Kinematics-Uniform Circular Motion

Kinematics- Uniform circular motion, time period

A uniform circular motion is defined as a body travelling in a circular direction at a constant speed. The body’s distance from the axis of rotation remains constant at all times. In a uniform circular motion, the body’s speed also remains constant. However, its velocity changes as a velocity is a vector quantity influenced by both the direction and speed. The existence of acceleration is shown by the fluctuating velocity; the centripetal acceleration is stable in amplitude and constantly pointed towards the rotation axis. A centripetal force, constant in magnitude but applied towards the axis of rotation, causes this acceleration.

Uniform circular motion formula

Considering = angular velocity, v = magnitude of the velocity, and r = radius of the given circle.

Radial or centripetal acceleration for uniform circular motion is: 

ar=v2/r=2r

Let us assume m = mass of the particle. Considering the second law of motion,

F=ma

mv2/r=m2r

The above equation formulates the origin of centripetal force.

Angular displacement

It is the angle that forms between the position vector and the centre. Let us assume, dS = linear displacement and r= circle radius.

d=dS/r(Radians)

Angular velocity

It is the rate at which the angular displacement changes.

=d/dt(Radian/sec)

In the case of uniform circular motion, the formula for angular velocity 

= v / r

Angular acceleration

It is the rate at which the angular velocity changes. Let us assume = angular acceleration, d= difference in angular velocity, and dt= time difference. 

=d/dt(Radian/sec2)

It is always zero regarding uniform circular motion because of the angular velocity, which is always constant.

Magnitude of acceleration

It can be represented as:

a=v2 / r

Uniform circular motion examples

There are several examples that help to illustrate uniform circular motion. These include:

• A watch’s second hand is always in motion on a circular dial.
• A nucleus is surrounded by its electrons. The electrons keep revolving and are in motion.
• A windmill has blades that are always in motion with the strokes of air.
• The artificial satellites are always in motion around the earth.
• Giant wheel
• A stone that is tied to a string
• Merry-go-round
• Stirring a batter in a circular direction
• All the planets that are revolving around the Sun

Uniform circular motion-time period

The period or the time taken to complete one oscillation of simple harmonic motion (SHM) can be calculated because it is a periodic motion. 

In SHM

F = -kx

In this equation, the negative sign (-) shows that the force is directed towards the opposite direction. The term ‘k’ is a constant called the force constant. The unit of ‘k’ is Newton per metre.

Let the mass attached be m. In that case, the acceleration (a) will be:

a=Fm

a=-kxm

a =–𝜔2x

By comparing coefficients, km=  𝜔2

The time period is the time required by the object to complete one oscillation. The frequency of simple harmonic motion is the total number of oscillations taken by the object per unit of time. Therefore, we can represent the frequency as:

f = 1/T

In the above equations:

a = acceleration 

T= time period

F = force

f = frequency 

m = mass

𝜔 = angular frequency

k = force constant 

Derivation of Time Period of simple pendulum.

Utilising the condition of motion, T – mg cosθ = mv2/L, where L is the length of the pendulum

The force has a tendency to carry the mass to its harmonious position,

τ = mgL × sinθ = mgsinθ × L = I × α

For little points of motions sin θ ≈ θ,

Thus, Iα = – mgLθ

α = -(mgLθ)/I

– ω02 θ = – (mgLθ)/I

ω02 = (mgL)/I

ω0 = √(mgL/I)

Utilising I = ML2, 

we get, ω0 = √(g/L)

Hence, the time span of a simple pendulum is given by,

T = 2π/ω0 = 2π × √(L/g)

Conclusion

The term “uniform circular motion” refers to the movement of an object in a circle at a constant speed. In a circle, an object’s path is continually shifting. The object’s direction is always tangent to the circle it is in. The angular velocity, denoted by the symbol ω(omega), refers to rotation per unit of time. The amount of displacement per unit of time is measured by velocity v. It is a vector with a specific direction. The linear velocity of an object in circular motion keeps changing because the object keeps on changing direction.