The angular velocity of a wheel increases from 120 to 480 rpm in 10 s .The number of revolutions made during this time is

Options

A. 10
B. 25
C. 50
D. 100

Explanation

To solve the problem of finding the number of revolutions made by a wheel as its angular velocity increases from 120 to 480 rpm over a period of 10 seconds, we can follow these steps:

### Step 1: Convert Angular Velocities from RPM to RPS
First, we need to convert the angular velocities from revolutions per minute (rpm) to revolutions per second (rps).

- Initial angular velocity, \( \omega_1 = 120 \text{ rpm} \)
- Final angular velocity, \( \omega_2 = 480 \text{ rpm} \)

To convert rpm to rps, we divide by 60 (since there are 60 seconds in a minute):

\[ \omega_1 = \frac{120}{60} = 2 \text{ rps} \]
\[ \omega_2 = \frac{480}{60} = 8 \text{ rps} \]

### Step 2: Calculate the Average Angular Velocity
Next, we calculate the average angular velocity (\( \omega_{avg} \)) during the time interval. The average angular velocity can be calculated using the formula:

\[ \omega_{avg} = \frac{\omega_1 + \omega_2}{2} \]

Substituting the values:

\[ \omega_{avg} = \frac{2 + 8}{2} = \frac{10}{2} = 5 \text{ rps} \]

### Step 3: Calculate the Total Time in Seconds
The total time \( T \) during which the acceleration occurs is given as 10 seconds.

### Step 4: Calculate the Total Number of Revolutions
The total number of revolutions (\( N \)) made during this time can be calculated using the formula:

\[ N = \omega_{avg} \times T \]

Substituting the average angular velocity and the time:

\[ N = 5 \text{ rps} \times 10 \text{ s} = 50 \text{ revolutions} \]

### Final Answer
The number of revolutions made during the time interval is **50 revolutions**.

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