Circular Motion and The Equations of Motion

Newton’s first law of motion states that if the body is in a state of rest or motion, it will continue to be in a state of rest or motion unless an external force is applied to it. This means that if a body is moving in a particular direction, it will continue to move in that direction unless a force acts upon it and changes its direction or slows it. When a body is in a circular motion, it is constantly changing its direction of motion. So it stands to reason that some force is acting upon it to keep it following the circular path. This force is the centripetal force, and it is essential for a body to maintain circular motion. There are various equations of motion associated with circular motion, which describe the ways in which the different factors in the motion are related to each other.

Angular frequency

If an object is moving along a circular path at a particular speed and it is completing one circuit of the path in a given unit of time, it is said to be in a circular motion. An example of this motion is the hour hand of a wall clock. The hand itself can be taken to be a vector. The tip of the hand undergoes a uniform circular motion. The hand or the vector itself makes an angle of 2π radians, and the time taken to complete this is one hour. This rate at which the vector completes the 2π radians is known as the angular frequency. It is usually denoted by ω = 2πf. So it can be inferred from the equation that T = 2π/ω.

Angular velocity

The angular velocity of circular motion is given by dividing the angular displacement by the time it takes to cover the angular displacement. It is conventionally denoted by the Greek letter ω, and its unit of measurement is rad/s. An object that is in circular motion has both angular velocity and linear velocity. In mathematical terms, the relation between angular velocity and linear velocity is given by the equation:

v = ωr

Here, v is the linear velocity, ω is the angular velocity, and r is the position vector with respect to the centre of the circular path.

Angular acceleration

By definition, acceleration is the change in velocity per unit of time. So angular acceleration is the change in angular velocity per unit of time. The unit of measurement of angular momentum is rad/s².

The angular acceleration of a body moving in circular motion is always directed toward the axis. When Newton’s second law is applied to it, the following equation for the resultant force is derived as  F = mv²/r. This is another equation of motion associated with circular motion.

Here, F is the centripetal force, m is the mass of the object, v is the velocity, and r is the position of the particle in relation to the centre of the path.

Equation of motion in circular motion:

The equation of motion for circular motion are given by following equations

θ=ω₀+αt

θ=ω₀t+1/2 αt²

ω²=ω_0²+2αθ

Where, ω₀is initial angular velocity,

ω is final angular velocity,

θ is the angular displacement,

α is the angular acceleration, and

t is the time.

Conclusion

The equations of motion associated with circular motion describe the way an object traversing a curved or circular path will move. These equations of motion, when derived from the factors involved with circular motion, give the different relations among the vectors and magnitudes of the velocity and acceleration and the forces that keep the body in a circular motion. These equations of motion of circular motion also help answer various questions about circular motion and help predict the trajectories of objects.