Consider the following four objects:
(i) a hoop
(ii) a flat disk
(iii) a solid sphere
(iv) a hollow sphere
Each of the objects has mass M and radius R. The axis of rotation passes through the center of each object and is perpendicular to the plane of the hoop and the plane of the flat disk. Which object requires the largest torque to give it the same angular acceleration?

Options

A. The solid sphere
B. The hollow sphere
C. The hoop
D. The flat disk

Answer

To determine which object requires the largest torque to achieve the same angular acceleration, we need to analyze the moment of inertia for each object and use the relationship between torque, moment of inertia, and angular acceleration.

### Step-by-Step Solution:

1. **Understand the Relationship**:
The torque (\( \tau \)) required to produce an angular acceleration (\( \alpha \)) is given by the equation:
\[ \tau = I \alpha \]
where \( I \) is the moment of inertia of the object.

2. **Identify the Moment of Inertia for Each Object**:
- For a **hoop** (about an axis through its center and perpendicular to the plane):
\[ I_H = M R^2 \]
- For a **flat disk** (about an axis through its center and perpendicular to the plane):
\[ I_D = \frac{1}{2} M R^2 \]
- For a **solid sphere** (about an axis through its center):
\[ I_S = \frac{2}{5} M R^2 \]
- For a **hollow sphere** (about an axis through its center):
\[ I_{HS} = \frac{2}{3} M R^2 \]

3. **Compare the Moments of Inertia**:
Now we will compare the values of \( I \):
- \( I_H = M R^2 \)
- \( I_D = \frac{1}{2} M R^2 \)
- \( I_S = \frac{2}{5} M R^2 \)
- \( I_{HS} = \frac{2}{3} M R^2 \)

To compare these, we can look at the coefficients:
- Hoop: \( 1 \)
- Disk: \( \frac{1}{2} \)
- Solid Sphere: \( \frac{2}{5} \)
- Hollow Sphere: \( \frac{2}{3} \)

4. **Determine the Maximum Moment of Inertia**:
The largest coefficient is for the hoop, which means:
\[ I_H > I_D, I_S, I_{HS} \]

5. **Conclusion**:
Since torque is directly proportional to the moment of inertia for a given angular acceleration, the object that requires the largest torque to achieve the same angular acceleration is the **hoop**.

### Final Answer:
The object that requires the largest torque to give it the same angular acceleration is the **hoop**.

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