Physics - Motion in a Circle

What is meant when an object is in uniform circular motion?

An object rotating at a steady rate is said to be in uniform circular motion.

What is the equation for the speed of a point on the perimeter of a wheel?

v = (2πr)/(T) = 2πrf

r is the radius of the wheel

T is the time period of one rotation

f is the frequency of rotation

What is angular displacement?

The angular displacement, θ of the object in time t is 2πt/T = 2πft, where T is the time for one rotation.

What is angular speed?

The angular speed, ω, is defined as the angular displacement per second.

ω = 2π/T = 2πf

What is centripetal acceleration?

The acceleration of the object is towards the centre of the circle and is called centripetal acceleration.

a = v²/r = ω²r

where v is velocity

where r is radius

where ω is angular speed

What is centripetal force?

The resultant force on an object around a circle at constant speed is called the centripetal force.

F = (mv²)/r = mω²r

where m is the mass of the object

where v is the velocity of the object

where r is the radius

where ω is the angular speed

Describe the forces acting on a car going over a bridge.

Consider a vehicle of mass m moving at speed v along a road that passes over the top of a hill or over the top of a curved bridge. At the top of the hill, the support force S from the road on the vehicle is directly upwards in the opposite direction to its weight, mg. The resultant force on the vehicle is the difference between the weight and the support force. The difference acts towards the centre of curvature of the hill as the centripetal force.

mg - S = mv²/r

The car will lose contact when the support force is 0.

Describe the motion of a car on a roundabout.

Consider a vehicle of mass m moving at speed v in a circle of radius r as it moves around a roundabout on a level road. The centripetal force is provided by the sideways force of friction between the vehicle’s tyres and the road surface.

Force of friction F = mv²/r

For no skidding or slipping to occur, the force of friction between the tyres and the road surface must be less than a limiting value F₀, that is proportional to the vehicle’s weight.

Limiting force of friction F₀ = mv₀²/r

Describe the motion of a car on a banked track.

Take N₁ and N₂ as the normal force at an angle θ to the vertical.

• (N₁ + N₂)sinθ acts as the centripetal force.

  • (N₁ + N₂)sinθ = mv²/r

• (N₁ + N₂)cosθ balances the weight (mg)

  • (N₁ + N₂)cosθ = mg

Therefore tanθ = v²/gr

There is no sideways friction if the speed v is such that v² = grtanθ