Circular Motion Physics Notes

Circular Motion Physics Notes

Circular Motion: When a body moves such that it always remains at a fixed distance from a fixed point, then its motion is said to be circular motion. The fixed distance is called the radius of the circular path and the fixed point is called the center of the circular path.

Uniform Circular Motion: Circular motion performed with a constant, speed is known as uniform circular motion.

Angular Displacement: Angle swept by the radius vector of a particle moving on a circular path is known as angular displacement of the particle.

Let the angular position of the particle at times t₁ and t₂ are θ₁ and θ₂ respectively. Then angular displacement.
Δθ = θ₂ – θ₁

Note:

• Angular displacement is a vector quantity.
• Its direction is perpendicular to the plane of rotation and given by the right-hand screw rule.
• Clockwise angular displacement is taken as negative and anticlockwise displacement as positive.
• Its unit is radian (in M.K.S.).
• Always change degree into radian. If it occurs in numerical problems, 1 radian = \(\frac{180^{\circ}}{\pi}\)
• It is a dimensionless quantity i.e., [M°L°T°].

Relation between angular displacement and linear displacement:
Since, angle = \(\frac{\text { Arc }}{\text { Radius }}\)
∴ Angular displacement = \(\frac{\text { Arc } P P^{\prime}}{\text { Radius }}\)
Δθ = \(\frac{\Delta S}{r}\)
∴ For circular motion
Δs = rΔθ

Angular Velocity: The rate of change of angular displacement of a body with respect to time is known as angular velocity. It is represented by ω (omega).

• It is a vector quantity.
• Its direction is the same as that of angular displacement i.e., perpendicular to the plane of rotation.
• If the particle is revolving in the clockwise direction then the direction of angular velocity is perpendicular to the plane downwards. Whereas in the case of the anticlockwise direction the direction will be upwards.
• Its unit is radian/sec.
• Its dimension is [M°L°T^-1].

Average Angular Velocity: It is defined as the ratio of total angular displacement to total time.
\(\vec{\omega}_{a v}=\frac{\text { Totalangulardisplacement }}{\text { Total timetaken }}\)
or \(\vec{\omega}_{a v}=\frac{\Delta \theta}{\Delta t}\)

Instantaneous Angular Velocity: Angular velocity of a body at some particular instant of time is known as instantaneous angular velocity.
or
Average angular velocity evaluated for a very short duration of time is known as instantaneous angular velocity.

Note: Instantaneous angular velocity can also be called simply angular as velocity.

Relation between linear velocity and angular velocity
We know that angular velocity

The period of uniform circular motion:
Total time taken by the particle performing uniform circular motion to complete one full circular path is known as time period.

In one time period total angle rotated by the particle is 2π and time period is T.
Hence, angular velocity
ω = \(\frac{2 \pi}{T}\)
or
T = \(\frac{2 \pi}{\omega}\)

Frequency: Number of revolutions made by the particle moving on a circular path in one second is
known as frequency.
f = \(\frac{1}{T}=\frac{\omega}{2 \pi}\)
⇒ ω = 2πf
Angular Acceleration: The rate of change of angular velocity is defined as angular acceleration
If Δω be the change in angular velocity in time Δt, then angular acceleration

• It is a vector quantity.
• Its direction is the same as that of change in angular velocity.
• Its unit is rad/sec.
• Dimension: [M°L°T^-2].

Relation between angular acceleration and linear acceleration
We know that
Linear acceleration = Rate of change of linear velocity
⇒ a = \(\frac{d v}{d t}\) …..(i)
Angular acceleration = Rate of change of angular velocity
⇒ a = \(\frac{d \omega}{d t}\) …….(ii)
From equation (i) and (ii)

Equation of Linear Motion and Rotational Motion

Physics Notes