A motor increases its speed from 500 rpm to 1000 rpm in 10 s. Calculate its angular acceleration and the number of revolutions made by it in this time.

Answer

To solve the problem step by step, we will first calculate the angular acceleration and then determine the number of revolutions made by the motor during the time interval.

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

- **Initial speed (ω₀)**: 500 rpm
- **Final speed (ω)**: 1000 rpm

The conversion formula from rpm to rad/s is:
\[ \text{rad/s} = \text{rpm} \times \frac{2\pi}{60} \]

Calculating the initial angular velocity:
\[ \omega_0 = 500 \times \frac{2\pi}{60} = \frac{500 \times 2\pi}{60} = \frac{1000\pi}{60} = \frac{50\pi}{3} \text{ rad/s} \]

Calculating the final angular velocity:
\[ \omega = 1000 \times \frac{2\pi}{60} = \frac{1000 \times 2\pi}{60} = \frac{2000\pi}{60} = \frac{100\pi}{3} \text{ rad/s} \]

### Step 2: Calculate Angular Acceleration
We can use the formula for angular acceleration (α):
\[ \alpha = \frac{\Delta \omega}{\Delta t} = \frac{\omega - \omega_0}{t} \]
Where:
- \( \Delta \omega = \omega - \omega_0 \)
- \( t = 10 \text{ s} \)

Substituting the values:
\[ \alpha = \frac{\frac{100\pi}{3} - \frac{50\pi}{3}}{10} = \frac{\frac{50\pi}{3}}{10} = \frac{50\pi}{30} = \frac{5\pi}{3} \text{ rad/s}^2 \]

### Step 3: Calculate the Angular Displacement
We can calculate the angular displacement (θ) using the formula:
\[ \theta = \omega_0 t + \frac{1}{2} \alpha t^2 \]

Substituting the known values:
\[ \theta = \left(\frac{50\pi}{3}\right) \times 10 + \frac{1}{2} \left(\frac{5\pi}{3}\right) \times (10)^2 \]
\[ \theta = \frac{500\pi}{3} + \frac{1}{2} \left(\frac{5\pi}{3}\right) \times 100 \]
\[ = \frac{500\pi}{3} + \frac{5\pi \times 100}{6} = \frac{500\pi}{3} + \frac{500\pi}{6} \]

To add these fractions, we need a common denominator:
\[ \frac{500\pi}{3} = \frac{1000\pi}{6} \]
So,
\[ \theta = \frac{1000\pi}{6} + \frac{500\pi}{6} = \frac{1500\pi}{6} = \frac{250\pi}{1} \text{ rad} \]

### Step 4: Calculate the Number of Revolutions
To find the number of revolutions (N), we use the relation:
\[ N = \frac{\theta}{2\pi} \]

Substituting the value of θ:
\[ N = \frac{250\pi}{2\pi} = \frac{250}{2} = 125 \text{ revolutions} \]

### Final Results
- **Angular acceleration (α)**: \(\frac{5\pi}{3} \text{ rad/s}^2\)
- **Number of revolutions (N)**: 125 revolutions