The density of a rod continuously increases from `A` to `B`. It is easier to set it into rotation by

Options

A. clamping the rod at `A` and applying a force `F` at `B`, perpendicular to the rod
B. clamping the rod at `B` and applying a force `F` at `A`, perpendicular to the rod
C. clamping the rod at mid point of `AB` and applying a force `F` at `A`, perpendicular to the rod
D. clamping the rod at mid-point of `AB` and applying force `F` at `B`, perpendicular to the rod.

Explanation

To solve the problem of determining where it is easier to set a rod with continuously increasing density into rotation, we can follow these steps:

### Step 1: Understand the Problem
The rod has a density that increases from point A to point B. We need to find out how to apply a force to make it easier to rotate the rod.

**Hint:** Visualize the rod and the density distribution along its length.

### Step 2: Identify the Center of Mass
Since the density increases from A to B, the center of mass of the rod will be closer to point B. This is because the mass distribution is not uniform; more mass is concentrated towards B.

**Hint:** Remember that the center of mass is affected by the distribution of mass along the length of the rod.

### Step 3: Understand Torque and Moment of Inertia
To rotate an object, torque (\( \tau \)) is applied, which is related to the moment of inertia (\( I \)) and angular acceleration (\( \alpha \)) by the equation:
\[ \tau = I \cdot \alpha \]
For easier rotation, we want to maximize \( \alpha \), which means we need to minimize \( I \).

**Hint:** Recall that moment of inertia depends on the mass distribution relative to the axis of rotation.

### Step 4: Determine the Moment of Inertia
The moment of inertia of the rod about an axis depends on how far the mass is from that axis. If we pivot the rod at point B, the mass distribution (and thus the moment of inertia) will be less compared to pivoting at point A.

**Hint:** Consider how the distance from the axis of rotation affects the moment of inertia.

### Step 5: Conclusion on Clamping Point
Since we want to minimize the moment of inertia to maximize angular acceleration, we should clamp the rod at point B and apply the force at point A. This configuration will allow for the easiest rotation of the rod.

**Hint:** Think about how the choice of pivot point influences the effectiveness of the applied force.

### Final Answer
To set the rod into rotation more easily, clamp the rod at point B and apply a force \( F \) at point A, perpendicular to the rod.