Circular Motion

Circular motion is a movement of an object along the circumference of a circle or rotation along a circular path.

Angular Displacement: The angle through which an object moves in a circular path is called angular displacement. Its unit is radian.

Angular velocity
The time rate of change of angular displacement is called angular velocity. It is denoted by ω. Its unit is rad s⁻¹.

↳ Average angular velocity
The change in the angular coordinate θ, expressed in radians, divided by the change in time, is called average angular velocity. It is denoted by ωₐᵥ. It is given by

ωₐᵥ = (θ₂ − θ₁) / (t₂ − t₁)
ωₐᵥ = Δθ / Δt [where Δθ = θ₂ − θ₁ and Δt = t₂ − t₁]

Instantaneous angular velocity
The limiting value of the average angular velocity of the body in a small time interval, as the time interval approaches zero. It is denoted by ωᵢₙₛ. It is given by

Angular Acceleration
The time rate of change of angular velocity is called angular acceleration. It is denoted by α. Its unit is rad·s⁻². Average Angular Acceleration

The ratio of change in angular velocity to the time taken by the particle to undergo that change is called average angular acceleration. It is denoted by αav.

Suppose a body has angular velocity ω₁ at time t₁ and ω₂ at time t₂, then

Instantaneous Angular Acceleration
The limiting value of the average angular acceleration of the body in a small time interval as the time interval approaches zero. It is denoted by αins.

Time Period
The time taken by a body to complete one revolution is called time period. It is denoted by T and given by:

 T=1/f; ​ where f is frequency.

Thus In time T, the particle completes one revolution, i.e., it describes an angle of 2π rad. Hence, angular velocity is given by:
 ω=2π/T

Frequency
The number of revolutions per second is called frequency (f).

f=1/T

Thus, in time T, the particle completes one revolution, i.e., it describes an angle θ = 2π rad. Hence, angular velocity is given by
𝜔 = 2π/T

Frequency
The number of revolutions completed per second is called frequency. It is denoted by f and given by

𝑓 = 1/T

Relation Between Time Period, Frequency, and Angular Velocity

Angular velocity (ω) = θ/t
For one complete rotation, θ = 2π and t = T (time period).
So, ω = 2π/T
∴ ω = 2πf [∵ f = 1/T]

Relation between Linear Velocity and Angular Velocity

Consider a body moving in a circular path of radius r and center at O in anti-clockwise direction. Suppose the body moves from its initial position A to B in a time interval t.
Let s be the displacement along the circular path i.e. AB = s, and θ be the angular displacement.

We have,
Linear velocity (v) = Δs/Δt –(i)
Angular velocity (ω) = Δθ/Δt
ω = Δ(s/r)/Δt [∵ θ = s/r]
ω = (1/r) Δs/Δt [∵ radius is constant]
ω= v/r     from eq. 1
ω = rω

Expression for centripetal Acceleration.

Consider a body moving in a circular path of radius r and center O with uniform speed in clockwise direction as shown in Fig.(a). Suppose v̅ₐ and v̅ᵦ be the velocities of the body at points A and B respectively. Let Δv̅ be change in velocity in small time Δt, AB = Δs i.e. Δs be the distance travelled, and Δθ be the angular displacement in small time Δt.
Let v = |v̅ₐ| = |v̅ᵦ| [: Speed is uniform]

From Fig (a),
Δθ = (AB / r) = Δs / r —- (i)

Again,
Δθ = PR / QR = Δv / v
Δθ = Δv / v — (ii)

From (i) and (ii),
Δs / r = Δv / v
Δv = v Δs / r
Dividing both sides by Δt and taking limΔt → 0

lim (Δv / Δt) = lim (v Δs / r Δt)
Δt → 0        Δt → 0

a = v·v / r
a = v² / r
Again,
a = (rω)² / r
∴ a = ω²r

Motion in Horizontal Circle (Conical Pendulum)

Consider a bob of mass m suspended from rigid support with the help of string of length l such that it moves along horizontal circle of radius r.

At instant t time, suppose the bob reaches point A. At this point, weight (mg) acts vertically downward and tension (T) acts in the string. Let T sinθ and T cosθ be the horizontal and vertical components, respectively. Let θ be the angle made by string with vertical. Let the time period be T, the bob and h be the height of the pendulum.

At the equilibrium condition, T cosθ balances the weight of the bob and
T sinθ provides the necessary centripetal force to the bob.

i.e.
T sinθ = mv² / r    —- (i)
T cosθ = mg    —- (ii)

Dividing (i) by (ii):
tanθ = v² / rg    —- (iii)

In ΔAOB,
tanθ = (AO / OB) = r / h    —- (iv)

From equations (iii) and (iv)

Motion in a Vertical Circle

Consider a body of mass m tied at the end of a string of length r that is moving in a vertical circle with constant speed v. Let mg be the weight of the body. At the lowest point A, a part of tension Tₐ balances the weight, and the remaining part provides the necessary centripetal force.

Applications of Centripetal Force:

1. Bending of a Cyclist.
Consider a cyclist of weight mg taking a turn of radius r with velocity v. In order to provide the necessary centripetal force, the cyclist leans through an angle θ inward from the vertical direction as shown in the figure. The weight mg acts vertically downward at the centre of gravity of the cyclist and the cycle. Let R be the normal reaction. At a balanced condition, the vertical component of R, i.e., Rcosθ, balances the weight of the cyclist, where the horizontal component Rsinθ provides the necessary centripetal force to the cyclist.

∴ Rsinθ = mv²/r — (i)

Rcosθ = mg — (ii)

Dividing (i) by (ii),

Tanθ = v²/rg

θ = Tan⁻¹(v²/rg)

Motion of the Car on a  level curved path: 

Consider a car of weight mg going around on a curved path of radius r with velocity v on a level road, as shown in the figure. Suppose F₁ and F₂ are the frictional forces acting on two tyres, respectively. Let R₁ and R₂ be the normal reactions of the ground on the two tyres.

If μ is the coefficient of friction,
At a balanced motion of the car,
R₁ + R₂ = mg — (i)

We know,
F₁ = μR₁ and F₂ = μR₂
∴ F₁ + F₂ = (R₁ + R₂)μ — (ii)
from (i) and (ii)
F₁ + F₂ = mgμ. Also. — (iii)
The total force of friction provides the necessary centripetal force i.e.
F₁ + F₂ = mv²/r — (iv)
From (iii) and (iv),
μmg = mv²/r

v² = μrg
v = √(μrg)

This gives the maximum velocity with which the car can take a turn of radius r and the coefficient. of friction μ between the tire and ground.

Compiled by Er.Basant Kumar Yadav