A body starts rolling down an inclined plane of length L and height h. This body reaches the bottom of the plane in time t. The relation between L and t is?

Options

A. `t prop L`
B. `t prop 1/L`
C. `L prop t^2`
D. `t prop 1/L^2`

Answer

To solve the problem of determining the relationship between the length \( L \) of an inclined plane and the time \( t \) it takes for a body to roll down, we can follow these steps:

### Step 1: Understand the motion
The body rolls down an inclined plane of length \( L \) and height \( h \). We need to find the relationship between \( L \) and \( t \).

### Step 2: Use the equations of motion
Since the body starts from rest, we can use the second equation of motion:
\[ L = ut + \frac{1}{2} a t^2 \]
where \( u \) is the initial velocity (which is 0) and \( a \) is the acceleration of the body down the incline.

### Step 3: Simplify the equation
Since the initial velocity \( u = 0 \), the equation simplifies to:
\[ L = \frac{1}{2} a t^2 \]

### Step 4: Determine the acceleration
The acceleration \( a \) of the rolling body can be expressed in terms of the gravitational acceleration \( g \) and the angle of the incline \( \theta \):
\[ a = \frac{g \sin \theta}{1 + \frac{I}{m r^2}} \]
where \( I \) is the moment of inertia of the body, \( m \) is its mass, and \( r \) is its radius.

### Step 5: Analyze the constants
In this scenario, \( g \), \( \sin \theta \), \( I \), \( m \), and \( r \) are constants for a given body and incline. Therefore, \( a \) is also a constant.

### Step 6: Relate \( L \) and \( t^2 \)
From the equation \( L = \frac{1}{2} a t^2 \), we can see that:
\[ L \propto t^2 \]
This means that the length \( L \) of the incline is directly proportional to the square of the time \( t \).

### Conclusion
Thus, the relationship between \( L \) and \( t \) is:
\[ L \propto t^2 \]