A ring, a disc a sphere and spherical shells are simutaneously released to roll down from the top of an inclined plane of height h. the four bodies will reach the bottom in the following order

Options

A. sphere,disc,shell and ring
B. ring,spherical,shell,disc and sphere
C. sphere,spherical shell, disc and ring.
D. sphere,spherical shell, ring and disc

Explanation

To solve the problem of determining the order in which a ring, a disc, a sphere, and a spherical shell roll down an inclined plane, we can follow these steps:

### Step 1: Understand the motion of the objects
When the objects are released from the top of the inclined plane, they will roll down due to gravity. Each object will have both translational and rotational motion.

### Step 2: Use the formula for acceleration
The acceleration \( a \) of a rolling object down an incline can be expressed as:
\[ a = \frac{g \sin \theta}{1 + \frac{k^2}{r^2}} \]
where:
- \( g \) is the acceleration due to gravity,
- \( \theta \) is the angle of the incline,
- \( k \) is the radius of gyration,
- \( r \) is the radius of the object.

### Step 3: Determine the radius of gyration for each object
The radius of gyration \( k \) for each object is:
- For the sphere: \( k^2 = \frac{2}{5} r^2 \)
- For the disc: \( k^2 = \frac{1}{2} r^2 \)
- For the ring: \( k^2 = r^2 \)
- For the spherical shell: \( k^2 = \frac{2}{3} r^2 \)

### Step 4: Calculate the value of \( x \) for each object
Using the formula for \( x \):
\[ x = 1 + \frac{k^2}{r^2} \]
we can calculate \( x \) for each object:

1. **Sphere**:
\[ x = 1 + \frac{\frac{2}{5} r^2}{r^2} = 1 + \frac{2}{5} = \frac{7}{5} = 1.4 \]

2. **Disc**:
\[ x = 1 + \frac{\frac{1}{2} r^2}{r^2} = 1 + \frac{1}{2} = \frac{3}{2} = 1.5 \]

3. **Ring**:
\[ x = 1 + \frac{r^2}{r^2} = 1 + 1 = 2 \]

4. **Spherical Shell**:
\[ x = 1 + \frac{\frac{2}{3} r^2}{r^2} = 1 + \frac{2}{3} = \frac{5}{3} \approx 1.67 \]

### Step 5: Compare the values of \( x \)
The values of \( x \) are:
- Sphere: \( 1.4 \)
- Disc: \( 1.5 \)
- Spherical Shell: \( 1.67 \)
- Ring: \( 2 \)

### Step 6: Determine the order of arrival
Since a lower value of \( x \) corresponds to higher acceleration, the order in which the objects reach the bottom of the incline is:
1. Sphere (1.4)
2. Disc (1.5)
3. Spherical Shell (1.67)
4. Ring (2)

### Final Answer
The order in which the bodies reach the bottom of the inclined plane is:
**Sphere, Disc, Spherical Shell, Ring.**

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