Three identical solid spheres (1, 2 and 3) are allowed to roll down from three inclined plane of angles `theta_(1), theta_(2) and theta_(3)` respectively starting at time t = 0, where `theta_(1) gt theta_(2) gt theta_(3)`, then

Options

A. sphere 1 will reach the bottom first
B. sphere 2 will reach the bottom first
C. sphere 3 will reach the bottom first
D. all the spheres will reach the bottom at the same time

Answer

To solve the problem of three identical solid spheres rolling down inclined planes with angles \( \theta_1, \theta_2, \) and \( \theta_3 \) (where \( \theta_1 > \theta_2 > \theta_3 \)), we need to analyze the motion of the spheres based on the physics of rolling motion.

### Step-by-Step Solution:

1. **Identify Forces Acting on the Spheres**:
Each sphere experiences gravitational force \( mg \) acting downwards. This force can be resolved into two components:
- Parallel to the incline: \( mg \sin \theta \)
- Perpendicular to the incline: \( mg \cos \theta \)

2. **Determine the Acceleration of Each Sphere**:
For a solid sphere rolling without slipping, the acceleration \( a \) down the incline can be derived from Newton's second law and the rotational motion equations. The net force acting down the incline is \( mg \sin \theta \), and the moment of inertia \( I \) for a solid sphere is \( \frac{2}{5}mr^2 \). The relationship between linear acceleration \( a \) and angular acceleration \( \alpha \) is given by \( a = r\alpha \).

The equation of motion for the sphere can be written as:
\[ mg \sin \theta - f = ma \]
where \( f \) is the frictional force.

The torque \( \tau \) about the center of mass is given by:
\[ \tau = I\alpha = fr \]

Substituting \( I \) and \( \alpha \):
\[ fr = \frac{2}{5}mr^2 \cdot \frac{a}{r} \]
Simplifying gives:
\[ f = \frac{2}{5}ma \]

Substituting \( f \) back into the equation of motion:
\[ mg \sin \theta - \frac{2}{5}ma = ma \]
Rearranging gives:
\[ mg \sin \theta = ma + \frac{2}{5}ma = \frac{7}{5}ma \]
Thus, the acceleration \( a \) is:
\[ a = \frac{5g \sin \theta}{7} \]

3. **Compare Accelerations for Each Sphere**:
The accelerations of the spheres are given by:
- For sphere 1: \( a_1 = \frac{5g \sin \theta_1}{7} \)
- For sphere 2: \( a_2 = \frac{5g \sin \theta_2}{7} \)
- For sphere 3: \( a_3 = \frac{5g \sin \theta_3}{7} \)

Since \( \theta_1 > \theta_2 > \theta_3 \), it follows that:
\[ \sin \theta_1 > \sin \theta_2 > \sin \theta_3 \]
Therefore:
\[ a_1 > a_2 > a_3 \]

4. **Conclusion**:
The sphere that rolls down the incline with the largest angle \( \theta \) will have the greatest acceleration and will reach the bottom first. Thus, sphere 1 (with angle \( \theta_1 \)) will reach the bottom first, followed by sphere 2, and then sphere 3.

### Final Answer:
Sphere 1 will reach the bottom first, followed by sphere 2, and then sphere 3.