Which of the following (if mass and radius are assumed to be same) have maximum percentage of total K.E. in rotational form while pure rolling ?

Options

A. Disc
B. Sphere
C. Ring
D. Hollow sphere

Answer

To solve the problem of determining which object has the maximum percentage of total kinetic energy in rotational form while pure rolling, we can follow these steps:

### Step 1: Understand the Kinetic Energy in Rolling Motion
The total kinetic energy (K.E.) of an object in pure rolling motion is the sum of its translational kinetic energy and rotational kinetic energy.
- Translational K.E. = \( \frac{1}{2} mv^2 \)
- Rotational K.E. = \( \frac{1}{2} I \omega^2 \)

### Step 2: Relate Angular Velocity to Linear Velocity
In pure rolling motion, the relationship between linear velocity (v) and angular velocity (ω) is given by:
\[ \omega = \frac{v}{R} \]
where R is the radius of the object.

### Step 3: Substitute ω in the Rotational K.E. Formula
Substituting \( \omega \) into the rotational kinetic energy formula gives:
\[ \text{Rotational K.E.} = \frac{1}{2} I \left(\frac{v}{R}\right)^2 = \frac{1}{2} \frac{I}{R^2} v^2 \]

### Step 4: Total K.E. in Terms of v
Now, the total kinetic energy can be expressed as:
\[ \text{Total K.E.} = \frac{1}{2} mv^2 + \frac{1}{2} \frac{I}{R^2} v^2 \]
Factoring out \( \frac{1}{2} v^2 \):
\[ \text{Total K.E.} = \frac{1}{2} v^2 \left(m + \frac{I}{R^2}\right) \]

### Step 5: Calculate the Percentage of Rotational K.E.
To find the percentage of kinetic energy that is in rotational form, we can use the formula:
\[ \text{Percentage of Rotational K.E.} = \frac{\text{Rotational K.E.}}{\text{Total K.E.}} \times 100 \]

Substituting the expressions we derived:
\[ \text{Percentage of Rotational K.E.} = \frac{\frac{1}{2} \frac{I}{R^2} v^2}{\frac{1}{2} v^2 \left(m + \frac{I}{R^2}\right)} \times 100 \]
This simplifies to:
\[ \text{Percentage of Rotational K.E.} = \frac{\frac{I}{R^2}}{m + \frac{I}{R^2}} \times 100 \]

### Step 6: Evaluate Moment of Inertia for Different Shapes
Now we need to evaluate the moment of inertia (I) for different objects:
1. **Disc**: \( I = \frac{1}{2} m R^2 \)
2. **Solid Sphere**: \( I = \frac{2}{5} m R^2 \)
3. **Ring**: \( I = m R^2 \)
4. **Hollow Sphere**: \( I = \frac{2}{3} m R^2 \)

### Step 7: Compare the Values of I
To maximize the percentage of rotational kinetic energy, we need to maximize the value of \( I \):
- For the ring, \( I = m R^2 \) is the largest.
- For the hollow sphere, \( I = \frac{2}{3} m R^2 \).
- For the disc, \( I = \frac{1}{2} m R^2 \).
- For the solid sphere, \( I = \frac{2}{5} m R^2 \).

### Step 8: Conclusion
Since the ring has the maximum moment of inertia, it will have the maximum percentage of total kinetic energy in rotational form while pure rolling.

**Final Answer**: The object with the maximum percentage of total K.E. in rotational form while pure rolling is the **ring**.

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