A flywheel of mass 10 kg and radius 10 cm is revolving at a speed of 240 r.p.m. Its kinetic energy is

Options

A. `32pi J`
B. `(32)/(pi) J`
C. `32pi^(2) J`
D. `3.2 pi^(2)J`

Answer

To find the kinetic energy of the flywheel, we will use the formula for the rotational kinetic energy, which is given by:

\[ KE = \frac{1}{2} I \omega^2 \]

where \( KE \) is the kinetic energy, \( I \) is the moment of inertia, and \( \omega \) is the angular velocity in radians per second.

### Step 1: Calculate the Moment of Inertia (I)
For a ring, the moment of inertia \( I \) is given by:

\[ I = m r^2 \]

where \( m \) is the mass and \( r \) is the radius.

Given:
- Mass \( m = 10 \, \text{kg} \)
- Radius \( r = 10 \, \text{cm} = 0.1 \, \text{m} \)

Substituting the values:

\[ I = 10 \, \text{kg} \times (0.1 \, \text{m})^2 = 10 \times 0.01 = 0.1 \, \text{kg m}^2 \]

### Step 2: Convert RPM to Radians per Second
The angular velocity \( \omega \) in radians per second can be calculated from the revolutions per minute (RPM) using the conversion factor:

\[ \omega = \frac{2\pi \times \text{RPM}}{60} \]

Given:
- RPM = 240

Substituting the value:

\[ \omega = \frac{2\pi \times 240}{60} = \frac{480\pi}{60} = 8\pi \, \text{rad/s} \]

### Step 3: Calculate the Kinetic Energy (KE)
Now we can substitute \( I \) and \( \omega \) into the kinetic energy formula:

\[ KE = \frac{1}{2} I \omega^2 \]

Substituting the values we found:

\[ KE = \frac{1}{2} \times 0.1 \, \text{kg m}^2 \times (8\pi)^2 \]

Calculating \( (8\pi)^2 \):

\[ (8\pi)^2 = 64\pi^2 \]

Now substituting this back into the equation:

\[ KE = \frac{1}{2} \times 0.1 \times 64\pi^2 = 0.05 \times 64\pi^2 = 3.2\pi^2 \, \text{J} \]

### Step 4: Final Calculation
Using \( \pi \approx 3.14 \):

\[ KE \approx 3.2 \times (3.14)^2 \approx 3.2 \times 9.8596 \approx 31.58 \, \text{J} \]

Thus, the kinetic energy of the flywheel is approximately:

\[ \boxed{31.58 \, \text{J}} \]