## Motion in Vertical Circle:

### Velocity of a body at Different Positions: • The velocity of a body moving in a vertical circle is given by #### Minimum Velocity of the Body at the Lowest Point L for Looping a Loop:

• This is the minimum velocity of the body required so that the body looping a loop i.e. to go round the circle once completely. Thus the minimum velocity of the body required at the lowest point L for looping a loop is 5gr  .

#### Minimum Velocity of the Body at the Highest Point H When Looping a Loop: Thus the minimum velocity of the body at the highest point H when looping a loop is gr  .

#### Velocity of the Body When String is Horizontal and When Looping a Loop: This is the velocity of the body at M when the string is horizontal and when the body is looping a loop is 3gr .

### Energy of a body at Different Positions:

#### Energy of a Body at the Lowest Point L When Looping a Loop:

• Energy of the body has two components a) kinetic and b) potential ### Energy of a Body at the Highest Point H When Looping a Loop:

• Energy of the body has two components a) kinetic and b) potential #### Energy of a Body When the String is Horizontal and When Looping a Loop:

• Energy of the body has two components a) kinetic and b) potential • we can see that the total energy of the body is conserved. Kinetic energy is maximum at the lowest point of the circular path and minimum at the highest point of the circular path. Thus as the body moves up the circle kinetic energy decreases and potential energy increases and when descends in the circle potential energy decreases and kinetic energy increases.

### Notes:

1. The tension in the string of a conical pendulum is constant, while that in simple pendulum varies with time. For a simple pendulum, the tension in the string depends on the position of the bob of the pendulum from mean position. It is maximum at mean position and minimum at the extreme position.
2. Motion of a body in vertical circle is not uniform circular motion because the speed of the body changes continuously. It is maximum at the bottom-most position and minimum at the uppermost position of the vertical circle.
3. The tension in the string at the lowest point L is greater than the tension at the highest point by six times the weight of the body.
4. If the velocity ‘v’ to the body is such that is v <= 2gr  , then the body oscillates about point L, the lowest point of the vertical circle.
5. If the velocity ‘v’ to the body is such that 2gr  < v < 5gr , then the body leaves the circular path and acts like a projectile. It will leave circular motion between M and H.