On Motion In A Vertical Circle

An object that moves or rotates in a circular path is said to have circular motion. Circular motion is classified into uniform and non-uniform circular motion based on its rate of rotation. A body with different motions at different points along its path of motion is called a non-uniform circular motion. Centripetal force plays a major role as it is the force due to which the object follows a circular path.

The effects of gravity as a force are also significant when considering all circular motions on Earth. Roller coasters or whirling a bucket of water are some examples of circular motion. In this article, we will describe the basics of vertical circular motion, followed by examples, typology and vertical circular motion derivation.

Vertical Circular Motion

If we measure the direction of motion of an object at any random point, it is observed that the direction changes at every point, further adding up to a motion that looks circular. Vertical motion is defined as the motion of an object in which the direction of movement or motion of the object is different at each point.

Types of Vertical Circular Motion

1) Imagine a bucket containing water. Whirling the bucket at a certain speed does not spill the water; however, as the speed is reduced, it is observed that the water spills. The speed at which the water is not spilt can easily be calculated using Newton’s second law, which states that the motion of the bucket depends upon two main factors. It is the net force that is acting on the bucket and the mass of the bucket. Two forces act on the bucket – tension and gravitational force.

2) Roller coasters are also a great example because we can experience the vertical circular motion. The reason why we don’t fall in the top position of the roller coaster is due to the vertical circular motion. The roller coaster undergoes a vertical circular loop that follows a standard speed, which is enough to keep us intact in our seats. There are seat belts because the roller coaster has speed limits. Gravity is the force that acts downwards while the normal force acts upward.

Variables in Vertical Circular Motion

Certain variables are related to vertical circular motion, and they help to understand the vertical circular motion derivation better. 

Angular Displacement

The angle is subtended by the position vector at the centre of the circular path.

Angular displacement = s/r

s→ Linear displacement

r→ Radius

Unit = Radian

Centripetal Acceleration

Acceleration acts on a body in circular motion whose direction is towards the centre of the circle.

Centripetal acceleration, a =  v2/r

Angular Velocity

Rate of change of angular displacement.

Angular velocity = d/dt (Angular displacement )

Unit: Rad/s

Angular Acceleration

Rate of change of angular velocity.

Linear acceleration = d/dt(d/dt (Angular velocity )

Unit: Rad/s2

Vertical Circular Motion Derivation

Vertical circular motion derivation is used to find out the tension at the high and the low points as well as the velocity of the object in motion. It is an expression for the difference in tensions at the highest and lowest points for a particle performing the vertical circular motion.   

Suppose a body of mass ‘m’ performs vertical circular motion on a circle of radius r.

Let, 

TL = Tension at the lowest point

TH = Tension at the highest point

VL = Velocity at the lowest point

VH = Velocity at the highest point

At the lowest point L, 

TL = mv L2/r + mg —(1)

At the highest point H,

TH = mvH2/r – mg —-(2)

Subtracting (1) by (2)

TL – TH = mvL2/r + mg – ( mv2H/r – mg)

             = m/r ( vL2 – vH2) + 2mg

Therefore, TL – TH = m/r ( vL2 – vH2) + 2mg -–(3)

By law of conservation of energy,

(P.E + K.E) at L = (P.E + K.E) at H 

0 + 1/2mvL2 = mg.2r + ½ mvH2

½ m(vL2 – vH2) = mg.2r

vL2 – vH2 = 4gr —(4)

From equation(3) and (4),

TL – TH = m/r(4gr) + 2mg = 4mg + 2mg

TL – TH = 6mg  

Conclusion

The applications and derivations in the above section help us to understand the concept of vertical circular motion. Circular motion is classified into vertical and horizontal circular motions based on the variation of the direction of speed. As the direction changes speed, the object under vertical circular motion also changes its path, which results in the formation of a circular path.