Circular motion: Class 11 Physics

Key points

The angle through which a body turns is its angular displacement θ. It is measured in radians. One radian is the angle at the centre of a circle that is subtended by an arc equal in length to the radius. The arc length is s = rθ, and 360° = 2π rad, so 1 rad = 57.3°.

Angular velocity ω is the rate of change of angular displacement, ω = θ/t, in rad/s. For one full turn in period T, ω = 2π/T = 2πf, where f is the frequency in hertz. A speed given in revolutions per minute (rpm) is changed to rad/s by multiplying by 2π/60. Angular acceleration α is the rate of change of ω, in rad/s². The linear and angular quantities are linked by v = rω and a_t = rα.

In uniform circular motion the speed is constant but the velocity keeps changing because its direction changes. So the body is accelerating. This acceleration points towards the centre and is called the centripetal acceleration: a_c = v²/r = rω². The force that causes it is the centripetal force F = mv²/r = mrω². It acts towards the centre and does no work, because it is always perpendicular to the motion.

Centripetal force is not a new kind of force. It is the name for the net inward force, which can be supplied by:

• tension in a string, for a stone whirled in a circle
• friction between the tyres and the road, for a car on a flat curve
• gravity, for a planet or satellite in orbit
• a component of the normal reaction on a banked road

For a car on a flat circular road, the maximum safe speed is found by equating friction to the centripetal force: μmg = mv²/r, so v_max = √(μgr). A banked curve of angle θ lets a car turn safely without relying on friction when tan θ = v²/(rg). The 'centrifugal force' felt by a passenger is only the effect of inertia, because the body tends to move in a straight line.

The moment of inertia I is the rotational equivalent of mass. For a particle, I = mr², and for a system, I = Σmr². It depends on the mass and on how the mass is spread about the axis. Some standard values: a thin ring about its central axis MR², a solid disc about its central axis ½MR², a solid sphere about a diameter ²⁄₅MR², and a thin rod about its centre ¹⁄₁₂ML².

Rotational dynamics parallels linear motion:

• Torque τ = Iα, like F = ma.
• Rotational kinetic energy = ½Iω², like ½mv².
• Angular momentum L = Iω, like p = mv. For a particle, L = mvr. The unit is kg m²/s.
• Torque is the rate of change of angular momentum, τ = ΔL/Δt.

The law of conservation of angular momentum: if no external torque acts, L = Iω stays constant. When a skater pulls in her arms, I decreases, so ω increases. A diver tucks into a ball to spin faster. A planet moves faster when it is nearer the Sun for the same reason.

Newton's law of gravitation: every two masses attract each other with a force that is directly proportional to the product of the masses and inversely proportional to the square of the distance between their centres: F = Gm₁m₂/r². G = 6.67 × 10⁻¹¹ N m² kg⁻² is the universal gravitational constant. The acceleration due to gravity at the surface is g = GM/R², and it decreases with height and is about 9.8 m/s² on Earth.

Satellites and orbits:

• A satellite in a circular orbit has gravity as its centripetal force: GMm/r² = mv²/r, so the orbital speed is v = √(GM/r).
• Close to the Earth's surface, v = √(gR) which is about 7.9 km/s, with a period near 84 minutes.
• Escape speed is the minimum speed needed to leave the Earth's gravity: v_e = √(2gR), about 11.2 km/s.
• A geostationary satellite has a period of 24 hours, orbits above the equator at a height of about 36 000 km, and stays above one point on Earth. It is used for communications.
• Objects in an orbiting satellite feel weightless because they and the satellite are in free fall together.

Study tip: In circular motion questions, first identify which force points to the centre, and set it equal to mv²/r. Convert all angles to radians and all speeds to SI units. Learn the derivations of orbital and escape speed, as they are frequently asked.

Definitions and flashcards

Say the meaning first, then open the card to check.

Formula sheet

Print this page

Short questions with model answers

These are the kind of questions asked for 2, 3 and 5 marks. Try each one before opening the answer.

Numericals with worked steps

Common mistakes

• Calling centripetal force an extra force on the free body diagram. It is the net inward force, not a separate one.
• Using degrees instead of radians in s = rθ and v = rω.
• Forgetting to convert rpm to rad/s, or using the diameter instead of the radius.
• Saying that astronauts feel weightless because gravity is zero. Gravity is still strong; they are in free fall.
• Using the height above the surface as r in the gravitation formula instead of the distance from the centre of the Earth.

Practise this chapter

Practice questions and timed test Printable worksheet