## Unit – V A |

## Motion in Vertical Circle:

- Consider a small body of mass ‘m’ attached to one end of a string and whirled in a vertical circle of radius ‘r’.

- In this case, the acceleration of the body increases as it goes down and decreases when goes up. Hence the motion of the body is not uniform circular motion.Irrespective of the position of the particle on the circle, the weight ‘mg’ always acts vertically downward.

### Velocity of Body:

- Let us consider a body performing a circular motion in a vertical circle. Let ‘v’ be the velocity of the body at any point P on the vertical circle. Let L be the lowest point of the vertical circle. Let ‘h’ be the height of point P above point L. let ‘u’ be the velocity of the body at L. By the law of conservation of energy

Energy at point P = Energy at point L

This is an expression for the velocity of a particle at any point performing a circular motion in a vertical circle.

### Tension in the String(Motion in Vertical Circle):

This is the expression for tension in the string.

### Special Cases(Motion in Vertical Circle):

**Case – I**(When the body is at the lowermost position i.e. body is at L)

**Case – II**(When the body is at the uppermost position i.e. body is at H)

**Case – IIi**(When the string is horizontal i.e. body is at M)

### Relation Between Tension at the Highest point and the Tension at the Lowest Point for a Body Moving in a Vertical Circle:

- The tension in the string at the highest point for a body moving in a vertical circle is

- The tension in the string at the lowest point for a body moving in a vertical circle is

- Thus the difference in tensions in the two positions

- Thus the tension in the string at the lowest point L is greater than the tension at the highest point H by six times the weight of the body.

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