Circular Motion

If energy is added, then where is the new radius of revolution and .

To solve the problem of finding the ratio of kinetic energies , we need to understand the physics involved in a simple pendulum executing circular motion.

The key points to consider are:

1. At the lowest point , the bob has maximum kinetic energy and zero potential energy.
2. At the highest point , the bob has maximum potential energy and minimum kinetic energy, just enough to maintain circular motion.

Let's go through the solution step-by-step:

1. The critical condition for the bob to just reach point is that its velocity at point should be such that it maintains the tension in the string. The minimum velocity required can be derived using the centripetal force equation:
2. The kinetic energy at point is given by:
3. Using conservation of energy from point to point , the total energy at point equals the total energy at point :
4. Substituting the expression for :
5. Simplifying the equation to find :
6. The kinetic energy at point is therefore:
7. Now, calculate the ratio of kinetic energies:

The correct answer is therefore .

To find the ratio of velocities of the two particles having the same mass, we need to consider the condition of constant centripetal force. The formula for centripetal force ( ) is given by:

where:

• is the mass of the particle,
• is the velocity of the particle,
• is the radius of curvature.

Since the mass of the two particles is the same and the centripetal force is constant, we can equate the expressions for the two particles and consider their ratio:

.

Simplifying, we get:

.

Rearranging gives the relation between their velocities and radii:

.

We know the radii are in the ratio . Substituting these values, we have:

.

Solve for :

.

Taking the square root of both sides, we get:

.

Hence, the velocities are in the ratio .

Given:
- The radius ,
- The initial velocity .

The particle’s normal and tangential accelerations are equal, so:

Rearrange the equation to separate variables:

Integrating both sides from to and to , we get:

Now calculating this we get:

This can be simplified as:

Where and for a complete revolution.

Now we integrate the expression for :

From this, solving for , we get:

Thus, the value of is 8.

The Correct Answer is: 8

To find the tension in the string when the stone is at the lowest point, we need to analyze the forces acting on the stone in circular motion.

First, let's outline the necessary parameters:

• Mass of stone,
• Radius of the circle,
• Revolutions per minute,
• Gravitational acceleration,

At the lowest point of the circle, the forces acting on the stone are the tension in the string and the gravitational force. The net centripetal force needed to keep the stone moving in a circle is provided by the tension in the string minus the gravitational force:

Where is the angular velocity in radians per second. We need to convert rpm to radians per second using the formula:

Now substitute back into the centripetal force formula:

Let's calculate each component:

Add these to find the total tension:

After rounding off based on significant figures provided in the options, the closest value is:

Correct Answer: 9.8 N

In this problem, we need to determine the speed of a particle on the rim of a wheel rolling on a plane surface, at the same level as the center of the wheel. Let's break down the problem step-by-step:

1. The given speed of the particle at the highest point of the rim is 8 m/s. When a wheel is rolling without slipping, the speed at the highest point of the rim is twice the velocity of the center of the wheel. Therefore, let the velocity of the center of the wheel be . We have:

1. Now, we need to find the speed of a particle on the rim of the wheel at the same level as the center of the wheel. At this point (horizontal from the center, either left or right on the rim), the velocity component due to rotation is perpendicular to the translational velocity of the wheel (i.e., center speed). The speed due to rotation at this point is . The wheel moves with uniform translational speed, also .
2. To find the resultant speed, we use the Pythagorean theorem since the translational and rotational velocities are perpendicular to each other:

1. Thus, the speed of the particle on the rim of the wheel at the same level as the center of the wheel is .

Displacement is the straight-line distance from the initial point to the final point.
Since the sportsman runs around the circular track and ends up at the same position (A), the displacement is the straight-line distance through the circle’s center. Therefore: The distance travelled is the total path length covered by the sportsman, which consists of two complete laps around the circular track. Thus, the total distance is: Thus, the correct answer is: