4.9 Circular Motion | Rotational And Circular Motion | Physics 11

Circular motion is the movement of an object along the circumference of a circle or rotation along a circular path. It is a fundamental type of motion that appears everywhere in the universe, from the orbits of planets and stars to the spinning of a wheel on a car and the motion of electrons in an atom. Understanding the principles of circular motion is key to analyzing a vast range of physical systems.

Definition

Circular motion describes a body's movement where its distance from a fixed central point remains constant. This constant distance is the radius of the circular path. A key feature of circular motion is that even if the object's speed is constant, its velocity is continuously changing because its direction of motion is always changing.

Types of Circular Motion

Uniform Circular Motion: This occurs when an object moves along a circular path at a constant speed. While the magnitude of the velocity vector (the speed) is constant, the direction of the vector is always changing, meaning the object is continuously accelerating.
Non-Uniform Circular Motion: This occurs when an object's speed changes as it moves along a circular path. In this case, the object has both a change in the direction of its velocity and a change in the magnitude of its velocity.

Type	Speed	Velocity	Acceleration
Uniform	Constant	Changing (direction only)	Centripetal acceleration only
Non-Uniform	Changing	Changing (magnitude & direction)	Centripetal and Tangential acceleration

Important Parameters in Circular Motion

Angular Displacement ( $θ$ )

The angle through which an object moves around a central axis. It is the rotational equivalent of linear displacement and is typically measured in radians.

$θ = \frac{s}{r}$

where $s$ is the arc length and $r$ is the radius.

Angular Velocity ( $ω$ )

The rate of change of angular displacement. It describes how quickly the object is rotating and is measured in radians per second (rad/s).

$ω = \frac{Δ θ}{Δ t}$

Angular Acceleration ( $α$ )

The rate of change of angular velocity. It is measured in radians per second squared (rad/s²).

$α = \frac{Δ ω}{Δ t}$

Angular acceleration is positive when the rotation is speeding up and negative (deceleration) when it is slowing down.

Tangential Velocity ( $v_{t}$ )

The instantaneous linear velocity of an object in circular motion. It is always directed tangent to the circular path and is perpendicular to the radius. Its magnitude is related to angular velocity by:

$v = r ω$

Centripetal Acceleration ( $a_{c}$ )

Since the direction of the velocity vector is always changing in circular motion, there must be an acceleration. This acceleration is called centripetal ("center-seeking") acceleration because it is always directed radially inward, towards the center of the circle. Its magnitude is given by:

$a_{c} = \frac{v ^{2}}{r} = r ω^{2}$

Centripetal Force ( $F_{c}$ )

According to Newton's Second Law, an acceleration must be caused by a net force. The centripetal force is the net force that produces the centripetal acceleration, keeping the object in its circular path. This force is also always directed towards the center of the circle.

$F_{c} = m a_{c} = \frac{m v ^{2}}{r} = m r ω^{2}$

Note: Centripetal force is not a new kind of force. It is the net result of other forces (like tension, gravity, or friction) that are causing the circular motion.

Centrifugal Force

This is an apparent or fictitious force that an object seems to experience, pushing it outward from the center of rotation. It is not a real force in an inertial (non-accelerating) frame of reference. Instead, it is a consequence of inertia — the object's tendency to continue moving in a straight line. An observer in a rotating (non-inertial) reference frame perceives this outward push as a pseudo-force.

Possible Questions/Answers

Q: What provides the centripetal force for the Earth to orbit the Sun? A: The gravitational force between the Earth and the Sun provides the necessary centripetal force that keeps the Earth in its (nearly) circular orbit.
Q: If you swing a ball on a string in a circle and the string breaks, what path does the ball take? A: The ball will fly off in a straight line, tangent to the circle at the point where the string broke. This is due to its inertia, as the centripetal force (the tension in the string) is no longer present to keep it turning.

Summary

Circular Motion is the movement of an object in a circular path at a constant distance from a center point.
It is characterized by a continuously changing velocity vector, which means there is always an acceleration.
This centripetal acceleration is caused by a centripetal force, both of which are directed towards the center of the circle.
The motion can be uniform (constant speed) or non-uniform (changing speed).

Parameter	Formula	Description
Angular Displacement ( $θ$ )	$θ = s / r$	Angle rotated through (rad).
Angular Velocity ( $ω$ )	$ω = Δ θ /Δ t$	Rate of rotation (rad/s).
Angular Acceleration ( $α$ )	$α = Δ ω /Δ t$	Rate of change of $ω$ (rad/s²).
Tangential Velocity ( $v_{t}$ )	$v = r ω$	Linear speed along the tangent.
Centripetal Acceleration ( $a_{c}$ )	$a_{c} = v^{2} / r = r ω^{2}$	Acceleration towards the center.
Centripetal Force ( $F_{c}$ )	$F_{c} = m v^{2} / r$	Net force towards the center.

The principles of circular motion are fundamental to understanding orbital mechanics, the design of rotating machinery, and many other areas of physics and engineering.

Definition

Types of Circular Motion

Uniform Circular Motion: This occurs when an object moves along a circular path at a constant speed. While the magnitude of the velocity vector (the speed) is constant, the direction of the vector is always changing, meaning the object is continuously accelerating.
Non-Uniform Circular Motion: This occurs when an object's speed changes as it moves along a circular path. In this case, the object has both a change in the direction of its velocity and a change in the magnitude of its velocity.

Type	Speed	Velocity	Acceleration
Uniform	Constant	Changing (direction only)	Centripetal acceleration only
Non-Uniform	Changing	Changing (magnitude & direction)	Centripetal and Tangential acceleration

Important Parameters in Circular Motion

Angular Displacement ( $θ$ )

The angle through which an object moves around a central axis. It is the rotational equivalent of linear displacement and is typically measured in radians.

$θ = \frac{s}{r}$

where $s$ is the arc length and $r$ is the radius.

Angular Velocity ( $ω$ )

The rate of change of angular displacement. It describes how quickly the object is rotating and is measured in radians per second (rad/s).

$ω = \frac{Δ θ}{Δ t}$

Angular Acceleration ( $α$ )

The rate of change of angular velocity. It is measured in radians per second squared (rad/s²).

$α = \frac{Δ ω}{Δ t}$

Angular acceleration is positive when the rotation is speeding up and negative (deceleration) when it is slowing down.

Q: What provides the centripetal force for the Earth to orbit the Sun? A: The gravitational force between the Earth and the Sun provides the necessary centripetal force that keeps the Earth in its (nearly) circular orbit.
Q: If you swing a ball on a string in a circle and the string breaks, what path does the ball take? A: The ball will fly off in a straight line, tangent to the circle at the point where the string broke. This is due to its inertia, as the centripetal force (the tension in the string) is no longer present to keep it turning.

Summary

Circular Motion is the movement of an object in a circular path at a constant distance from a center point.
It is characterized by a continuously changing velocity vector, which means there is always an acceleration.
This centripetal acceleration is caused by a centripetal force, both of which are directed towards the center of the circle.
The motion can be uniform (constant speed) or non-uniform (changing speed).

Parameter	Formula	Description
Angular Displacement ( $θ$ )	$θ = s / r$	Angle rotated through (rad).
Angular Velocity ( $ω$ )	$ω = Δ θ /Δ t$	Rate of rotation (rad/s).
Angular Acceleration ( $α$ )	$α = Δ ω /Δ t$	Rate of change of $ω$ (rad/s²).
Tangential Velocity ( $v_{t}$ )	$v = r ω$	Linear speed along the tangent.
Centripetal Acceleration ( $a_{c}$ )	$a_{c} = v^{2} / r = r ω^{2}$	Acceleration towards the center.
Centripetal Force ( $F_{c}$ )	$F_{c} = m v^{2} / r$	Net force towards the center.

The principles of circular motion are fundamental to understanding orbital mechanics, the design of rotating machinery, and many other areas of physics and engineering.