Table of Contents
Circular Motion and Centripetal Force
1. Introduction
Circular motion occurs when an object moves along a circular path around a fixed centre or axis.
Examples include the rotation of a fan, the movement of a satellite in orbit, the turning of a vehicle around a bend and the revolution of the Moon around Earth.
Circular motion is important in mechanics, astronomy, engineering, transportation and the design of rotating machinery.
2. Types of Circular Motion
2.1 Uniform circular motion
Uniform circular motion occurs when an object moves along a circular path at constant speed.
Although speed remains constant, velocity changes continuously because its direction changes.
Therefore, uniform circular motion is accelerated motion.
2.2 Non-uniform circular motion
Non-uniform circular motion occurs when an object moves along a circular path while its speed changes.
In this situation, both the magnitude and direction of velocity may change.
3. Angular Displacement
Angular displacement is the angle through which an object rotates about a fixed axis.
It is commonly measured in radians.
One complete revolution is equal to:
[
2\pi\ \text{radians}=360^\circ
]
One radian is the angle subtended at the centre of a circle by an arc equal in length to the radius.
[
\theta=\frac{s}{r}
]
Where:
- (\theta) = angular displacement in radians
- (s) = arc length
- (r) = radius
4. Angular Velocity
Angular velocity is the rate of change of angular displacement with time.
[
\omega=\frac{\Delta\theta}{\Delta t}
]
For uniform circular motion:
[
\omega=\frac{2\pi}{T}=2\pi f
]
Where:
- (\omega) = angular velocity in rad/s
- (T) = period
- (f) = frequency
Angular velocity is a vector quantity directed along the axis of rotation, according to the right-hand rule.
5. Period and Frequency
Period (T) is the time required to complete one revolution or cycle.
Its SI unit is the second (s).
Frequency (f) is the number of complete revolutions or cycles per second.
Its SI unit is the hertz (Hz).
[
f=\frac{1}{T}
]
For example, if a wheel completes 5 revolutions in one second, its frequency is 5 Hz and its period is 0.2 seconds.
6. Linear or Tangential Speed
Linear speed in circular motion is the distance travelled along the circular path per unit time.
For one complete revolution, the distance travelled is the circumference:
[
2\pi r
]
Therefore:
[
v=\frac{2\pi r}{T}
]
Since (\omega=2\pi/T):
[
v=r\omega
]
The velocity at any point is directed tangentially to the circular path.
7. Centripetal Acceleration
Centripetal acceleration is the acceleration directed towards the centre of a circular path.
It is required to continuously change the direction of velocity during circular motion.
[
a_c=\frac{v^2}{r}
]
It can also be expressed as:
[
a_c=r\omega^2
]
Where:
- (a_c) = centripetal acceleration
- (v) = tangential speed
- (r) = radius
- (\omega) = angular velocity
7.1 Important characteristics
- It is directed towards the centre of the circular path.
- It is perpendicular to the instantaneous tangential velocity in uniform circular motion.
- It increases with the square of speed.
- It decreases as the radius increases when speed is constant.
- It is present even when speed remains constant.
8. Centripetal Force
Centripetal force is the net inward force required to maintain circular motion.
[
F_c=\frac{mv^2}{r}
]
Where:
- (F_c) = centripetal force
- (m) = mass
- (v) = tangential speed
- (r) = radius
It can also be expressed as:
[
F_c=mr\omega^2
]
Centripetal force is not a separate fundamental type of force. It is the name given to the resultant inward force responsible for circular motion.
Depending on the situation, centripetal force may be supplied by:
- Gravitational force.
- Friction.
- Tension.
- Normal reaction.
- Electrostatic force.
- Magnetic force.
8.1 Example: Calculating centripetal force
A 2 kg object moves in a circle of radius 4 metres at a speed of 6 m/s.
[
F_c=\frac{mv^2}{r}
]
[
F_c=\frac{2\times6^2}{4}
]
[
F_c=\frac{72}{4}=18\ N
]
The required centripetal force is 18 N, directed towards the centre.
9. Centripetal Force in Everyday Life
9.1 A vehicle turning around a bend
Friction between the tyres and road provides the centripetal force needed to turn a vehicle.
If friction is insufficient, the vehicle may skid outwards relative to the curved path.
9.2 A stone tied to a string
When a stone is rotated in a circle using a string, tension in the string supplies the centripetal force.
If the string breaks, the stone moves approximately along the tangent to the circular path at the point of release, neglecting other forces.
9.3 Satellite motion
The gravitational attraction between Earth and a satellite supplies the centripetal force needed to maintain its orbit.
9.4 Rotating machinery
Fans, turbines, centrifuges and rotating wheels involve circular motion.
The design must account for the forces and stresses associated with rotation.
10. Banking of Roads
A road is banked when its outer edge is raised above its inner edge on a curved section.
Banking helps provide the required centripetal force through the horizontal component of the normal reaction.
For an ideal frictionless banked road:
[
\tan\theta=\frac{v^2}{rg}
]
Where:
- (\theta) = banking angle
- (v) = design speed
- (r) = radius of the curve
- (g) = gravitational acceleration
Banking reduces dependence on friction alone to provide the required centripetal force.
In real road design, friction, vehicle variation, drainage, visibility and safety are also considered.
11. Centrifugal Force
Centrifugal force is an apparent outward force observed when motion is described from a rotating, non-inertial reference frame.
It has the same magnitude as the centripetal force in the corresponding simple circular-motion analysis, but acts in the opposite direction within that rotating frame.
In an inertial frame, an object moving in a circle requires an inward net force. No real outward force is required to explain the motion.
11.1 Examples of apparent centrifugal effects
- Passengers feel pushed outward when a car turns.
- Water appears to move towards the outer edge of a rotating container.
- Objects in a rotating centrifuge experience an outward apparent effect in the rotating frame.
12. Vertical Circular Motion
Vertical circular motion occurs when an object moves along a vertical circular path.
The gravitational force affects the motion differently at different points.
For an object moving in a vertical circle on a string:
At the highest point: Both tension and weight act towards the centre.
[
T+mg=\frac{mv^2}{r}
]
At the lowest point: Tension acts towards the centre while weight acts away from it.
[
T-mg=\frac{mv^2}{r}
]
These equations assume a vertical circle with a taut string and the indicated directions.
For a complete vertical circle, the object must retain sufficient speed at the highest point to prevent the string from becoming slack.
13. Applications of Circular Motion
Circular motion is important in:
- Design of roads and railway curves.
- Satellite and planetary orbits.
- Centrifuges used in laboratories.
- Rotating machinery and turbines.
- Washing machine spin cycles.
- Amusement park rides.
- Vehicle wheel design.
- Industrial separation processes.
36 Key Takeaways for Exams
- Circular motion occurs when an object follows a circular path.
- Uniform circular motion has constant speed but changing velocity.
- Non-uniform circular motion involves changing speed.
- Angular displacement measures rotation through an angle.
- One complete revolution equals (2\pi) radians or 360 degrees.
- One radian subtends an arc equal in length to the radius.
- Angular velocity is the rate of change of angular displacement.
- The SI unit of angular velocity is radian per second.
- Period is the time required for one complete revolution.
- Frequency is the number of revolutions per second.
- Period and frequency are reciprocals.
- The SI unit of frequency is hertz.
- Linear speed in circular motion is (2\pi r/T).
- Tangential speed is related to angular velocity by (v=r\omega).
- Instantaneous velocity is directed tangentially to the circular path.
- Centripetal acceleration is directed towards the centre.
- Centripetal acceleration is (v^2/r).
- Centripetal acceleration can also be expressed as (r\omega^2).
- Uniform circular motion involves acceleration despite constant speed.
- Centripetal force is the net inward force required for circular motion.
- Centripetal force is (mv^2/r).
- Centripetal force is not a separate fundamental force.
- Friction can provide centripetal force for a vehicle turning on a flat road.
- Tension provides centripetal force for a stone rotated on a string.
- Gravity provides centripetal force for an orbiting satellite.
- Banking of roads helps provide centripetal force through the normal reaction.
- For an ideal frictionless banked road, (\tan\theta=v^2/(rg)).
- Centrifugal force is an apparent force in a rotating reference frame.
- In an inertial frame, the net force required for circular motion acts inward.
- When a string breaks, a rotating object initially moves along the tangent to its circular path.
- Vertical circular motion involves changes in the contribution of gravity and tension.
- At the top of a vertical circle, gravity acts towards the centre.
- At the bottom of a vertical circle, gravity acts away from the centre.
- Centripetal acceleration increases with the square of tangential speed.
- Increasing the radius reduces centripetal acceleration when tangential speed is constant.
- Circular motion is important in transport, astronomy, engineering and industrial machinery.