Rotational and Circular Motion notes

MDCAT Physics

This chapter explains angular displacement, angular velocity, angular acceleration, and the connection between angular and linear quantities. It also covers circular motion, centripetal force, torque, rotational inertia, conservation of angular momentum, and circular orbits.

Angular Displacement

Angular displacement is the angle through which a body rotates about a fixed axis. It is represented by θ and is measured in radians. Angular displacement is a vector quantity when the axis of rotation is fixed. Its direction is found using the right-hand rule.

One complete revolution is equal to 360° or 2π radians. In calculations, angles should usually be expressed in radians because the relations between angular and linear quantities use radians.

Angular displacement: θ = arc length/radius, so θ = s/r.
One complete revolution = 2π radian = 360°.
Half revolution = π radian = 180°.
Quarter revolution = π/2 radian = 90°.
For anticlockwise rotation, angular displacement is usually taken as positive.
For clockwise rotation, angular displacement is usually taken as negative.
For a small angular displacement, the angular displacement vector is directed along the axis of rotation according to the right-hand rule.

Angular Velocity and Angular Acceleration

Angular velocity is the rate of change of angular displacement with time. Average angular velocity is ω = Δθ/Δt, while instantaneous angular velocity is ω = dθ/dt. Its SI unit is radian per second, written as rad/s.

Angular acceleration is the rate of change of angular velocity. It is represented by α and has SI unit rad/s2. If angular velocity is constant, angular acceleration is zero.

Angular velocity: ω = Δθ/Δt.
Angular acceleration: α = Δω/Δt.
For uniform angular velocity, α = 0.
Anticlockwise angular velocity is positive and clockwise angular velocity is negative, when anticlockwise is selected as the positive direction.
Frequency f is the number of revolutions per second, measured in hertz.
Time period T is the time for one complete revolution.
Relations: ω = 2πf and T = 1/f.
The period in terms of angular velocity is T = 2π/ω.

Rotational Equations of Motion

For constant angular acceleration, rotational motion has equations similar to the equations of linear motion. Angular displacement replaces displacement, angular velocity replaces velocity, and angular acceleration replaces linear acceleration.

These equations can be used when the angular acceleration remains constant throughout the motion.

ω = ω0 + αt.
θ = ω0t + 1/2 αt2.
ω2 = ω0² + 2αθ.
θ = (ω0 + ω)t/2.
Angular velocity in revolutions per minute must be converted using 1 revolution = 2π radians.
For 540 rpm, angular speed = 540 × 2π/60 = 18π rad/s.
If a flywheel reaches 18π rad/s from rest in 6 s, α = 18π/6 = 3π rad/s2.

Angular and Linear Quantities

A point on a rotating body has a linear or tangential velocity along the tangent to its circular path. The angular velocity of every point in a rigid rotating body is the same, but the linear speed depends on the distance from the axis.

The relation between tangential speed and angular speed is v = rω. Therefore, points farther from the axis move faster even though all points complete each revolution in the same time.

Arc length: s = rθ, when θ is measured in radians.
Tangential speed: v = rω.
Tangential acceleration: at = rα.
For the same angular velocity, v is directly proportional to r.
A particle at r = 2 m with ω = 8 rad/s has v = 16 m/s.
All points of a rigid rotating body have the same angular velocity.
Points at different distances from the axis have different tangential speeds.
If angular acceleration is zero, tangential acceleration is zero.

Circular Motion and Acceleration

A body moving with constant speed in a circle is still accelerating because the direction of its velocity continuously changes. This acceleration is directed towards the centre of the circular path and is called centripetal acceleration.

If speed changes as well as direction, the body has both tangential acceleration and centripetal acceleration. Tangential acceleration changes the magnitude of velocity, while centripetal acceleration changes its direction.

Centripetal acceleration: ac = v²/r.
Using v = rω, centripetal acceleration is also ac = rω².
Centripetal acceleration is directed radially inward, towards the centre.
Tangential acceleration: at = dv/dt = rα.
For constant speed, at = 0 and ac = v²/r.
For constant speed circular motion, the total acceleration has magnitude v²/r and points towards the centre.
The velocity of a particle in circular motion is tangent to the circle and perpendicular to the radius at that instant.
Centripetal acceleration is not zero during uniform circular motion.

Centripetal and Centrifugal Force

Centripetal force is the inward force required to keep a body moving in a circular path. It is not a new type of force. It may be supplied by tension, friction, gravity, electric force, or another appropriate force.

The vector form is Fc = (mv²/r) r̂ when r̂ is defined as the inward radial unit vector. If r̂ is defined outward, a negative sign is required. The magnitude is always mv²/r.

Centripetal force: Fc = mv²/r = mrω².
The direction of centripetal force is towards the centre.
If speed is doubled while radius remains constant, centripetal force becomes four times.
If radius is increased four times and speed is doubled, Fc remains unchanged because (2v)²/(4r) = v²/r.
For equal masses and equal speeds, forces on circular paths of radii r1 and r2 satisfy F1/F2 = r2/r1.
A satellite remains in circular orbit because Earth’s gravitational force provides the centripetal force.
Centrifugal force has magnitude mv²/r and appears outward in a rotating reference frame.
Centrifugal force is a fictitious or pseudo force in an inertial frame, whereas centripetal force is the real inward resultant force.

Torque and Rotational Inertia

Torque is the turning effect of a force about an axis. Its magnitude is τ = rF sin θ, where r is the perpendicular distance from the axis to the line of action of the force. Torque is maximum when the force is perpendicular to the radius.

Rotational inertia, or moment of inertia, measures resistance to angular acceleration. It depends on the mass of the body and how far its mass is distributed from the axis.

Torque: τ = rF sin θ.
Rotational form of Newton’s second law: τ = Iα.
SI unit of torque and moment of inertia is N m and kg m2, respectively.
For uniform angular velocity, α = 0, so the net torque is zero.
Zero net torque does not mean that every individual torque is zero.
For particles, I = Σmr².
For a mass m placed at distance r, its moment of inertia is mr².
If a disk has I = 0.70 kg m2 and a 2.0 kg mass is added at 0.40 m, added inertia = 2.0(0.40)² = 0.32 kg m2, giving total I = 1.02 kg m2.
For particles on an angular ring rotating with the same ω, centripetal force is proportional to radius. Thus F1/F2 = R1/R2 for particles of equal mass at R1 and R2.

Angular Momentum and Conservation

Angular momentum is the rotational counterpart of linear momentum. For a particle moving in a circle, its magnitude is L = mvr. For a rigid body rotating about a fixed axis, L = Iω.

Angular momentum changes when an external torque acts. The relation is τ = ΔL/Δt. If the net external torque is zero, angular momentum remains constant.

Angular momentum of a particle: L = mvr for perpendicular radius and velocity.
Angular momentum of a rigid body: L = Iω.
Conservation law: if external torque is zero, I1ω1 = I2ω2.
A spinning person rotates faster when pulling the arms or legs inward because rotational inertia decreases.
A spinning person rotates more slowly when spreading the arms or legs outward because rotational inertia increases.
The direction of angular momentum is along the rotation axis according to the right-hand rule.
Angular momentum has SI unit kg m2/s.
Conservation of angular momentum applies when the resultant external torque is negligible.

Circular Orbits and Kepler’s Third Law

In a circular orbit, gravitational attraction provides the centripetal force. The orbital speed and period depend on the radius of the orbit. A larger orbital radius gives a longer period.

For bodies orbiting the same central object, Kepler’s third law gives T² proportional to r³, or T proportional to r3/2. This relation can be used to compare orbital periods.

For a circular orbit, gravitational force acts as centripetal force.
Kepler’s third law: T²/r³ = constant for bodies orbiting the same central body.
If orbital radius changes from r to r/2, the new period is T' = T/(2√2).
For Earth’s year of about 365 days, the period at half the orbital radius is approximately 129 days.
The Moon rotates once about its own axis in approximately the same time that it revolves once around Earth.
Therefore, the ratio of the Moon’s angular speed about Earth to its angular speed about its own axis is 1:1.
A circular orbit requires a continuously inward acceleration even when orbital speed is constant.

Key terms

Angular displacement: The angle through which a body rotates about a fixed axis, measured in radians.
Radian: The angle subtended at the centre by an arc whose length equals the radius.
Angular velocity: The rate of change of angular displacement with time, given by ω = Δθ/Δt.
Angular acceleration: The rate of change of angular velocity with time, given by α = Δω/Δt.
Frequency: The number of complete revolutions or cycles made per second.
Time period: The time required to complete one revolution or cycle.
Tangential velocity: The linear velocity of a point moving along the tangent to a circular path.
Tangential acceleration: Acceleration along the tangent that changes the magnitude of linear velocity.
Centripetal acceleration: The inward acceleration required for circular motion, equal to v²/r.
Centripetal force: The inward resultant force that keeps a body moving in a circular path.
Centrifugal force: An apparent outward force observed from a rotating reference frame.
Torque: The turning effect of a force about an axis, equal to rF sin θ.
Moment of inertia: The rotational inertia of a body, depending on its mass distribution about an axis.
Angular momentum: The rotational momentum of a body, equal to Iω for a rigid body.
Uniform circular motion: Motion in a circle with constant speed but continuously changing velocity direction.
Right-hand rule: A rule used to determine the direction of angular vectors from the direction of rotation.

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Physics shortcuts

Comparing distance and displacement

Distance equals the magnitude of displacement only when the particle travels along a straight path without reversing direction.

Check whether the path is straight and one-directional.
If yes, distance = |displacement|.
Example: A particle moves 5 m east in a straight line. Distance = 5 m and displacement magnitude = 5 m.

This shortcut does not apply to a curved path or to motion involving a change of direction.

Projectile range and components

For a projectile launched and landing at the same level, use R = u² sin 2θ/g. Resolve the initial velocity into horizontal and vertical components when needed.

Write ux = u cos θ and uy = u sin θ.
For the same launch and landing level, R = u² sin 2θ/g.
Example: u = 20 m/s, θ = 30°, g = 10 m/s². R = 400 sin 60°/10 = 34.6 m.

The range formula does not apply directly when the projectile lands at a different height.

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