Circular Motion Calculator Centripetal Force & Velocity
Calculate angular velocity, centripetal acceleration, centripetal force, and banked curves with step-by-step solutions. Perfect for GCSE and A-Level Physics.
Angular velocity relates to frequency, period, and linear velocity
Solve for:
Calculate from:
Quick examples:
Angular velocity from frequency
GCSEω in rad/s, f in Hz
Angular velocity from period
GCSET is the period in seconds
Linear velocity
GCSERelates linear and angular velocity
Centripetal acceleration
GCSEAlways toward the centre
Centripetal force
GCSENewton's 2nd law for circular motion
What is Circular Motion?
Circular motion is movement along a circular path at constant speed. Although the speed is constant, the velocity is always changing because direction changes continuously. This change in velocity means there must be acceleration - called centripetal acceleration - directed toward the centre.
Key Concepts
- • Constant speed but changing velocity
- • Centripetal means "centre-seeking"
- • Velocity is tangent to the circle
- • Acceleration points toward the centre
Real World Examples
- • Car on roundabout - friction provides F
- • Ball on string - tension provides F
- • Planet orbiting star - gravity provides F
- • Roller coaster loop - normal + gravity
Angular vs Linear Velocity
In circular motion, we use two types of velocity: linear velocity (v) in m/s, and angular velocity (ω) in rad/s.
ω = 2πf
From frequency
ω = 2π/T
From period
v = ωr
Link between v and ω
Key insight: All points on a rotating object have the same angular velocity ω, but points further from the centre have higher linear velocity v = ωr.
Centripetal Force Explained
Centripetal force is the resultant force acting toward the centre of the circle. It is NOT a separate type of force - it is provided by existing forces!
F = mv²/r = mω²r
What Provides Centripetal Force?
- • Car turning → Friction
- • Ball on string → Tension
- • Satellite orbit → Gravity
- • Banked curve → Normal force component
- • Centrifuge → Normal force from wall
Common Misconceptions
- • ❌ Centripetal force is NOT a separate force
- • ❌ Centrifugal force is NOT real (in inertial frames)
- • ❌ Objects are NOT pushed outward
- • ✓ Inertia makes objects tend to go straight
Vertical Circles
A-LevelIn vertical circles, gravity affects the motion, so tension varies with position. The tension is maximum at the bottom and minimum at the top.
Minimum speed at top
v = √(gr)
When T = 0
Tension at top
T = m(v²/r - g)
Weight helps!
Tension at bottom
T = m(v²/r + g)
Must support weight
Exam tip: At the top, both tension AND weight act toward the centre (down). At the bottom, tension acts up (toward centre) while weight acts down (away from centre).
Banked Curves
A-LevelOn a banked curve, the road is tilted at angle θ. The horizontal component of the normal force provides centripetal force, allowing vehicles to turn without needing friction at the ideal speed.
tan θ = v²/(rg)
The "ideal" banking angle for speed v
At Ideal Speed
- • No friction needed
- • Normal force provides all centripetal force
- • Most comfortable for passengers
- • Used in velodrome, NASCAR tracks
At Other Speeds
- • Too fast: friction needed up the slope
- • Too slow: friction needed down the slope
- • Maximum safe speed depends on μ
- • Ice/rain reduces safe speed range
Worked Examples
Example 1: Angular velocity from frequency
The UK mains frequency is 50 Hz. Find the angular velocity.
Example 2: Centripetal acceleration
A car travels at 10 m/s around a curve of radius 5 m. Find the centripetal acceleration.
Example 3: Centripetal force
A 2 kg ball on a string moves at 10 m/s in a circle of radius 5 m. Find the tension.
Example 4: Minimum speed at top of loop
A roller coaster loop has radius 10 m. What is the minimum speed at the top?
Example 5: Banked curve angle
A car travels at 20 m/s around a curve of radius 100 m. Find the banking angle.
θ = arctan(0.408) ≈ 22.2°
Example 6: Tension at top of vertical circle
A 0.5 kg ball swings at 15 m/s at the top of a 10 m radius circle. Find the tension.
T = 0.5 × (22.5 - 9.81) = 0.5 × 12.69 ≈ 6.3 N
Common Mistakes to Avoid
Treating centripetal force as a separate force - it's the resultant!
Confusing rpm with rad/s. Always convert: ω = 2π × (rpm/60).
Thinking centrifugal force is real. It's only apparent in rotating frames.
Using degrees instead of radians in angular velocity calculations.
Forgetting that velocity is tangent to the circle, not radial.
Wrong signs in vertical circle equations: check directions carefully!
Frequently Asked Questions
What is centripetal force?
The resultant force toward the centre of a circle. It's provided by tension, friction, gravity, or normal force.
Is centrifugal force real?
No! It's a fictitious force that appears only in rotating reference frames.
What is angular velocity?
Rate of rotation in radians per second (rad/s). One rotation = 2π radians.
How are v and ω related?
Linear velocity v = ωr, where r is the radius.
What is the minimum speed at the top?
v = √(gr), when only gravity provides centripetal force (T = 0).
Why is tension greatest at the bottom?
Tension must overcome weight AND provide centripetal force.
What do banked curves do?
Allow turning without friction by using the normal force component.
How do I convert rpm to rad/s?
ω = 2π × (rpm/60). For example, 60 rpm = 2π rad/s.
Explore More Free Tools
All our tools are 100% free with step-by-step learning