The angular velocity of a wheel increases from 100 to 300 in 10 s. The number of revolutions made during that time is

Options

A. 600
B. 1500
C. 1000
D. 2000

Explanation

To find the number of revolutions made by the wheel as its angular velocity increases from 100 to 300 in 10 seconds, we can follow these steps:

### Step 1: Convert Angular Velocities to Radians per Second
The initial angular velocity \( \omega_1 \) is given as 100 revolutions per minute (rpm). We need to convert this to radians per second.

\[ \omega_1 = 100 \text{ rpm} = 100 \times \frac{2\pi \text{ radians}}{1 \text{ revolution}} \times \frac{1 \text{ minute}}{60 \text{ seconds}} = \frac{200\pi}{60} = \frac{10\pi}{3} \text{ radians/second} \]

The final angular velocity \( \omega_2 \) is 300 rpm.

\[ \omega_2 = 300 \text{ rpm} = 300 \times \frac{2\pi \text{ radians}}{1 \text{ revolution}} \times \frac{1 \text{ minute}}{60 \text{ seconds}} = \frac{600\pi}{60} = 10\pi \text{ radians/second} \]

### Step 2: Calculate Angular Acceleration
We can find the angular acceleration \( \alpha \) using the formula:

\[ \alpha = \frac{\omega_2 - \omega_1}{t} \]

Substituting the values:

\[ \alpha = \frac{10\pi - \frac{10\pi}{3}}{10} = \frac{10\pi \left(1 - \frac{1}{3}\right)}{10} = \frac{10\pi \cdot \frac{2}{3}}{10} = \frac{2\pi}{3} \text{ radians/second}^2 \]

### Step 3: Calculate the Angular Displacement
Now we can calculate the angular displacement \( \theta \) using the formula:

\[ \theta = \omega_1 t + \frac{1}{2} \alpha t^2 \]

Substituting the values:

\[ \theta = \left(\frac{10\pi}{3}\right) \cdot 10 + \frac{1}{2} \cdot \left(\frac{2\pi}{3}\right) \cdot (10)^2 \]

Calculating each term:

\[ \theta = \frac{100\pi}{3} + \frac{1}{2} \cdot \frac{2\pi}{3} \cdot 100 = \frac{100\pi}{3} + \frac{100\pi}{3} = \frac{200\pi}{3} \text{ radians} \]

### Step 4: Convert Angular Displacement to Revolutions
To find the number of revolutions, we divide the angular displacement by \( 2\pi \):

\[ \text{Number of revolutions} = \frac{\theta}{2\pi} = \frac{\frac{200\pi}{3}}{2\pi} = \frac{200}{6} = \frac{100}{3} \approx 33.33 \]

### Final Answer
The number of revolutions made during that time is approximately 33.33 revolutions.

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