Circular Motion
Understand how objects move in circular paths, the forces that keep them there, and how these principles apply to banked curves and orbital mechanics.
Uniform Circular Motion
An object moving in a circle at constant speed is undergoing uniform circular motion. Although the speed is constant, the velocity is continuously changing because the direction of motion changes at every point. This means the object is always accelerating, even if it never speeds up or slows down.
Velocity and Acceleration in Circular Motion
Centre
v (tangent)
ac (inward)
Velocity is always tangent to the circle. Centripetal acceleration always points toward the centre.
Key Equations
Centripetal acceleration: ac = v2 / r
Period of revolution: T = 2πr / v
Angular velocity: ω = 2π / T = v / r
Centripetal acceleration (angular form): ac = ω2r
Important: Centripetal acceleration is not a new type of force. It is the net acceleration directed toward the centre, caused by whatever force (tension, gravity, friction, normal force) keeps the object on its circular path.
Centripetal Force and Banked Curves
By Newton's second law, the net inward force producing circular motion is the centripetal force: Fc = mv2/r. This is not a separate force but the resultant of real forces acting on the object.
Flat Road Turns
On a flat road, friction provides the centripetal force.
Fc = f = μsmg
Maximum speed: vmax = √(μsgr)
Banked Curves
Banking the road tilts the normal force inward, providing centripetal force even without friction.
Ideal banking angle: tan(θ) = v2 / (rg)
At the ideal speed, no friction is needed.
Force Diagram: Banked Curve (no friction)
Normal Force (N)
Perpendicular to banked surface, tilted inward
N sin(θ) = centripetal
N cos(θ) = mg
Weight (mg)
Acts vertically downward
Orbits and Satellite Motion
For an object in orbit, gravity provides the centripetal force. Setting the gravitational force equal to the centripetal force requirement gives us the orbital speed and period. This applies to satellites, the Moon, and planets.
Deriving Orbital Speed
Gravitational force = Centripetal force
GMm/r2 = mv2/r
Orbital speed
v = √(GM/r)
Orbital period
T = 2πr / v = 2π√(r3/GM)
Key Insight: Weightlessness in Orbit
Astronauts in the International Space Station feel weightless not because there is no gravity, but because both they and the station are in free fall together. They are continuously falling toward Earth but moving sideways fast enough to miss it -- this is what an orbit is. The apparent weightlessness is because there is no normal force acting on them.
Key Vocabulary
Centripetal Acceleration
The acceleration directed toward the centre of a circular path, with magnitude ac = v2/r. It changes the direction of velocity, not its magnitude.
Centripetal Force
The net inward force (Fc = mv2/r) required to maintain circular motion. It is provided by real forces such as gravity, tension, or friction.
Angular Velocity
The rate of change of angular position, measured in rad s-1. Related to linear speed by v = ωr.
Banked Curve
A curved road or track tilted at an angle so that the normal force has a horizontal component directed toward the centre of the curve.
Worked Examples
A 1200 kg car travels around a flat circular track of radius 50 m at 15 m s-1. Find the centripetal acceleration and the required friction force.
Step 1: Centripetal acceleration: ac = v2/r = (15)2/50 = 225/50 = 4.5 m s-2
Step 2: Centripetal force (friction): Fc = mac = 1200 × 4.5 = 5400 N
Answer: The centripetal acceleration is 4.5 m s-2 and friction must provide 5400 N directed toward the centre.
A road curve has radius 200 m and is banked at 12 degrees. Find the ideal speed (no friction needed). Use g = 9.8 m s-2.
Step 1: Use tan(θ) = v2/(rg).
Step 2: v2 = rg tan(θ) = 200 × 9.8 × tan(12°) = 200 × 9.8 × 0.2126 = 416.7
Step 3: v = √416.7 ≈ 20.4 m s-1 (about 73 km h-1)
A satellite orbits Earth at an altitude of 400 km. Find its orbital speed. (MEarth = 5.97 × 1024 kg, REarth = 6.37 × 106 m, G = 6.67 × 10-11 N m2 kg-2)
Step 1: Orbital radius r = REarth + altitude = 6.37 × 106 + 4.00 × 105 = 6.77 × 106 m
Step 2: v = √(GM/r) = √(6.67 × 10-11 × 5.97 × 1024 / 6.77 × 106)
Step 3: v = √(5.88 × 107) ≈ 7670 m s-1 (about 7.7 km s-1)
Answer: The satellite orbits at approximately 7.7 km s-1, completing one orbit in about 92 minutes.
Knowledge Check
Select the correct answer for each question. Click "Check Answer" to see if you are right.
Question 1
In uniform circular motion, the centripetal acceleration is directed:
Question 2
If the radius of a circular path is doubled while speed remains constant, the centripetal acceleration:
Question 3
On a banked curve with no friction, the centripetal force is provided by:
Question 4
For a satellite in a stable circular orbit, if the orbital radius increases, the orbital speed:
Question 5
A ball on a string is whirled in a horizontal circle. If the string breaks, the ball will:
Key Concepts Summary
- ●In uniform circular motion, speed is constant but velocity changes direction continuously, producing centripetal acceleration ac = v2/r directed toward the centre.
- ●The centripetal force Fc = mv2/r is the net inward force, not a separate force -- it is provided by tension, gravity, friction, or the normal force.
- ●On banked curves, the horizontal component of the normal force provides centripetal force: tan(θ) = v2/(rg).
- ●For orbital motion, gravity provides the centripetal force, giving orbital speed v = √(GM/r) and showing that more distant orbits are slower.
- ●If the centripetal force is removed, the object moves in a straight line tangent to the circle (Newton's first law).