A uniform thin rod of mass m and length L is standing vertically along the y-axis on a smooth horizontal surface, with its lower end at the origin (0,0). A slight disturbane at t = 0 causes the lower end t slip on the smooth surface along the positive x-axis, and the rod starts falling.
(i) What is the path followed by the centr of mass of the rod during its fall?
(ii) Find the equaiton to the trajectory of a point on the rod shape of the path of this point?

Explanation

To solve the problem step by step, we will address both parts of the question regarding the motion of the uniform thin rod.

### Step 1: Analyze the Motion of the Rod
The rod is initially standing vertically along the y-axis with its lower end at the origin (0,0). When a slight disturbance occurs, the lower end of the rod begins to slip along the x-axis while the rod starts to fall.

### Step 2: Determine the Path of the Center of Mass
1. **Identify Forces**: The only forces acting on the rod are its weight (mg) acting downward through its center of mass and the normal force from the ground at the lower end. There are no horizontal forces acting on the rod.

2. **Horizontal Motion**: Since there are no horizontal forces acting on the rod, the horizontal acceleration is zero. Therefore, the horizontal velocity of the center of mass remains constant, which is zero at the start.

3. **Vertical Motion**: The center of mass will only move vertically downward under the influence of gravity. Thus, the path followed by the center of mass is a straight vertical line downwards.

**Conclusion for Part (i)**: The path followed by the center of mass of the rod during its fall is a straight vertical line downwards.

### Step 3: Determine the Trajectory of a Point on the Rod
1. **Define the Point on the Rod**: Let’s consider a point P on the rod that is at a distance r from the lower end. As the rod falls, this point will trace a path.

2. **Use Geometry**: At any instant, let θ be the angle that the rod makes with the vertical. The coordinates of point P can be expressed in terms of r and θ:
- The vertical position (y-coordinate) of point P is given by:
\[ y = r \sin(\theta) \]
- The horizontal position (x-coordinate) of point P is given by:
\[ x = r \cos(\theta) \]

3. **Relate θ to the Position**: The length of the rod is L, and the center of mass is at a distance L/2 from the lower end. The relationship between the angle and the coordinates can be established using trigonometric identities.

4. **Establish the Equation**: Using the Pythagorean theorem:
\[ \sin^2(\theta) + \cos^2(\theta) = 1 \]
Substituting for y and x:
\[ \left(\frac{y}{r}\right)^2 + \left(\frac{x}{r}\right)^2 = 1 \]
Rearranging gives:
\[ y^2 + x^2 = r^2 \]
This indicates that the point P traces a circular path of radius r.

5. **Shape of the Path**: Since the rod is falling while the lower end moves horizontally, the trajectory of point P will describe an elliptical path as the angle θ changes continuously.

**Conclusion for Part (ii)**: The trajectory of a point on the rod is elliptical in shape.

### Summary of Solutions:
- (i) The path followed by the center of mass is a straight vertical line downwards.
- (ii) The trajectory of a point on the rod is elliptical in shape.