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# Vertical Circular Motion: Definition, Equation for VCM, Practice Problems and FAQs

Imagine a baby girl holding a string in her hand. One end of a string is tied to a stone. She stretches her arm upfront. Keeping her hand fixed at the center, she revolves the string around in a vertical plane. If you look closely, you will notice that the stone assumes different velocities at different points of the motion; the same applies to a roller coaster ride. Such a motion can also be described as a “non-uniform” motion.

### Definition of Vertical Circular Motion

Vertical circular motion (VCM) is a special type of circular motion in which the body travels in a vertical plane.

In the case we are discussing here, velocity of the body is different at different points. It is an example of non-uniform circular motion.

## Equations for VCM

Consider a ball of mass connected with a light inextensible string of length , which is performing a vertical circular motion. The tension ( in the string acts radially inwards.The ball starts from point and completes half a circle at point . The weight is resolved into two components when it makes an angle ; if indicates its velocity, then

## Tension and velocity in VCM

Case (i):

i.e. Tension is maximum at point A .

represents velocity at point

Case (ii) : When

i.e. Tension is minimum at point C.

represents velocity at point .

Substituting ;

Applying energy conservation between points A and C ,

Kinetic energy at point = Kinetic energy at point + Potential energy at point

Substituting equation in equation ,

The above equation gives an expression for the maximum tension at point In this condition, the string is said to be taut.

Velocity at point for just completing the circle

Using energy conservation between the points and ,

Let represent velocity at point .

;

Substituting the value of from equation

= ;

Tension at point B for critical VCM

; at point B

Substituting the value of in the above equation; we get

Since tension is ⟂ to the direction of velocity at every instant, the work done by it is zero.

Conditions for looping

1)The ball will oscillate about the lower point if

2)The ball will leave the circle and behave like a projectile, if .

## Practice problems

1) A mass is attached to a thin wire and whirled in a vertical circle. The wire is most likely to break under which of the following conditions?

(a) When the mass is at the lowest point (b) When it is inclined at an angle of to the wire

(c) When the mass is at the highest point (d) The wire is horizontal

Solution) a

The tension at the lowermost point in a VCM is maximum. , ∴ it will break when the mass is at the lowermost point.

2) A ball which is initially at rest, starts sliding along a frictionless track from a height (as shown in the figure) and just completes a vertical circle of diameter, . Find the height

1. (b ) (c) (d)

Solution) d

For a body to complete a VCM , the minimum velocity at point must be

Now by energy conservation

solving, we get

3) A block of mass at the end of the string is whirled round in a vertical circle of radius . The critical speed of the block at the top of its swing below which the string would slacken before the block reaches the bottom is ?

Solution)d

At the top most point, the velocity of the particle is .

4) One end of a string of length is tied to a mass of It is whirled in a vertical circle with initial angular frequency Find the tension in the string when the ball is at the lowermost point of its motion. (Take

Solution) d

Given,

Mass,

( indicates the centripetal force)

## FAQs

Q. Give two examples for motion in a vertical circle?

Ans) Roller Coaster rides and Ferris wheel are two examples of VCM.

Q. What is the expression for tension of the string when it is horizontal in case of critical VCM ?

Ans)

Q. What is the expression for velocity at the top most point in a VCM of radius in case of critical VCM?

Ans)

Q. What is the expression for the velocity at the bottom of a VCM in case of critical VCM?

Ans)

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