Circular Motion
3. Centripetal Acceleration
Learning outcomes
- I can define centripetal acceleration.
- I can calculate centripetal acceleration.
- I can relate acceleration to speed and radius.
- I can compare circular systems with different radii and speeds.
- I can solve problems involving centripetal acceleration.
What Is Centripetal Acceleration?
An object moving around a circular path is constantly changing its direction.
Because velocity includes direction, changing direction means changing velocity.
A change in velocity means that the object is accelerating.
The acceleration directed toward the centre of a circular path is called centripetal acceleration.
The word centripetal means:
centre-seeking
Therefore:
centripetal acceleration = acceleration directed toward the centre of a circular path.
Constant Speed but Changing Velocity
An object can have:
constant speed
while still having:
changing velocity
This happens during uniform circular motion.
Imagine a car travelling around a circular track at exactly 15 m/s.
Its speed remains 15 m/s.
However:
- at one point it travels north
- later it travels west
- later it travels south
- later it travels east
Its direction changes continuously.
Therefore, its velocity changes continuously.
So the car is accelerating.
Direction of Centripetal Acceleration
Centripetal acceleration always points:
toward the centre of the circular path
At the same time, the object's instantaneous velocity points:
tangent to the circular path
Therefore, in uniform circular motion:
velocity → tangent
centripetal acceleration → centre
The two vectors are perpendicular at each instant.
Why Does Acceleration Point Inward?
Acceleration describes how velocity changes.
Consider an object at two nearby points on a circular path.
Its speed may be the same at both points, but the velocity vectors point in slightly different directions.
The change in velocity, Δv, points toward the inside of the circular path.
Since:
a = Δv/Δt
the acceleration also points inward.
This inward acceleration continuously turns the velocity vector and keeps the object following the curved path.
What Happens Without Centripetal Acceleration?
Newton's First Law tells us that an object with no resultant force continues moving with constant velocity.
That means:
straight-line motion
If the inward acceleration disappeared, the object would no longer follow the circle.
It would initially move along a line tangent to the circle.
Centripetal acceleration is therefore what continually changes the direction of the object's velocity.
Calculating Centripetal Acceleration
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 shows that centripetal acceleration depends on:
speed
and:
radius
Notice that mass does not appear in the equation.
Mass Does Not Affect Centripetal Acceleration Directly
Suppose two objects travel around the same circular path at the same speed.
Object A has a mass of 2 kg.
Object B has a mass of 20 kg.
Because:
a_c = v²/r
both objects have the same centripetal acceleration.
However, the heavier object requires a greater centripetal force because:
F = ma
This is an important difference between centripetal acceleration and centripetal force.
Effect of Speed
From:
a_c = v²/r
centripetal acceleration is proportional to the square of speed:
a_c ∝ v²
This means speed has a very strong effect.
If speed doubles:
acceleration becomes 4 times greater
If speed triples:
acceleration becomes 9 times greater
If speed quadruples:
acceleration becomes 16 times greater
A relatively small increase in speed can therefore produce a large increase in centripetal acceleration.
Example 1: Basic Calculation
A car travels at:
10 m/s
around a circular curve of radius:
20 m
Calculate its centripetal acceleration.
Use:
a_c = v²/r
Substitute:
a_c = 10²/20
a_c = 100/20
a_c = 5 m/s²
Therefore:
a_c = 5 m/s² toward the centre
Example 2: Doubling the Speed
The same car travels around the same 20 m radius curve at:
20 m/s
Calculate the centripetal acceleration.
a_c = 20²/20
a_c = 400/20
a_c = 20 m/s²
Compare:
At 10 m/s:
a_c = 5 m/s²
At 20 m/s:
a_c = 20 m/s²
The speed doubled.
The centripetal acceleration became:
4 times greater.
Why Speed Is Squared
The v² relationship means that faster circular motion becomes increasingly demanding.
Consider the same curve:
5 m/s → v² = 25
10 m/s → v² = 100
15 m/s → v² = 225
20 m/s → v² = 400
The speed increases evenly, but v² increases much more rapidly.
This is one reason high-speed cornering produces much larger accelerations than low-speed cornering.
Effect of Radius
Centripetal acceleration is inversely proportional to radius:
a_c ∝ 1/r
Therefore:
larger radius → smaller centripetal acceleration
smaller radius → larger centripetal acceleration
assuming speed remains constant.
If radius doubles:
acceleration becomes half as large
If radius triples:
acceleration becomes one-third as large
Example 3: Changing Radius
A car travels at 12 m/s around a curve of radius 24 m.
a_c = 12²/24
a_c = 144/24
a_c = 6 m/s²
Now suppose the radius doubles to:
48 m
a_c = 144/48
a_c = 3 m/s²
Doubling the radius reduced the acceleration by half.
Tight Curves vs Wide Curves
Imagine two cars travelling at the same speed.
Car A travels around a tight curve.
Car B travels around a wide curve.
The car on the tighter curve has the greater centripetal acceleration.
Why?
Its direction must change more rapidly.
A larger-radius curve changes the direction of motion more gradually.
Comparing Circular Systems
The equation:
a_c = v²/r
allows us to compare circular systems without always calculating exact values.
For example:
System A:
v = 10 m/s
r = 20 m
System B:
v = 20 m/s
r = 40 m
For System A:
a_A = 10²/20 = 5 m/s²
For System B:
a_B = 20²/40 = 10 m/s²
Even though System B has twice the radius, its doubled speed has a larger effect because speed is squared.
Therefore:
System B has twice the centripetal acceleration.
Ratio Method
We can compare two systems using:
a₂/a₁ = (v₂²/r₂) ÷ (v₁²/r₁)
which can be rearranged to:
a₂/a₁ = (v₂/v₁)²(r₁/r₂)
This is useful when the question asks:
"How many times larger?"
rather than asking for an exact acceleration.
Example 4: Compare Two Systems
System B has:
- twice the speed of System A
- twice the radius of System A
How do their centripetal accelerations compare?
Speed effect:
2² = 4
Radius effect:
÷ 2
Therefore:
4 ÷ 2 = 2
System B has:
2 times the centripetal acceleration of System A.
Example 5: Same Speed, Different Radius
Object A travels around a circle of radius:
2 m
Object B travels around a circle of radius:
8 m
Both travel at:
4 m/s
Object A:
a_A = 4²/2 = 8 m/s²
Object B:
a_B = 4²/8 = 2 m/s²
Therefore:
Object A has four times the centripetal acceleration.
Example 6: Same Radius, Different Speed
Object A:
v = 3 m/s
Object B:
v = 6 m/s
Both move around circles with radius:
4 m
Object A:
a_A = 3²/4 = 2.25 m/s²
Object B:
a_B = 6²/4 = 9 m/s²
Object B moves twice as fast but experiences:
four times the centripetal acceleration.
Cars Turning
A car travelling around a curved road experiences centripetal acceleration toward the centre of the curve.
If the car travels faster:
a_c increases strongly
If the curve becomes tighter:
a_c increases
This helps explain why tight curves often require lower speeds.
Example 7: Car on a Curve
A car travels at:
18 m/s
around a curve of radius:
54 m
Calculate the centripetal acceleration.
a_c = 18²/54
a_c = 324/54
a_c = 6 m/s²
The acceleration is:
6 m/s² toward the centre of the curve.
Satellites and Centripetal Acceleration
A satellite in a circular orbit constantly changes direction.
Therefore, it experiences centripetal acceleration.
The acceleration points toward Earth's centre.
For a satellite, this centripetal acceleration is produced by:
gravity
The satellite can maintain approximately constant speed while its velocity continuously changes direction.
Why Satellites Are Accelerating
A satellite may appear to move smoothly around Earth at nearly constant speed.
But acceleration does not require a change in speed.
It requires a change in:
velocity
Because the satellite's direction changes continuously:
velocity changes
Therefore:
the satellite accelerates continuously.
Planets and Circular Motion
Planetary orbits are elliptical, but a circular orbit can be used as a useful simplified model.
For an ideal circular orbit:
velocity → tangent
centripetal acceleration → toward the Sun
gravity → provides the required inward acceleration.
Ferris Wheel
A rider on a Ferris wheel moves around a circular path.
The rider's centripetal acceleration always points toward the centre.
At the top:
acceleration points downward.
At the bottom:
acceleration points upward.
At the right side:
acceleration points left.
At the left side:
acceleration points right.
The magnitude may remain constant during uniform circular motion, but its direction continuously changes.
Rotating Wheels
A point on the edge of a rotating wheel experiences centripetal acceleration toward the centre.
A point closer to the centre follows a smaller circular path.
How the accelerations compare depends on what is held constant.
If the points have the same linear speed, the smaller-radius point has greater centripetal acceleration.
However, points fixed on the same rigid rotating wheel share the same angular speed, and points farther from the axis have greater linear speed. In that case, their centripetal acceleration increases with radius.
This distinction is important in more advanced circular-motion analysis.
Period and Centripetal Acceleration
The period, T, is the time required for one complete revolution.
The distance travelled in one revolution is:
2πr
Therefore:
v = 2πr/T
Substituting this into:
a_c = v²/r
gives:
a_c = 4π²r/T²
This equation allows us to calculate centripetal acceleration using radius and period.
Example 8: Using Period
An object moves in a circle of radius:
2 m
and completes one revolution every:
4 s
Use:
a_c = 4π²r/T²
Substitute:
a_c = 4π²(2)/4²
a_c = 8π²/16
a_c ≈ 4.93 m/s²
Therefore:
a_c ≈ 4.9 m/s² toward the centre.
Frequency and Centripetal Acceleration
Frequency is:
f = 1/T
Since:
v = 2πrf
we can substitute into the centripetal acceleration equation:
a_c = (2πrf)²/r
Therefore:
a_c = 4π²rf²
This is useful for rotating systems where frequency is known.
Example 9: Using Frequency
A point moves in a circle of radius:
0.50 m
at a frequency of:
2 Hz
Use:
a_c = 4π²rf²
a_c = 4π²(0.50)(2²)
a_c = 8π²
a_c ≈ 79 m/s²
This is much larger than Earth's gravitational acceleration.
Rapid rotation can therefore produce very large centripetal accelerations.
Centrifuges
Laboratory centrifuges demonstrate this dramatically.
A centrifuge rotates samples at high speed.
Because:
a_c ∝ v²
high rotational speeds can produce accelerations many times greater than gravitational acceleration near Earth's surface.
This allows substances with different properties to separate more quickly.
Centrifuges are widely used in:
- biology
- medicine
- biotechnology
- chemistry
Comparing Centripetal Acceleration with g
Near Earth's surface:
g ≈ 9.8 m/s²
Suppose a rotating system produces:
a_c = 49 m/s²
Then:
49/9.8 = 5
The acceleration is approximately:
5g
This means its magnitude is about five times Earth's gravitational acceleration.
Centripetal Acceleration and Centripetal Force
Centripetal acceleration and centripetal force are closely connected but are not the same quantity.
Centripetal acceleration:
a_c = v²/r
Centripetal force:
F_c = ma_c
Therefore:
F_c = mv²/r
The acceleration depends on:
- speed
- radius
The required force also depends on:
- mass
A more massive object does not automatically have greater centripetal acceleration, but it requires more force to produce the same acceleration.
Example 10: Finding Speed
An object moves around a circle of radius:
5 m
with centripetal acceleration:
20 m/s²
Find its speed.
Start with:
a_c = v²/r
Rearrange:
v² = a_cr
Substitute:
v² = (20)(5)
v² = 100
v = 10 m/s
Example 11: Finding Radius
An object travels at:
12 m/s
and experiences centripetal acceleration:
8 m/s²
Find the radius.
Start with:
a_c = v²/r
Rearrange:
r = v²/a_c
Substitute:
r = 12²/8
r = 144/8
r = 18 m
Example 12: Comparing Two Cars
Car A:
speed = 10 m/s
radius = 25 m
Car B:
speed = 15 m/s
radius = 45 m
Car A:
a_A = 10²/25
a_A = 4 m/s²
Car B:
a_B = 15²/45
a_B = 5 m/s²
Therefore:
Car B has the greater centripetal acceleration.
Even though Car B travels around a wider curve, its greater speed produces the larger acceleration.
Example 13: Predict Without Calculating
An object moves around a circle.
Its speed triples while the radius becomes three times larger.
How does the centripetal acceleration change?
Speed effect:
3² = 9
Radius effect:
÷ 3
Combined effect:
9 ÷ 3 = 3
Therefore:
centripetal acceleration becomes 3 times greater.
Example 14: Another Comparison
System B has half the speed of System A but the same radius.
Because:
a_c ∝ v²
half the speed gives:
(1/2)² = 1/4
Therefore:
System B has one-quarter of the centripetal acceleration of System A.
Example 15: Radius and Speed Both Change
System A:
v = 4 m/s
r = 2 m
System B:
v = 8 m/s
r = 8 m
System A:
a_A = 4²/2 = 8 m/s²
System B:
a_B = 8²/8 = 8 m/s²
Therefore:
both systems have the same centripetal acceleration.
The increase in speed is balanced by the increase in radius.
Circular Motion in Sports
Centripetal acceleration occurs whenever athletes or sporting equipment follow curved paths.
Examples include:
- cyclists travelling around curved tracks
- speed skaters turning
- hammer throwers rotating before release
- gymnasts performing circular movements
- swinging bats and rackets
Higher speeds and tighter turns produce greater centripetal accelerations.
Amusement Rides
Rotating amusement rides can produce significant centripetal accelerations.
Designers must consider:
- speed
- radius
- acceleration
- forces on riders
- structural limits
Because acceleration depends on speed squared, increasing ride speed can substantially increase the required forces.
Washing Machines
During the spin cycle, clothes move in approximately circular paths around the drum.
The drum provides the inward acceleration needed to continually redirect the clothes.
Water that passes through holes in the drum is no longer constrained to follow the same circular path.
This is another everyday application of circular-motion principles.
Why Radius Can Be Tricky
It is important to ask what is being held constant.
At constant linear speed:
a_c = v²/r
so increasing radius decreases acceleration.
But for points on the same rigid rotating object, they have the same angular speed rather than the same linear speed.
Since:
v = ωr
we can write:
a_c = ω²r
In that situation:
larger radius → greater centripetal acceleration.
Always check what the problem tells you is constant.
Common Mistakes
Mistake 1: Saying an object at constant speed has zero acceleration
If its direction changes, its velocity changes.
Therefore, it accelerates.
Mistake 2: Pointing centripetal acceleration tangent to the circle
Velocity is tangent.
Centripetal acceleration points inward.
Mistake 3: Pointing acceleration outward
Centripetal acceleration always points toward the centre.
Mistake 4: Forgetting to square speed
Use:
v²
not v.
Mistake 5: Using diameter instead of radius
If the diameter is given:
r = diameter/2
Mistake 6: Including mass in the centripetal acceleration equation
Mass is not needed for:
a_c = v²/r
Mass becomes important when calculating force.
Mistake 7: Assuming doubling speed doubles acceleration
Doubling speed produces:
4 times the acceleration.
Mistake 8: Assuming doubling radius doubles acceleration
At constant speed, doubling radius:
halves the acceleration.
Mistake 9: Comparing only the speeds
When comparing two circular systems, consider both:
v² and r
Mistake 10: Confusing centripetal acceleration with centripetal force
Acceleration is measured in:
m/s²
Force is measured in:
N
Did You Know?
Centripetal acceleration can become enormous in rapidly rotating systems.
Because acceleration depends on the square of speed, high-speed centrifuges can produce accelerations many thousands of times greater than g.
This makes centrifuges useful for separating very small particles and biological materials.
The same equation used to analyze a car travelling around a curve can therefore also help describe equipment used in advanced scientific laboratories.
Key Terms
Centripetal acceleration: Acceleration directed toward the centre of a circular path.
Circular motion: Motion along a circular path.
Uniform circular motion: Circular motion at constant speed.
Velocity: Speed in a specified direction.
Tangential velocity: Instantaneous velocity directed tangent to a circular path.
Radius: Distance from the centre of the circle to the moving object.
Period: Time required for one complete revolution.
Frequency: Number of complete revolutions per second.
Centripetal force: The net inward force responsible for centripetal acceleration.
Revolution: One complete trip around a circular path.
Key Equations
Centripetal acceleration:
a_c = v²/r
Finding speed:
v = √(a_cr)
Finding radius:
r = v²/a_c
Speed from period:
v = 2πr/T
Centripetal acceleration using period:
a_c = 4π²r/T²
Frequency:
f = 1/T
Speed using frequency:
v = 2πrf
Centripetal acceleration using frequency:
a_c = 4π²rf²
Connection to force:
F_c = ma_c
Key Relationships
At constant radius:
a_c ∝ v²
Therefore:
2× speed → 4× acceleration
3× speed → 9× acceleration
½ speed → ¼ acceleration
At constant speed:
a_c ∝ 1/r
Therefore:
2× radius → ½ acceleration
3× radius → ⅓ acceleration
½ radius → 2× acceleration
Key Takeaways
- Centripetal acceleration is the acceleration directed toward the centre of a circular path.
- An object moving at constant speed around a circle is still accelerating.
- This happens because its velocity changes direction continuously.
- Velocity points tangent to the circular path.
- Centripetal acceleration points toward the centre.
- The magnitude of centripetal acceleration is a_c = v²/r.
- Centripetal acceleration does not directly depend on mass.
- Increasing speed increases centripetal acceleration according to the square of speed.
- Doubling speed quadruples centripetal acceleration.
- Tripling speed increases centripetal acceleration by a factor of nine.
- At constant speed, increasing radius decreases centripetal acceleration.
- Doubling radius halves centripetal acceleration.
- Tight curves produce greater centripetal acceleration than wide curves at the same speed.
- When both speed and radius change, both effects must be considered.
- Period and frequency can also be used to calculate centripetal acceleration.
- Centripetal acceleration is related to centripetal force through F = ma.
- Cars, satellites, planets, Ferris wheels, centrifuges, rotating machinery, sports, and amusement rides all involve centripetal acceleration.
- When comparing circular systems, a useful reasoning process is:
compare speed → square the speed change → compare radius → combine the effects → determine the relative centripetal acceleration.