Circular motion flashcards

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Radian
The angle at the centre of a circle subtended by an arc equal in length to the radius. 2π rad = 360°.
Angular displacement
The angle through which a radius line turns about the centre, in rad or degrees.
Angular velocity ω
Angle turned through per unit time, ω = Δθ/Δt. Unit rad s⁻¹.
Period T of circular motion
Time for one complete revolution. T = 2π/ω.
Centripetal acceleration
The acceleration of a body moving in a circle, directed towards the centre; a = v²/r = rω².
Centripetal force
The resultant force towards the centre needed to keep a body moving in a circle; F = mv²/r. It is provided by real forces.
Arc length from angle
s = rθ, with θ in radians. (No — learn it)
Linear speed from angular velocity
v = ωr (Yes)
Period from angular velocity
T = 2π/ω (so ω = 2πf, with f = 1/T). (Yes)
Two forms of the centripetal acceleration
a = v²/r = rω² (Yes)
Two forms of the centripetal force
F = ma = mv²/r = mrω² (Yes)
Minimum speed at the top of a vertical circle
v = √(gr): tension or reaction is zero and mg = mv²/r. (No — derive it)
Conical pendulum angle θ to the vertical
tan θ = v²/(rg), from T cos θ = mg and T sin θ = mv²/r. (No — derive it)
Vector diagram for a = v²/r
Draw v_A and v_B tail to tail with angle Δθ between them; Δv joins their tips and points to the centre; |Δv| = vΔθ.
Direction of velocity and acceleration in uniform circular motion
Velocity along the tangent; acceleration along the radius, towards the centre — always at 90° to each other.
Free-body diagram: car on a banked track
Weight down; normal reaction perpendicular to the track. N cos θ balances the weight, N sin θ points to the centre.
Graph to test F = mrω² (m and r fixed)
F against ω²: straight line through the origin, gradient = mr.
Path of a whirled ball when the string breaks
A straight line along the tangent at the point of release, not along the radius.
Why is a body moving in a circle at constant speed accelerating?
Its velocity changes direction; a change in velocity is an acceleration (towards the centre).
Why is a resultant force needed for circular motion?
With no resultant force a body moves in a straight line (first law); to accelerate towards the centre, F = ma requires a force towards the centre.
Why does the centripetal force do no work?
It is perpendicular to the velocity, so there is no displacement along the force; speed and kinetic energy stay constant.
What provides the centripetal force for a car on a flat bend?
Sideways friction between the tyres and the road, acting towards the centre.
How does banking help a vehicle round a bend?
The tilted normal reaction has a horizontal component towards the centre, so less friction is needed; at one speed none is needed.
Forces at the top and bottom of a vertical loop (rider inside)
Top: N + mg = mv²/r. Bottom: N − mg = mv²/r. The reaction is largest at the bottom.
Same rigid body, different radii: what is the same?
Angular velocity ω is the same everywhere; speed v = ωr is larger further from the axis.
Convert 54° to radians.
54 × π/180 = 0.942 rad.
A record turns at 33.3 rev per minute. Angular velocity?
ω = 2π × 33.3 ÷ 60 = 3.49 rad s⁻¹.
A point 0.30 m from an axis has period 0.50 s. Its speed?
v = 2πr/T = 2π × 0.30 ÷ 0.50 = 3.8 m s⁻¹. (Yes)
A 0.15 kg ball moves at 6.0 m s⁻¹ in a horizontal circle of radius 0.90 m. Tension?
F = mv²/r = 0.15 × 36 ÷ 0.90 = 6.0 N. (Yes)
Minimum speed at the top of a vertical loop of radius 4.0 m?
v = √(gr) = √(9.81 × 4.0) = 6.3 m s⁻¹. (No — derive it)
Centripetal acceleration 0.12 m from the axis of a shaft turning at 50 rev s⁻¹?
ω = 2π × 50 = 314 rad s⁻¹; a = rω² = 0.12 × 314² = 1.2 × 10⁴ m s⁻². (Yes)
A 900 kg car takes a flat bend of radius 60 m at 15 m s⁻¹. Friction needed?
F = mv²/r = 900 × 225 ÷ 60 = 3.4 kN, towards the centre. (Yes)
Angle in degrees in s = rθ or ω = Δθ/Δt
Wrong: these need radians. Convert first (× π/180).
Revolutions per minute given
ω = 2π × (rev min⁻¹) ÷ 60. Do not forget either the 2π or the 60.
Adding a “centripetal force” arrow to a free-body diagram
Don't. Draw only real forces; their resultant towards the centre is the centripetal force.
Radius of a conical pendulum or swing ride
Use the horizontal distance from the axis to the body, not the length of the string or chain.
Speeds at different points on a vertical circle
The speed is not constant: link speeds at different heights with conservation of energy.

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