## Sunday, January 10, 2016

### Minimum velocity at the bottom and at Horizontal of Vertical Circular Motion

Body moving against the gravitational force in circular motion is called vertical circular motion. To continue the body in vertical circular motion, we shall give some minimum velocity at each point of motion. It is like moving a bucket of water in air with your hand and it demands some minimum velocity at each point. What is the required velocity at any point depends on the weight and the tension at any point of vertical circular motion. The effective force acting towards the center is called centripetal force and it is the effective value of the weight and tension.

We have all ready proved in the previous post that the minimum velocity at the top of the vertical circle as the square root of radius of circular motion and acceleration due to gravity. Taking this value into consideration, we can measure the minimum velocity at the bottom of vertical circular motion.

We have to consider the bottom of vertical circular motion as the reference point and from that point, we do measure the potential energy. If the body is at the reference point, potential energy is treated as zero.

To measure the minimum velocity at the bottom of vertical of vertical circle, we cannot take tension as zero.. It is because, at that point gravity acts in the downward direction and if the body has to continue its vertical circular motion, it shall have some force acting towards the center. If there is no tension, then there is no force acting towards the center and hence it cannot have circular motion.

Thus there shall be some tension at the bottom of the vertical circular motion and to measure the minimum velocity, we shall use conservation of energy concept. As per this concept, energy is neither created nor destroyed. It just converts from one form to other and the total energy of the system remains constant.

Mechanical energy is in two formats called potential energy and kinetic energy and for any system sum of these two always remains constant. If one energy increases, other energy decreases but the total energy of the system shall remains constant.

We can apply law of conservation of energy at the bottom and top of the vertical circular motion. At the bottom, body has only kinetic energy and no potential energy as the bottom is the reference point.

When the body has gone to the top of the vertical circular motion, it is at some height equal to the diameter of circular motion and it has some kinetic energy also at that point.

By equating energies at the two points bottom and top of vertical circular motion and by substituting the value of minimum velocity at the top of vertical circular motion, we can get the velocity at the bottom  of vertical circular motion as proved in the following video lesson.

Minimum velocity at Horizontal position of vertical circular motion

In the similar way, we can also find the velocity of the body in the horizontal position.We can also use the concept of law of conservation of energy.By equating the conservation of energy at the bottom and at the horizontal position, we can get the velocity at the horizontal position. In solving this, we need to use the value of the velocity that we have derived from the above.

At the horizontal position, there will be potential energy and kinetic energy. At the bottom there is only kinetic energy. We can get the value of velocity at the bottom of the vertical circular motion as shown in the video lesson below.

Related Posts