Two uniform solid spheres having unequal radii are released from rest from the same height on a rough incline. If the spheres roll without slipping

Options

A. the heavier sphere reaches the bottom first
B. the bigger sphere reaches the bottom first
C. the two spheres reach the bottom together
D. the information given is not sufficient to tell whidch sphere will reach the bottom first.

Answer

To solve the problem of two uniform solid spheres with unequal radii rolling down a rough incline without slipping, we can follow these steps:

### Step-by-Step Solution:

1. **Identify Forces Acting on the Sphere:**
- The gravitational force acting down the incline can be expressed as \( F = mg \sin \theta \), where \( m \) is the mass of the sphere and \( \theta \) is the angle of the incline.
- There is also a frictional force \( f \) acting up the incline.

2. **Apply Newton's Second Law:**
- For the linear motion of the sphere, we can write the equation:
\[ mg \sin \theta - f = ma \quad \text{(1)} \]
- Here, \( a \) is the linear acceleration of the center of mass of the sphere.

3. **Consider the Rotational Motion:**
- The torque \( \tau \) due to the frictional force about the center of mass is given by:
\[ \tau = f \cdot r \]
- This torque causes an angular acceleration \( \alpha \) of the sphere:
\[ \tau = I \alpha \quad \text{(2)} \]
- For a solid sphere, the moment of inertia \( I \) is given by:
\[ I = \frac{2}{5} m r^2 \]

4. **Relate Angular and Linear Acceleration:**
- Since the sphere rolls without slipping, the relationship between linear acceleration \( a \) and angular acceleration \( \alpha \) is:
\[ a = \alpha r \quad \text{(3)} \]

5. **Substituting for Angular Acceleration:**
- From equation (3), we can express \( \alpha \) as:
\[ \alpha = \frac{a}{r} \]
- Substituting this into equation (2):
\[ f \cdot r = \left(\frac{2}{5} m r^2\right) \left(\frac{a}{r}\right) \]
- Simplifying gives:
\[ f = \frac{2}{5} m a \quad \text{(4)} \]

6. **Substituting Back into the Linear Motion Equation:**
- Substitute equation (4) into equation (1):
\[ mg \sin \theta - \frac{2}{5} m a = ma \]
- Rearranging gives:
\[ mg \sin \theta = ma + \frac{2}{5} m a \]
- This simplifies to:
\[ mg \sin \theta = \frac{7}{5} m a \]

7. **Solving for Linear Acceleration:**
- Dividing both sides by \( m \) and rearranging gives:
\[ a = \frac{5}{7} g \sin \theta \]

8. **Conclusion:**
- The acceleration \( a \) is independent of the radius of the spheres. Therefore, both spheres will reach the bottom of the incline at the same time, regardless of their differing radii.

### Final Answer:
Both spheres reach the bottom together.

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