A solid sphere, a ring and a disc all having same mass and radius are placed at the top of an incline and released. The friction coefficient between the objects and the incline are same but not sufficient to allow pure rolling. Least time will be taken in reaching the bottom by

Options

A. the solid sphere
B. the ring
C. the disc
D. all will take the same time

Answer

To solve the problem step by step, we will analyze the motion of the solid sphere, ring, and disc on the incline.

### Step 1: Understand the Problem
We have three objects: a solid sphere, a ring, and a disc, all with the same mass (m) and radius (R). They are placed at the top of an incline and released. The friction coefficient (μ) between the objects and the incline is the same but not sufficient to allow pure rolling.

### Step 2: Identify Forces Acting on the Objects
When the objects are released, the following forces act on each of them:
- The gravitational force acting down the incline: \( F_{\text{gravity}} = mg \sin \theta \)
- The frictional force acting up the incline: \( F_{\text{friction}} = \mu mg \cos \theta \)

### Step 3: Write the Equation of Motion
The net force acting on each object along the incline can be expressed as:
\[ F_{\text{net}} = mg \sin \theta - \mu mg \cos \theta \]
Using Newton's second law, \( F = ma \), we can write the equation for acceleration (a) of each object:
\[ ma = mg \sin \theta - \mu mg \cos \theta \]
Dividing through by m, we get:
\[ a = g \sin \theta - \mu g \cos \theta \]

### Step 4: Analyze the Acceleration
Since the mass (m), gravitational acceleration (g), angle of incline (θ), and friction coefficient (μ) are the same for all three objects, the acceleration (a) will be the same for the solid sphere, ring, and disc:
\[ a = g (\sin \theta - \mu \cos \theta) \]

### Step 5: Determine the Time to Reach the Bottom
All three objects start from rest (initial velocity \( u = 0 \)) and have the same acceleration. The distance (d) they travel down the incline is the same. Using the equation of motion:
\[ d = ut + \frac{1}{2} a t^2 \]
Substituting \( u = 0 \):
\[ d = \frac{1}{2} a t^2 \]
Rearranging gives:
\[ t^2 = \frac{2d}{a} \quad \Rightarrow \quad t = \sqrt{\frac{2d}{a}} \]
Since \( a \) is the same for all three objects, the time \( t \) taken to reach the bottom will also be the same for the solid sphere, ring, and disc.

### Conclusion
All three objects will take the same time to reach the bottom of the incline. Therefore, the answer is that they will all take the same time.

### Final Answer
**All will take the same time.**

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