Revision summary

Identify the centre and radius, distinguish speed from velocity, and name the actual forces before writing an inward-force equation.

Angle, period and frequency

T is time per complete revolution; f is revolutions per second. In these magnitude equations, ω is angular speed. For signed angular velocity, state the viewing direction and positive rotation sense, then use the rate of signed angular displacement. Convert degrees to radians and revolutions/min to revolutions/s as needed.

Two points on one rigid rotating platform have equal angular speed and period. Their tangential speeds are proportional to their distances from the axis.

With angle measured anticlockwise from +x, x/r = cos θ and y/r = sin θ. Coordinates can be signed even though radius is positive. For small |θ| in radians, sin θ ≈ tan θ ≈ θ and cos θ ≈ 1. A small chord approximates its arc; these approximations are unsuitable for a 135° turn.

Time many complete turns between same-sense crossings to obtain T. Check the radius from the axis, counting, timing, rate stability and repeatability. A calibrated video chord gives an interval estimate, not an exact finite-arc distance; calculate an independent speed if testing v = rω.

Changing velocity at constant speed

Instantaneous velocity is tangent; acceleration is inward and perpendicular to velocity for uniform circular motion. Their directions change around the circle. Constant speed does not mean zero acceleration.

Form Δv by vector subtraction, then divide by Δt for average acceleration. A finite-interval average approaches the instantaneous value as the interval shrinks; do not label the finite construction as exact instantaneous acceleration.

A purely perpendicular resultant changes direction without changing speed and does no instantaneous work. If speed changes on a curved path, a tangential component is also present, so the total resultant need not be perpendicular.

Use the inward resultant, not an extra force

Draw actual interactions on the selected body, resolve them inward and account for other directions. Tension, gravity, friction or contact forces can contribute. There is no additional centripetal interaction to add or outward balancing force to invent in the stated laboratory frame.

For a horizontal string on a smooth table, tension supplies the radial resultant and support balances weight. For a conical pendulum with angle θ from vertical, S cos θ = mg and S sin θ = mv²/r. Use the horizontal orbit radius, not the slanted string length.

At fixed m and r, F is proportional to v². At fixed m and v it is proportional to 1/r; at fixed m and ω it is proportional to r. State what is held fixed. If the horizontal tether is removed and no horizontal force remains, the body initially follows the tangent.

A plot of F against v² has gradient m/r in kg/m. A plot of ln(F/F₀) against ln(v/v₀) has dimensionless gradient 2 for the same fixed-mass, fixed-radius law. Use positive dimensionless ratios and the same logarithm base on both axes. Model-data agreement alone is not experimental verification.

For positive ratios, logarithms turn multiplication into addition, division into subtraction and powers into factors. lg means log₁₀: a difference of 3 in lg values represents a factor of 1000. Do not apply the product rule to a sum.

Gravity and the near-surface approximation

Gravity is attractive, along the joining line. The equal and opposite forces act on different bodies. Use centre separation, not surface gap. G is universal; g depends on location. Doubling centre distance at fixed masses quarters the force and field magnitude.

At height h above spherical Earth, r = R + h. When h is small compared with R, g changes very little; local inward directions are also nearly parallel within a small surface region. The external inverse-square graph applies at and outside the surface, not inside Earth.

When gravity alone acts, a = g inward. Field strength in N/kg and free-fall acceleration in m/s² have equal magnitudes and equivalent units, but different definitions. Support or drag changes the acceleration. An orbiting body's lack of usual support does not mean gravity is zero.

Circular and geostationary orbits

These use a small satellite, a much larger spherical central body treated as fixed, circular motion and negligible drag, thrust or other gravitational sources. Satellite mass cancels from speed and period but still affects the gravitational force. Larger circular radius gives lower speed and longer period.

A geostationary orbit is circular, equatorial, west-to-east with Earth, and has Earth's rotation period. All conditions matter. Use the supplied period consistently: about 86164 s for the sidereal value, or 86400 s when that approximation is given. Subtract R from the calculated r to obtain altitude.

The satellite shares Earth's angular speed while moving and accelerating in an Earth-centred inertial description. Its fixed apparent position supports a fixed-pointing communication antenna and continuing weather observation of a region. It is not overhead at every latitude or equally suited to polar coverage.

Read the named quantity and units: θ and T can denote other quantities elsewhere. Here T is period, while S labels tension. Angular velocity includes a stated rotation sense; angular speed is its magnitude.