Introduction to Circular Motion
A circular motion involves an object whose path is curved or circular and is fixed about a point called the centre. A circular motion is always directed towards its centre. However, the object keeps changing its direction. Hence, the change of direction makes it a vector quantity. The speed may be constant or keep changing during a circular motion. But, it is to be remembered that the direction is always varying. In simple terms, we can say that the motion is like moving around the circumference of a circle.
In daily life we see many familiar situations like a rotating fan, a motor, planets moving around the sun etc. It is a natural phenomena and is very essential to explain many unsolved problems in physics. For quantum mechanics and other fields also, it is studied to find the spin angular momenta of minute particles. Circular motion is hence a major concept in physics and engineering applications.
Types of Circular Motion
Circular motion is also of two types: uniform and non-uniform circular motion.
Uniform Circular Motion
An object will have a uniform circular motion, if it travels along a circular path with the same speed or let’s say constant speed. We know, velocity means the rate of change of displacement. So a question may arise, if there is no change in speed then how a change in velocity arises? The answer is that, as continuous change in direction occurs, it changes the velocity. For example, planets revolving around the sun.
Non-Uniform Circular Motion
An object will have non-uniform circular motion, if it travels with varying speed along a circular path. Here, at different instant the velocity remains different. A man running around a circular path can have different velocity and hence different acceleration at different instant of time. However, the path remains fixed.
Terms Used in Circular Motion
There are various terms which are to be understood before studying circular motion. Below are given some important terms:
Angular Displacement (θ):
It is the angle made by the object moving along a circular path. When an object moves from one point to another point in a circle, it makes an arc. Therefore, angular displacement is measured as (θ) = l/r [Equation 1]
In equation 1, l is the distance moved by the object from one point to another. Similarly, r is the radius of the circle. For a body making one full revolution, the total displacement will be 2π.
Angular Velocity (ω):
It is measured to find how fast an object is rotating or turning along a circular path. Mathematically, it is defined as the rate of change of angular displacement with respect to time. It is denoted by omega (ω) and is measured in radian/second.
ω = change in displacement/change in time = θ2-θ1 /t2-t1 [Equation 2] .
Instantaneous Angular Velocity :
It appears whenever the circular motion is of non-uniform type. It is defined as that small velocity picked at a certain interval, when the velocity is continuously varying. It is denoted by ‘dω’ which is calculated as:
dω = Δx→limit 0 dx/dt [Equation 3]
Angular Acceleration (α):
It is the rate of change of angular velocity with time. It also occurs when the velocity is changing and is denoted by alpha (α). Mathematically,
α = v2 – v1/t2-t1 [Equation 4]
Instantaneous Angular Acceleration (α)
As like instantaneous angular velocity, it is the small angular acceleration picked at any instant of time. It is denoted as ‘dα’. Mathematically,
dα = Δv →limit 0 dv/dt [Equation 5]
Centripetal Force
To get it simple, a force arises to keep an object moving in a circular path. This force is constantly acting towards the center of the path and is known as centripetal force. The meaning of ‘centripetal’ is center-directed. It is always pointed inward and hence keeps the body holding in motion.
For example, when a car is turning in a curved path, the friction between the tire and the road provides the force to the car to make it able to turn. This force is the centripetal force. To make the car keep moving along the path, without toppling, the centripetal force must be less than or equal to the maximum static friction.
The formula for centripetal force is:
Fc=mv^2/r [Equation 6]
where m is the mass of a moving object, v is its speed and r is radius of that circular path.
Centrifugal Force:
This is only an apparent force which doesn’t exist physically. However, the rotating frame of reference can give an illusion to an observer. This virtual or pseudo force is known as the centrifugal force. For example, in a merry-go round, two friends are rotating, while an observer is watching from the ground. One friend taking the round sees another being pushed outward, while the observer from the ground finds them rotating in an inward motion. Here, the friend inside, experiences a centrifugal force while the observer in the ground observes centripetal force. Centripetal and centrifugal forces are opposite in nature.
Centrifugal force = -mv^2/r
For example, when a car suddenly takes a turn, our body tries to continue in the same straight but the car starts to rotate. In this case, we feel an outward push due to our inertia of the motion. But actually we are never pushed outward but travelling in the inward direction directed towards the centre. This gives us a faulty experience of a centrifugal force.
Relation Between Linear and Angular Velocities/Acceleration
Circular motion can also be related to linear motion by considering very small displacement at a small instant of time such that a circular path can also appear as a small dot of a line. Therefore a larger circular path and hence larger circular motion can also be studied by studying a very small time frame. The relation between linear velocity (v) and angular velocity (ω) is given as:
v = rω [Equation 7]
Where, r is the radius of the circular path.
Similarly, a linear acceleration has two components: tangential and centripetal acceleration, which are related to angular acceleration (α) as:
- Tangential acceleration (at) = rα [Equation 8] (This appears due to non-uniform velocity of the object)
- Centripetal acceleration (ac) = v^2/r OR rω^2 [Equation 9] (This appears due to the change in direction of the object with uniform speed.)
Hence, a linear motion is also important in studying a circular motion.
Circular Motion in Vertical and Horizontal Planes
Circular motion acts differently when an object is moving along horizontal and vertical circles which is briefly described below:
Motion in Horizontal Plane
When an object is continuously moving in its circular path along a horizontal plane, gravity remains silent and hence motion is not affected by the gravity. However, it is still acting downward but is balanced by other forces like tension or friction that gives a support for rotation. If the centripetal force is lesser than the object may be released tangentially outward. Here, the centripetal force acts towards the centre.
For example, a stone tied with a rope and rotated overhead by hand.
Motion in Vertical Plane
In case of vertical circular motion, gravity comes into action and hence changes according to the direction of motion. At the top, bottom and sides of the circle, the gravity acts towards the centre, away from the centre and perpendicular to the force (centripetal) respectively. Hence, the tension also keeps changing. For example, roller-coster round, bucket of water swung vertically etc.
Torque in Circular Motion
Torque is a rotational force. It makes an object rotate about an axis. When force is applied far from the axis of rotation, it creates a turning force. This turning force is called the torque. Torque determines how quickly an object starts rotating or changes its rotational speed. Thus, it is very important to calculate the torque in a circular motion. Torque is denoted by ‘ τ’ and is calculated as:
τ=r×F [Equation 8]
where r is the distance from the axis which is also called the torque arm and F is the force applied.
Work, Energy and Angular Momentum in Circular Motion
- Rotational Kinetic Energy:Just like in linear motion, kinetic energy in circular motion is given by rotational kinetic energy and is calculated as:
KE=½ Iω^2 [Equation 9]
In equation 9, I is the moment of inertia and ω is the angular velocity. The mass distribution of a rotating object along its axis of rotation is called the moment inertia and is calculated as I = ∑mr^2. Thus the rotational kinetic energy also depends upon the distribution of the mass around its axis of rotation.
- Work Done by Torque:
The work done in rotational motion is the work done by the torque and is calculated as:
W = τθ [Equation 10]
Here, θ is the angular displacement.
These equations help to calculate the transfer of energy in various simple to complex systems like wheels or wind turbines.
- Angular momentum (L)
As like in linear motion we have the conservation of linear momentum, there is angular momentum in circular motion. It is given as:
L=Iω [Equation 11]
Also, conservation of angular momentum applies to every system and states that, the total angular momentum remains conserved if no external torque acts on it. From equation 11 it can be seen that reducing angular velocity can be increased by decreasing moment of inertia.
Examples of Circular Motion in Daily Life
Some of the examples of circular motion observed in real practices are given below:
- Ceiling Fan: The blades of ceiling fans move in a circular path fixed at the center. This is a good example to demonstrate circular motion.
- Merry-go-round: The motion of the merry-go-round in parks and circuses is also centrally fixed and is an example of circular motion. It is a circular motion in a horizontal plane.
- Satellites Revolving Around Planets: Satellites follow a circular path around the planets. Hence, they follow the principle of circular motion.
- Stone Tied to a String: When a stone is tied to a string andd swung in a circular path the centripetal force comes in action and holds a circular motion that id directed to the center.
- Stirring a Batter:When we stir a batter like that of cake or any other liquid or semi-liquid, the stirrer follows a circular path and hence showcases a circular motion.
- Motion of Athletes in Circular Path: Athletes running on a circular track are in circular motion. They maintain a constant distance from the center of the track.
- Movement of Electrons Around the Nucleus: Atomic models claim that electrons are moving in circular orbits around the nucleus. This also illustrates circular motion at a microscopic level.
Friction plays a big role to hold a real-world circular motion. For example, we make a turn around curved roads, friction our foot and road provide us the necessary centripetal force. Hence, no or less friction then the centripetal force can cause injury.
Applications in Engineering and Technology
The principle of circular motion are also used to design many engineering systems and engines. It ensures safety, efficiency and reliability for the systems. Some of the common applications are listed below:
- Designing gear systems and engines in vehicles
- Stabilizing the path of a spacecraft
- Mechanical clocks and watches
- Creating centrifugal pumps and separators
- Satellite and planetary mission design
Conclusion
Every motion is not simple and concentrated in a linear path. Even earth is rotating around the sun and following a circular path. Therefore, broadly observing, every object on earth is following a circular path. Thus, circular motion forms a basis of nature. The idea of the association of linear motion with circular motion has helped in solving various complex problems of physics and engineering. The physics of celestial objects is studied on the basis of this technique. Thus, from a simple athlete or a gymnast to a space motion, everything is concentrated on a fixed axis, held by the centripetal force. Circular motion is also important to design various tools and engines for crafting orbital motions in space. Hence, studying circular motion can give a good concept to understand daily activities related to complex natural phenomena.
References
Erlichson, H. (1991). Motive force and centripetal force in Newton’s mechanics. American Journal of Physics, 59(9), 842-849.
Signell, P. (2002). Acceleration and force in circular motion. Project Physnet. Documento digital em formato PDF. East Lasing/MI: Michigan University.
https://en.wikipedia.org/wiki/Circular_motion
https://byjus.com/jee/circular-motion
https://www.vedantu.com/jee-main/physics-circular-motion
