1.A thin ruler stands vertically along a wall at the edge of a horizontal floor. The ruler starts sliding down due to slight disturbance. At the moments during the slide, the ends of the ruler remain in contact with the vertical wall and horizontal floor. What will be the locus of the centre of the ruler as the ruler falls?(mention the shape.)

To visualize the motion of the ruler as it slides down the wall, we can analyze the situation using some basic principles of geometry and physics. The ruler, which we can think of as a straight line segment, maintains contact with both the wall and the floor as it falls. This creates a unique path for the center of the ruler. Let's break this down step by step.

Understanding the Setup

Imagine the ruler is initially standing straight up, perfectly vertical against the wall. As it begins to slide down, the bottom end remains fixed on the floor while the top end slides down the wall. The key point to focus on is the center of the ruler, which is the midpoint of the segment.

Geometric Analysis

Let’s denote the length of the ruler as L. The center of the ruler will be located at a distance of L/2 from either end. As the ruler tilts, the angle it makes with the floor changes, but the lengths from the center to the wall and the floor remain constant.

Path of the Center

As the ruler slides down, the center will trace a specific path. To determine this path, we can use the concept of a right triangle formed by the wall, the floor, and the ruler itself. The vertical distance from the floor to the center of the ruler is h = L/2 * sin(θ), where θ is the angle between the ruler and the floor. The horizontal distance from the wall to the center is d = L/2 * cos(θ).

Identifying the Locus

As the ruler falls, the center moves in a way that can be described mathematically. The relationship between h and d can be expressed as:

• h^2 + d^2 = (L/2)^2

This equation represents a circle with a radius of L/2. Therefore, the locus of the center of the ruler as it falls is a circular arc. The center of this circle is located at the point where the ruler was initially vertical, specifically at the height of L/2 from the floor and at a distance of L/2 from the wall.

Visualizing the Motion

To further illustrate this, think of the center of the ruler as a point moving along the circumference of a circle as the ruler falls. Initially, when the ruler is vertical, the center is at the top of the circle. As it slides down, the center moves along the circular path until the ruler lies flat on the floor.

Conclusion

In summary, the locus of the center of the ruler as it slides down the wall is a circular arc. This motion is governed by the geometric constraints of the ruler maintaining contact with both the wall and the floor, leading to a fascinating interplay of angles and distances. Understanding this concept not only helps in visualizing the motion but also reinforces the principles of geometry in real-world applications.