### Step 3: Calculate the Total Revolutions
Now, we can use the angular displacement formula to find the total number of revolutions (θ):
\[ \theta = \omega_0 t + \frac{1}{2} \alpha t^2 \]
Substituting the known values:
\[ \theta = (40\pi)(20) + \frac{1}{2}(\pi)(20^2) \]
Calculating each term:
1. First term: \(40\pi \times 20 = 800\pi\)
2. Second term: \(\frac{1}{2} \times \pi \times 400 = 200\pi\)

Adding both terms:
\[ \theta = 800\pi + 200\pi = 1000\pi \, \text{radians} \]

### Step 4: Convert Radians to Revolutions
To find the number of revolutions, we convert radians back to revolutions:
\[ \text{Revolutions} = \frac{\theta}{2\pi} = \frac{1000\pi}{2\pi} = 500 \]

### Final Answer
The motor makes **500 revolutions** in the 20 seconds.

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