A solid iron sphere `A` rolls down an inclined plane. While an identical hollow sphere `B` of same mass sides down the plane in a frictionless manner. At the bottom of the inclined plane, the total kinetic energy of sphere `A` is.

Options

A. less than that of `B`
B. equal to that of `B`
C. more than that of `B`
D. Sometimes more and sometimes less

Explanation

To find the total kinetic energy of the solid iron sphere A at the bottom of the inclined plane, we can follow these steps:

### Step 1: Understand the situation
We have two spheres: a solid sphere A and a hollow sphere B, both with the same mass and radius. Sphere A rolls down the inclined plane while sphere B slides down without friction.

### Step 2: Identify the types of kinetic energy
The total kinetic energy (KE) of a rolling object consists of two parts:
1. Translational kinetic energy (TKE) given by \( KE_{trans} = \frac{1}{2} mv^2 \)
2. Rotational kinetic energy (RKE) given by \( KE_{rot} = \frac{1}{2} I \omega^2 \)

Where:
- \( m \) is the mass of the sphere,
- \( v \) is the linear velocity of the center of mass,
- \( I \) is the moment of inertia,
- \( \omega \) is the angular velocity.

### Step 3: Calculate the moment of inertia
For the solid sphere A, the moment of inertia \( I \) is given by:
\[ I_A = \frac{2}{5} m r^2 \]

For the hollow sphere B, the moment of inertia \( I \) is given by:
\[ I_B = \frac{2}{3} m r^2 \]

### Step 4: Relate linear and angular velocity
Since sphere A rolls without slipping, we have the relationship:
\[ v = r \omega \]
Thus, we can express \( \omega \) in terms of \( v \):
\[ \omega = \frac{v}{r} \]

### Step 5: Substitute \( \omega \) into the rotational kinetic energy
For sphere A, the rotational kinetic energy becomes:
\[ KE_{rot, A} = \frac{1}{2} I_A \omega^2 = \frac{1}{2} \left(\frac{2}{5} m r^2\right) \left(\frac{v^2}{r^2}\right) = \frac{1}{5} mv^2 \]

### Step 6: Calculate total kinetic energy for sphere A
Now, we can find the total kinetic energy for sphere A:
\[ KE_A = KE_{trans, A} + KE_{rot, A} = \frac{1}{2} mv^2 + \frac{1}{5} mv^2 \]
To combine these, we need a common denominator:
\[ KE_A = \frac{5}{10} mv^2 + \frac{2}{10} mv^2 = \frac{7}{10} mv^2 \]

### Step 7: Conclusion
Thus, the total kinetic energy of sphere A at the bottom of the inclined plane is:
\[ KE_A = \frac{7}{10} mv^2 \]