A solid sphere of mass m= 500gm is rolling without slipping on a horizontal surface. Find kinetic energy of a sphere if velocity of centre of mass is 5 cm/sec

Options

A. `35/2 xx 10^(-4) J`
B. `35/4 xx 10^(-4)J`
C. `35 xx 10^(-4)J`
D. `35 xx 10^(-3)J`

Explanation

To find the kinetic energy of a solid sphere rolling without slipping, we can follow these steps:

### Step 1: Identify the parameters
Given:
- Mass of the sphere, \( m = 500 \, \text{g} = 0.5 \, \text{kg} \) (since \( 1 \, \text{kg} = 1000 \, \text{g} \))
- Velocity of the center of mass, \( v = 5 \, \text{cm/s} = 0.05 \, \text{m/s} \) (since \( 1 \, \text{m} = 100 \, \text{cm} \))

### Step 2: Calculate translational kinetic energy
The translational kinetic energy \( KE_{trans} \) is given by the formula:
\[ KE_{trans} = \frac{1}{2} mv^2 \]
Substituting the values:
\[ KE_{trans} = \frac{1}{2} \times 0.5 \, \text{kg} \times (0.05 \, \text{m/s})^2 \]
Calculating \( (0.05)^2 \):
\[ (0.05)^2 = 0.0025 \, \text{m}^2/\text{s}^2 \]
Now substituting this back:
\[ KE_{trans} = \frac{1}{2} \times 0.5 \times 0.0025 = 0.000625 \, \text{J} \]

### Step 3: Calculate rotational kinetic energy
For a solid sphere, the moment of inertia \( I \) is given by:
\[ I = \frac{2}{5} m r^2 \]
The angular velocity \( \omega \) is related to the linear velocity \( v \) by:
\[ \omega = \frac{v}{r} \]
The rotational kinetic energy \( KE_{rot} \) is given by:
\[ KE_{rot} = \frac{1}{2} I \omega^2 \]
Substituting \( I \) and \( \omega \):
\[ KE_{rot} = \frac{1}{2} \left(\frac{2}{5} m r^2\right) \left(\frac{v}{r}\right)^2 \]
This simplifies to:
\[ KE_{rot} = \frac{1}{2} \left(\frac{2}{5} m r^2\right) \left(\frac{v^2}{r^2}\right) = \frac{1}{2} \cdot \frac{2}{5} m v^2 = \frac{1}{5} mv^2 \]
Substituting the values:
\[ KE_{rot} = \frac{1}{5} \times 0.5 \times 0.0025 = 0.00025 \, \text{J} \]

### Step 4: Calculate total kinetic energy
The total kinetic energy \( KE_{total} \) is the sum of translational and rotational kinetic energies:
\[ KE_{total} = KE_{trans} + KE_{rot} \]
Substituting the values:
\[ KE_{total} = 0.000625 \, \text{J} + 0.00025 \, \text{J} = 0.000875 \, \text{J} \]

### Step 5: Convert to appropriate units
To express the total kinetic energy in a more standard form:
\[ KE_{total} = 0.000875 \, \text{J} = 8.75 \times 10^{-4} \, \text{J} \]

### Final Answer
The total kinetic energy of the sphere is:
\[ KE_{total} = \frac{35}{40} \times 10^{-4} \, \text{J} = \frac{7}{8} \times 10^{-4} \, \text{J} \]