Question: A particle initially at rest moves from point A on the surface of a fixed smooth hemisphere of radius r. The particle loses contact with the hemisphere at point B. C is the center of the hemisphere. What is the equation relating the variables involved in this scenario?

Solution:

To solve this problem, let's consider the various variables involved:

1. Distance traveled by the particle: Let's denote this by "d".
2. Angle between the vertical line and the line joining points A and B: Let's denote this by "θ".
3. Radius of the hemisphere: Denoted by "r".

We need to find the equation relating these variables.

Step 1: Analyzing the forces acting on the particle

When the particle is at point A, it is in contact with the hemisphere, so the normal force and gravitational force balance each other. The normal force provides the centripetal force required for circular motion.

At point B, the particle loses contact with the hemisphere, so the normal force is zero. The only force acting on the particle is the gravitational force, which provides the centripetal force required for circular motion.

Step 2: Finding the centripetal force at point A

At point A, the normal force provides the centripetal force. The centripetal force is given by the equation:

F_c = m * a_c

where F_c is the centripetal force, m is the mass of the particle, and a_c is the centripetal acceleration.

The centripetal acceleration is given by:

a_c = v^2 / r

where v is the velocity of the particle at point A.

Since the particle is initially at rest, v = 0. Therefore, the centripetal acceleration is also zero.

Thus, the centripetal force at point A is zero.

Step 3: Finding the centripetal force at point B

At point B, the only force acting on the particle is the gravitational force. The gravitational force provides the centripetal force required for circular motion.

The gravitational force is given by the equation:

F_g = m * g

where F_g is the gravitational force and g is the acceleration due to gravity.

The centripetal force is given by:

F_c = m * a_c

where a_c is the centripetal acceleration.

Since the particle is moving in a circular path, the gravitational force provides the centripetal force. Therefore, we can equate the gravitational force and the centripetal force:

m * g = m * a_c

Simplifying, we get:

g = a_c

Step 4: Relating the variables

From step 2, we know that the centripetal force at point A is zero. Therefore, the gravitational force at point A is also zero.

From step 3, we know that the gravitational force at point B is equal to the centripetal force. Therefore, we have:

m * g = m * a_c

Since mass cancels out, we are left with:

g = a_c

We can also relate the centripetal acceleration to the distance traveled and the angle θ.

The distance traveled, d, is given by:

d = r * θ

The centripetal acceleration, a_c, can be expressed as:

a_c