A solid sphere of diameter 0.1 m and 5 kg is rolling down an inclined plane with a speed of 4 m/s. The total kinetic energy of the sphere is :

Options

A. 28 J
B. 56 J
C. 84 J
D. 112 J

Answer

To find the total kinetic energy of a solid sphere rolling down an inclined plane, we need to consider both its translational and rotational kinetic energy. Here’s how to solve the problem step by step:

### Step 1: Identify the given values
- Diameter of the sphere (d) = 0.1 m
- Radius of the sphere (r) = d/2 = 0.1 m / 2 = 0.05 m
- Mass of the sphere (m) = 5 kg
- Speed of the sphere (v) = 4 m/s

### Step 2: Calculate the moment of inertia (I) of the solid sphere
The moment of inertia (I) for a solid sphere is given by the formula:
\[ I = \frac{2}{5} m r^2 \]
Substituting the values:
\[ I = \frac{2}{5} \times 5 \, \text{kg} \times (0.05 \, \text{m})^2 \]
\[ I = \frac{2}{5} \times 5 \times 0.0025 \]
\[ I = \frac{2}{5} \times 0.0125 \]
\[ I = 0.005 \, \text{kg m}^2 \]

### Step 3: Calculate the angular velocity (ω)
The angular velocity (ω) is related to the linear velocity (v) by the formula:
\[ \omega = \frac{v}{r} \]
Substituting the values:
\[ \omega = \frac{4 \, \text{m/s}}{0.05 \, \text{m}} \]
\[ \omega = 80 \, \text{rad/s} \]

### Step 4: Calculate the rotational kinetic energy (KE_rot)
The rotational kinetic energy (KE_rot) is given by:
\[ KE_{\text{rot}} = \frac{1}{2} I \omega^2 \]
Substituting the values:
\[ KE_{\text{rot}} = \frac{1}{2} \times 0.005 \, \text{kg m}^2 \times (80 \, \text{rad/s})^2 \]
\[ KE_{\text{rot}} = \frac{1}{2} \times 0.005 \times 6400 \]
\[ KE_{\text{rot}} = 0.0025 \times 6400 \]
\[ KE_{\text{rot}} = 16 \, \text{J} \]

### Step 5: Calculate the translational kinetic energy (KE_trans)
The translational kinetic energy (KE_trans) is given by:
\[ KE_{\text{trans}} = \frac{1}{2} m v^2 \]
Substituting the values:
\[ KE_{\text{trans}} = \frac{1}{2} \times 5 \, \text{kg} \times (4 \, \text{m/s})^2 \]
\[ KE_{\text{trans}} = \frac{1}{2} \times 5 \times 16 \]
\[ KE_{\text{trans}} = 2.5 \times 16 \]
\[ KE_{\text{trans}} = 40 \, \text{J} \]

### Step 6: Calculate the total kinetic energy (KE_total)
The total kinetic energy is the sum of the translational and rotational kinetic energies:
\[ KE_{\text{total}} = KE_{\text{trans}} + KE_{\text{rot}} \]
Substituting the values:
\[ KE_{\text{total}} = 40 \, \text{J} + 16 \, \text{J} \]
\[ KE_{\text{total}} = 56 \, \text{J} \]

### Final Answer
The total kinetic energy of the sphere is **56 Joules**.

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