A cylinder of mass M and radius R rolls on an inclined plane. The gain in kinetic energy is

Options

A. (a)`1/2Mv^2`
B. (b)`1/2Iomega^2`
C. (c)`3/4Mv^2`
D. (d)`3/4Iomega^2`

Answer

To find the gain in kinetic energy of a cylinder of mass \( M \) and radius \( R \) rolling down an inclined plane, we can follow these steps:

### Step 1: Understand the Kinetic Energy Components
The total kinetic energy (\( KE \)) of a rolling object consists of two parts:
1. Translational Kinetic Energy (\( KE_{trans} \))
2. Rotational Kinetic Energy (\( KE_{rot} \))

The formulas for these are:
- Translational: \( KE_{trans} = \frac{1}{2} M v^2 \)
- Rotational: \( KE_{rot} = \frac{1}{2} I \omega^2 \)

### Step 2: Determine the Moment of Inertia and Angular Velocity
For a solid cylinder, the moment of inertia \( I \) about its axis is given by:
\[ I = \frac{1}{2} M R^2 \]

The angular velocity \( \omega \) is related to the linear velocity \( v \) by the equation:
\[ \omega = \frac{v}{R} \]

### Step 3: Substitute \( \omega \) into the Rotational Kinetic Energy Formula
Now, substituting \( \omega \) into the rotational kinetic energy formula:
\[ KE_{rot} = \frac{1}{2} I \omega^2 = \frac{1}{2} \left(\frac{1}{2} M R^2\right) \left(\frac{v^2}{R^2}\right) \]
This simplifies to:
\[ KE_{rot} = \frac{1}{4} M v^2 \]

### Step 4: Calculate Total Kinetic Energy
Now, we can find the total kinetic energy by adding the translational and rotational kinetic energies:
\[ KE_{total} = KE_{trans} + KE_{rot} = \frac{1}{2} M v^2 + \frac{1}{4} M v^2 \]

### Step 5: Combine the Terms
To combine these terms, we can find a common denominator:
\[ KE_{total} = \frac{2}{4} M v^2 + \frac{1}{4} M v^2 = \frac{3}{4} M v^2 \]

### Conclusion
Thus, the gain in kinetic energy of the rolling cylinder is:
\[ \text{Gain in Kinetic Energy} = \frac{3}{4} M v^2 \]