Circular motion and gravitation

Use H1 Physics angular motion and centripetal force, then apply gravitation to circular orbits and geostationary satellites.

GCE A-Level H1 Physics 2027
5 lessons

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Before you begin

A car on a bend, a ball on a string and a satellite in orbit all move in circles, and they all need a resultant force towards the centre. The lessons start by measuring rotation in radians and linking angular velocity to speed. They then find the centripetal acceleration and the force needed to produce it, introduce Newton’s law of gravitation, and use gravity as the centripetal force to work out the speed and period of an orbit, finishing with geostationary satellites.

Be comfortable with: Newton’s laws of motion, because every circular motion problem starts by finding the resultant of the real forces.

Learning goals

Express angular displacement in radians and use s = rθ.
Relate angular velocity, period, frequency and tangential speed using v = rω.
Explain and apply centripetal acceleration and resultant-force relationships.
Apply Newton's law of gravitation to point and spherical masses.
Derive and apply gravitational field strength, including the near-surface model.
Analyse circular gravitational orbits and geostationary satellite conditions.

Syllabus statements covered

express angular displacement in radians
show an understanding of and use the concept of angular velocity
recall and use v = rω to solve problems
show an understanding of centripetal acceleration in the case of uniform motion in a circle, and qualitatively describe motion in a curved path (arc) as due to a resultant force that is both perpendicular to the motion and centripetal in direction
recall and use centripetal acceleration a = rω2, and a = v2/r to solve problems
recall and use F = mrω², and F = mv²/r to solve problems
recall and use Newton’s law of gravitation in the form F = Gm₁m₂/r²
show an understanding that near the surface of the Earth, gravitational field strength is approximately constant and equal to the acceleration of free fall
analyse circular orbits in inverse square law fields by relating the gravitational force to the centripetal acceleration it causes
show an understanding of satellites in geostationary orbit and their applications

Lessons

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Topic reference

Quick reference

Idea	Relationship	Note
angle in radians	θ = s/r	2π rad = 360°
angular velocity	ω = Δθ/(Δ t) = 2π/T = 2π f	unit rad s⁻¹
speed on a circle	v = rω	ω in rad s⁻¹
centripetal acceleration	a = rω² = v²/r	towards the centre
centripetal force	F = mrω² = mv²/r	the inward resultant of the real forces
Newton’s law of gravitation	F = Gm₁m₂/r²	r from centre to centre
gravitational field strength	g = GM/r²	about 9.81 N kg⁻¹ at the Earth’s surface
circular orbit	GMm/r² = mv²/r, so v = square root of (GM/r)	satellite’s mass cancels
orbital period	T = (2π r)/v, so T² = 4π²r³/GM	same central mass: T² ∝ r³

Core knowledge to remember

Angular displacement is expressed in radians. One radian is the angle subtended at the centre of a circle by an arc equal in length to the radius.
Angular velocity is the rate of change of angular displacement.
In uniform motion in a circle, the centripetal acceleration is directed towards the centre. Motion in a curved path is due to a resultant force that is both perpendicular to the motion and centripetal in direction.
Newton’s law of gravitation: two point masses attract each other with a force proportional to the product of their masses and inversely proportional to the square of their separation.
Near the surface of the Earth, the gravitational field strength is approximately constant and equal to the acceleration of free fall.
In a circular orbit, the gravitational force provides the centripetal acceleration.
A geostationary satellite has a circular orbit in the plane of the Equator, moves in the same direction as the Earth’s rotation and has a period equal to the Earth’s rotation period, so it stays above one point on the Equator. It is used for communications and weather monitoring.

Common mistakes

Using degrees, or revolutions per minute, in v = rω or s = rθ.
Saying an object moving in a circle at constant speed is not accelerating.
Drawing a “centripetal force” as an extra force alongside tension, friction or gravity.
Using the height above a planet’s surface instead of the distance from its centre in F = Gm₁m₂/r².
Saying there is no gravity in orbit. Orbiting astronauts feel weightless because they are in free fall.
Giving only the 24-hour period as the condition for a geostationary orbit.