Circular Motion
1. Motion in a Circle
Learning outcomes
- I can describe uniform circular motion.
- I can explain why velocity changes in circular motion.
- I can identify acceleration in circular motion.
- I can distinguish circular motion from linear motion.
- I can analyze examples of circular motion.
What Is Circular Motion?
Circular motion occurs when an object moves along a circular path around a fixed point or axis.
Examples include:
- a car travelling around a circular track
- a satellite orbiting Earth
- a rider on a Ferris wheel
- a point on a rotating fan blade
- a ball attached to a string being swung in a circle
- a point on the edge of a spinning wheel
- Earth moving approximately around the Sun
In each case, the direction of the object's motion is continually changing.
That changing direction is the key to understanding circular motion.
Uniform Circular Motion
Uniform circular motion occurs when an object moves around a circular path at constant speed.
The word uniform refers to the speed remaining constant.
For example, a car could travel around a circular track at:
10 m/s
without speeding up or slowing down.
However, something important is still changing:
the direction of motion.
Therefore, the object's velocity is changing even though its speed is constant.
Speed and Velocity Are Different
Speed tells us how fast an object is moving.
Speed is a scalar quantity.
Velocity tells us both:
- how fast an object is moving
- the direction in which it is moving
Velocity is a vector quantity.
Therefore:
velocity = speed + direction
This distinction is essential for understanding circular motion.
Constant Speed Does Not Mean Constant Velocity
Imagine a car travelling around a circular track at exactly:
20 m/s
At the top of the circle it may be travelling left.
At the bottom it may be travelling right.
On one side it may be travelling upward.
On the other side it may be travelling downward.
Its speed remains:
20 m/s
but its direction continually changes.
Therefore:
its velocity continually changes.
Velocity Is Tangent to the Circle
At any instant, the velocity of an object in circular motion points tangent to the circular path.
A tangent is a line that touches a circle at one point.
Therefore:
velocity → tangent to circle
while, as we will see:
acceleration → toward centre
These directions are perpendicular in uniform circular motion.
What Happens If the Circular Motion Stops?
Imagine a ball attached to a string being swung in a circle.
If the string suddenly breaks, the ball does not continue travelling around the circle.
Instead, it moves approximately along the tangent to the circle at the instant the string breaks.
This happens because the ball's instantaneous velocity was already pointing along the tangent.
Without the inward force, there is nothing to continually change the direction of that velocity.
Newton's First Law then describes the subsequent motion.
Circular Motion Involves Acceleration
Acceleration means:
the rate of change of velocity
An object accelerates whenever its velocity changes.
Velocity can change because:
- speed changes
- direction changes
- both speed and direction change
Therefore, an object moving at constant speed around a circle is still accelerating because its direction changes continuously.
Centripetal Acceleration
The acceleration of an object moving in a circle is directed:
toward the centre of the circle
This acceleration is called centripetal acceleration.
The word centripetal means:
centre-seeking
At every point around the circle:
velocity → tangent
acceleration → centre
Velocity and Acceleration Directions
This relationship is one of the most important ideas in circular motion.
Imagine an object at the right side of a circle moving counterclockwise.
Its velocity points:
upward
Its acceleration points:
left, toward the centre
At the top of the circle:
velocity points left
acceleration points down toward the centre.
At the left side:
velocity points down
acceleration points right.
At the bottom:
velocity points right
acceleration points up.
The directions continually change as the object moves.
Why Does Inward Acceleration Produce a Circle?
Imagine an object travelling forward.
If no force acts on it, Newton's First Law predicts that it will continue moving in a straight line.
Now imagine that its velocity is continually redirected slightly toward one side.
Its path bends.
If the direction changes continuously toward one fixed centre, the object can follow a circular path.
Circular motion therefore requires continuous inward acceleration.
Centripetal Acceleration and Speed
The magnitude of centripetal acceleration is:
a_c = v² / r
where:
- a_c = centripetal acceleration in m/s²
- v = speed in m/s
- r = radius of the circular path in m
This equation tells us two important things.
Higher speed → greater centripetal acceleration
Larger radius → smaller centripetal acceleration
Effect of Speed
Because:
a_c ∝ v²
centripetal acceleration depends on the square of speed.
If speed doubles:
a_c becomes 4 times greater
If speed triples:
a_c becomes 9 times greater
This is why travelling around a curve much faster requires a much greater inward acceleration.
Effect of Radius
Because:
a_c ∝ 1/r
increasing the radius decreases centripetal acceleration when speed remains constant.
For example:
double the radius → half the centripetal acceleration.
This helps explain why gentle, wide curves are easier to travel around at high speed than tight curves.
Example 1: Centripetal Acceleration
A car travels around a circular track at:
10 m/s
The radius is:
20 m
Calculate the centripetal acceleration.
a_c = v²/r
a_c = 10²/20
a_c = 100/20
a_c = 5 m/s²
The acceleration is:
5 m/s² toward the centre of the circle.
Example 2: Faster Motion
The same car now travels at:
20 m/s
around the same 20 m radius curve.
a_c = 20²/20
a_c = 400/20
a_c = 20 m/s²
The speed doubled from 10 m/s to 20 m/s.
But the acceleration increased from:
5 m/s² to 20 m/s²
That is four times greater.
Centripetal Force
Acceleration requires a resultant force.
Newton's Second Law tells us:
F = ma
Therefore, circular motion requires a resultant force directed toward the centre.
This inward resultant force is called the centripetal force.
Centripetal Force Is Not a New Type of Force
This is extremely important.
"Centripetal force" is not an additional force such as gravity, friction, tension, or normal force.
It is the name given to the resultant inward force that produces circular motion.
Different real forces can provide the centripetal force.
For example:
- tension can provide it
- gravity can provide it
- friction can provide it
- normal force can provide it
- a combination of forces can provide it
Ball on a String
Consider a ball attached to a string and swung horizontally in a circle.
The string pulls the ball toward the centre.
The inward force is provided by:
tension
Without the tension, the ball would no longer follow the circular path.
Car Turning on a Flat Road
When a car travels around a curve on a flat road, the tyres need an inward force.
That force is generally supplied by:
static friction between the tyres and the road
The friction force points toward the centre of the curved path.
If there is not enough friction, the car cannot follow the required circular path.
Why a Car May Skid
Suppose a car enters a curve too quickly.
The required centripetal acceleration increases with:
v²
Therefore, the required inward force also increases rapidly.
If the available tyre-road friction is insufficient, the car cannot turn as sharply as required and deviates from the intended circular path.
This is one reason excessive speed is particularly important on tight curves.
Satellites in Orbit
A satellite travelling around Earth is another example of circular or approximately circular motion.
The satellite has a tangential velocity.
Gravity pulls it toward Earth.
Gravity therefore provides the inward acceleration required for the orbit.
For an ideal circular orbit:
gravity provides the centripetal force.
Why Doesn't the Satellite Fall Straight Down?
A satellite is falling toward Earth.
However, it also has a large sideways velocity.
As it falls, Earth's curved surface falls away beneath it.
The result is continuous free fall around Earth.
This produces an orbit.
The satellite's velocity remains tangent to its path while gravitational acceleration points approximately toward Earth's centre.
Planets and Circular Motion
Planetary orbits are generally elliptical rather than perfectly circular.
However, circular motion provides a useful introductory model.
Gravity provides the inward force that continually changes the direction of a planet's velocity.
For a perfectly circular orbit, the speed would remain constant while the velocity direction continually changed.
Ferris Wheel
A rider on a Ferris wheel follows a circular path.
At every point:
velocity is tangent to the wheel.
Centripetal acceleration points toward the centre.
At the top:
acceleration points downward.
At the bottom:
acceleration points upward.
At the sides:
acceleration points horizontally toward the centre.
Rotating Fan
Consider a point near the end of a fan blade.
As the fan rotates:
- the point follows a circular path
- its velocity continually changes direction
- its acceleration points toward the axis of rotation
Points farther from the centre travel through a larger distance during each rotation.
Washing Machine Spin Cycle
During a washing machine's spin cycle, the drum rotates rapidly.
The drum exerts forces on the clothes that help keep them moving along a circular path.
Water can pass through holes in the drum and is no longer constrained in the same way.
This everyday device provides a useful example of circular motion and the need for inward force.
Amusement-Park Rides
Many amusement-park rides involve circular motion.
Examples include:
- Ferris wheels
- rotating swings
- looped tracks
- spinning platforms
In each case, some real force or combination of forces must provide the required inward resultant force.
The exact forces depend on the particular ride.
Circular Motion vs Linear Motion
Linear motion occurs along a straight path.
If an object moves in a straight line at constant speed:
- speed is constant
- direction is constant
- velocity is constant
- acceleration is zero
Uniform circular motion occurs along a circular path.
If speed remains constant:
- speed is constant
- direction changes
- velocity changes
- acceleration is not zero
This is the key difference.
Comparing the Two
For constant-velocity linear motion:
velocity → same magnitude and same direction
For uniform circular motion:
velocity → same magnitude but continuously changing direction
Therefore:
Linear constant velocity:
a = 0
Uniform circular motion:
a ≠ 0
even though speed is constant.
Is the Object Accelerating or Decelerating?
Students sometimes describe centripetal acceleration as "speeding up."
That is not necessarily correct.
In uniform circular motion:
speed remains constant
The acceleration changes the direction of the velocity rather than its magnitude.
Therefore, the object accelerates without speeding up.
Velocity Vector Changes
Consider two nearby points on a circular path.
At the first point, the velocity points in one direction.
A short time later, the velocity points in a slightly different direction.
The difference between these velocity vectors is:
Δv
For uniform circular motion, the change in velocity points inward.
Since:
a = Δv/Δt
the acceleration also points inward.
The Centre Is Special
In uniform circular motion, centripetal acceleration always points toward the instantaneous centre of curvature.
For a perfect circle, this is simply the centre of the circle.
As the object moves, the direction of acceleration therefore changes continuously.
The acceleration vector is not fixed in one compass direction.
It continually follows the object around the circle while pointing inward.
Example 3: Identify the Directions
A car moves clockwise around a circular track.
At the top of the track:
Velocity points:
right
Centripetal acceleration points:
down toward the centre
At the right side:
Velocity points:
down
Centripetal acceleration points:
left toward the centre
At the bottom:
Velocity points:
left
Centripetal acceleration points:
up toward the centre
At the left side:
Velocity points:
up
Centripetal acceleration points:
right toward the centre
Period and Circular Motion
The period, T, is the time required for one complete revolution.
It is measured in:
seconds
If an object completes one circle every 4 seconds:
T = 4 s
The distance travelled in one complete revolution is the circumference:
distance = 2πr
Therefore, for uniform circular motion:
v = 2πr/T
Example 4: Speed from Period
A rider travels around a circular ride of radius:
5 m
One revolution takes:
10 s
Distance per revolution:
2πr
= 2π(5)
≈ 31.4 m
Speed:
v = 31.4/10
v ≈ 3.14 m/s
Frequency
Frequency describes how many complete revolutions occur each second.
It is measured in:
hertz (Hz)
1 Hz means:
1 revolution per second
Period and frequency are related:
f = 1/T
and:
T = 1/f
For example, if:
T = 0.5 s
then:
f = 1/0.5
f = 2 Hz
The object completes two revolutions every second.
Speed and Frequency
Since:
v = 2πr/T
and:
f = 1/T
we can also write:
v = 2πrf
Therefore, at a fixed radius:
higher frequency → greater speed.
This is useful for rotating wheels, motors, fans, laboratory centrifuges, and other rotating systems.
Example 5: A Rotating Wheel
A point on a wheel is:
0.40 m
from the centre.
The wheel completes:
2 revolutions each second
Therefore:
f = 2 Hz
Speed:
v = 2πrf
v = 2π(0.40)(2)
v ≈ 5.0 m/s
The point moves at approximately:
5.0 m/s
along its circular path.
Different Points on a Rotating Object
Consider a rotating disc.
A point near the centre and a point near the edge complete each revolution in the same amount of time.
However, the point near the edge travels a greater distance.
Therefore, points farther from the centre have greater tangential speed when they share the same angular rotation rate.
This explains why the outer edge of a large rotating object can move very quickly.
Example 6: Circular Motion Analysis
A ball moves around a circle at constant speed.
Which quantities remain constant?
Mass: constant
Speed: constant
Radius: constant
Which quantities change?
Velocity: changes because direction changes
Acceleration direction: changes continuously while always pointing toward the centre
Therefore, uniform circular motion is accelerated motion despite having constant speed.
Example 7: What Happens If Speed Doubles?
An object moves around a circle of fixed radius.
Its original centripetal acceleration is:
3 m/s²
Its speed doubles.
Because:
a_c ∝ v²
the acceleration becomes:
4 × 3
= 12 m/s²
Doubling speed quadruples centripetal acceleration.
Example 8: What Happens If Radius Doubles?
An object moves at constant speed around a circle.
Its radius doubles.
Because:
a_c ∝ 1/r
the centripetal acceleration becomes:
half as large.
If the original acceleration was:
8 m/s²
the new acceleration is:
4 m/s²
Circular Motion and Newton's First Law
Newton's First Law states that an object remains at rest or continues moving with constant velocity unless acted upon by a resultant force.
An object moving in a circle does not have constant velocity.
Its direction changes.
Therefore:
there must be a resultant force.
That resultant force points inward.
If the inward resultant force disappears, the object moves approximately tangent to the circle.
Circular Motion and Newton's Second Law
Newton's Second Law states:
F_net = ma
In circular motion, the acceleration is inward.
Therefore, the resultant force must also point inward.
For uniform circular motion:
F_c = mv²/r
where:
- F_c = inward resultant or centripetal force
- m = mass
- v = speed
- r = radius
This equation will become especially important when solving quantitative circular-motion problems.
Example 9: Required Inward Force
A 2 kg object moves at:
6 m/s
around a circle of radius:
3 m
Centripetal acceleration:
a_c = 6²/3
a_c = 12 m/s²
Required inward force:
F = ma
F = 2 × 12
F = 24 N
Therefore, the system must provide a resultant force of:
24 N toward the centre.
Where Does Centripetal Force Come From?
The phrase centripetal force tells us the direction and role of the resultant force.
We still need to identify the real physical force producing it.
Examples:
Ball on string:
tension
Satellite:
gravity
Car on flat curve:
friction
Object against the wall of a rotating container:
normal force
Roller coaster:
often a combination of normal force and gravity
Always identify the actual forces in a free-body diagram.
Is There an Outward Force?
When sitting in a turning car, you may feel as though you are being pushed outward.
However, in an inertial frame, the force needed to make your body follow the circular path is directed inward.
Your body's inertia tends to keep it moving along its previous straight-line direction.
The car changes direction beneath and around you.
This can create the sensation of being pushed outward.
Centrifugal Force
You may encounter the term centrifugal force.
In a rotating reference frame, centrifugal force can be introduced as an apparent or inertial force to describe observations from within that rotating frame.
But when analyzing circular motion from an ordinary inertial reference frame, the physical resultant force on the object points:
inward
toward the centre.
For introductory free-body diagrams, do not automatically add an outward "centrifugal force" as another real interaction force.
Analyzing Circular Motion Examples
When you encounter a circular-motion situation, ask:
1. What object is moving?
2. Where is the centre of the circle?
3. Which direction is the velocity?
The velocity is tangent to the path.
4. Which direction is the acceleration?
Toward the centre.
5. What real force or forces produce the inward resultant?
Examples include gravity, friction, tension, or normal force.
6. Is the speed constant?
If yes, the motion may be uniform circular motion.
Example 10: Satellite
Object:
satellite
Path:
approximately circular orbit
Velocity:
tangent to orbit
Acceleration:
toward Earth's centre
Force providing acceleration:
gravity
Speed:
constant for an ideal circular orbit
Therefore:
uniform circular motion is a useful model.
Example 11: Car on a Circular Track
Object:
car
Velocity:
tangent to track
Acceleration:
toward centre of curve
Force providing horizontal inward acceleration on a flat road:
friction
If the car maintains constant speed:
uniform circular motion
If the car speeds up or slows down while turning:
the motion is circular but not uniform circular motion.
Non-Uniform Circular Motion
An object can move in a circle while changing its speed.
This is called non-uniform circular motion.
In this case, the object can have:
- inward centripetal acceleration
- tangential acceleration
The inward component changes direction.
The tangential component changes speed.
Uniform circular motion has constant speed, so it has no tangential acceleration.
Its acceleration is purely inward.
Real-World Example: Centrifuge
A centrifuge spins samples rapidly around a circular path.
High rotational speeds can produce very large centripetal accelerations.
Centrifuges are used in:
- scientific laboratories
- medicine
- biotechnology
- chemistry
They can help separate materials with different densities.
Real-World Example: Earth
Earth rotates around its axis approximately once every 24 hours.
A point on Earth's surface therefore follows a circular path around the rotation axis.
Points near the equator travel through a larger circle than points closer to the poles.
This means tangential speed due to Earth's rotation depends on distance from the rotation axis.
Common Mistakes
Mistake 1: Saying constant speed means zero acceleration
In circular motion, direction changes.
Therefore, velocity changes and acceleration exists.
Mistake 2: Pointing velocity toward the centre
Velocity points:
tangent to the circle
Acceleration points:
toward the centre
Mistake 3: Pointing centripetal acceleration outward
Centripetal acceleration always points inward.
Mistake 4: Treating centripetal force as an extra force
Centripetal force is the resultant inward force, not a separate interaction force.
Mistake 5: Saying a released object flies radially outward
If the inward force disappears, the object initially follows the tangent to its circular path.
Mistake 6: Confusing speed with velocity
Speed can remain constant while velocity changes because direction changes.
Mistake 7: Thinking circular motion violates Newton's First Law
Circular motion requires a resultant inward force precisely because the object would otherwise continue in a straight line.
Mistake 8: Assuming every circular motion is uniform
If speed changes, the circular motion is not uniform.
Mistake 9: Assuming gravity always points downward on a diagram
For an orbiting satellite, gravity points toward the centre of Earth.
Mistake 10: Automatically adding an outward centrifugal force to a free-body diagram
In an inertial reference frame, identify the actual physical forces and determine their inward resultant.
Did You Know?
A satellite in circular orbit is constantly accelerating even if its speed remains almost perfectly constant.
Its velocity continuously turns as gravity pulls it toward Earth.
If gravity suddenly disappeared, the satellite would not continue following its curved orbit.
It would initially move along the tangent to its orbit.
The same basic idea applies to a ball on a string, a car on a circular track, and a planet in orbit:
circular motion requires continuous inward acceleration.
Key Terms
Circular motion: Motion along a circular path.
Uniform circular motion: Circular motion at constant speed.
Speed: The rate at which distance is travelled.
Velocity: Speed in a specified direction.
Vector: A quantity with both magnitude and direction.
Acceleration: Rate of change of velocity.
Centripetal acceleration: Acceleration directed toward the centre of a circular path.
Centripetal force: The resultant inward force responsible for centripetal acceleration.
Tangent: A line touching a circle at one point; instantaneous velocity is tangent to the circular path.
Radius: Distance from the centre of a circle to the moving object.
Period: Time required for one complete revolution.
Frequency: Number of complete revolutions per second.
Revolution: One complete trip around a circular path.
Key Equations
Centripetal acceleration:
a_c = v²/r
Centripetal force:
F_c = mv²/r
Newton's Second Law:
F_net = ma
Circumference:
C = 2πr
Speed in uniform circular motion:
v = 2πr/T
Period and frequency:
f = 1/T
T = 1/f
Speed using frequency:
v = 2πrf
Key Takeaways
- Circular motion occurs when an object follows a circular path.
- Uniform circular motion means the object moves around the circle at constant speed.
- Constant speed does not mean constant velocity.
- Velocity changes continuously because its direction changes.
- Therefore, an object in uniform circular motion is accelerating.
- The velocity vector is always tangent to the circular path.
- Centripetal acceleration always points toward the centre.
- The magnitude of centripetal acceleration is a_c = v²/r.
- Increasing speed greatly increases centripetal acceleration because acceleration depends on speed squared.
- Increasing radius decreases centripetal acceleration when speed is unchanged.
- Circular motion requires a resultant inward force.
- The inward resultant is called the centripetal force.
- Centripetal force is not a new type of physical force.
- Tension, friction, gravity, normal force, or combinations of forces can provide the required inward force.
- If the inward force disappears, the object initially travels tangent to the circular path.
- Uniform circular motion differs from constant-velocity linear motion because circular motion has continuously changing direction and non-zero acceleration.
- Non-uniform circular motion involves changing speed as well as changing direction.
- The period is the time for one revolution, while frequency is the number of revolutions per second.
- Satellites, planets, cars turning, Ferris wheels, fan blades, centrifuges, and rotating machinery can all involve circular motion.
- A useful circular-motion reasoning chain is:
identify the circular path → locate the centre → draw velocity tangent to the path → draw acceleration toward the centre → identify the real inward force → determine whether speed is constant → analyze the motion.