Circular motion
Use radians, angular velocity and centripetal acceleration to solve H2 Physics circular-motion problems, including banked bends and vertical circles.
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Before you begin
A body moving in a circle at constant speed is still accelerating, because the direction of its velocity keeps changing. The lessons start by measuring angles in radians, define angular velocity and link it to the speed of a point on the circle. They then derive the centripetal acceleration and use Newton’s second law to find the resultant force needed, and apply it to banked bends, conical pendulums and vertical circles.
Be comfortable with: Newton’s laws of motion, because every circular-motion problem is Newton’s second law with an acceleration towards the centre.
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.
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
Lessons
Work through them in order.
Core lessons
- Angular displacement in radiansDefine the radian, convert between degrees and radians, and use s = rθ and angular displacement for turns that may reverse.
- Angular velocity and tangential speedDefine angular velocity, link it to period and frequency, and use v = rω for points on a wheel and for wheels joined by a belt.
- Centripetal acceleration and centripetal forceDerive the centripetal acceleration v²/r = rω², and use F = mv²/r with the real forces on a body moving in a circle.
- Banked bends and the conical pendulumResolve the normal force on a frictionless banked bend, or the tension in a conical pendulum, to find the speed from tan θ = v²/rg.
- Forces in a vertical circleWrite the force equation at the top, bottom and side of a vertical circle, and find the slowest speed at which a string stays taut at the top.
Practise and check
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Topic reference
Quick reference
| Idea | Relation | Notes |
|---|---|---|
| angle in radians | θ = s/r | 2π rad = 360° |
| angular velocity | ω = Δθ/(Δ t) = 2π/T = 2π f | rad s⁻¹ |
| speed of a point on the circle | v = rω | along the tangent |
| centripetal acceleration | a = v²/r = rω² | towards the centre |
| centripetal force | F = mv²/r = mrω² | resultant of the real forces |
| banked bend, no friction | tan θ = v²/rg | also the conical pendulum |
| top of a vertical circle | FT + mg = mv²/r | string taut if v ≥ square root of gr |
| bottom of a vertical circle | FT-mg = mv²/r | tension greatest here |
Core knowledge to remember
- One radian is the angle at the centre of a circle when the arc length equals the radius.
- Angular velocity is the rate of change of angular displacement.
- In uniform circular motion the speed is constant but the velocity is not, so the body has an acceleration, v²/r, directed towards the centre.
- A body moves along a curved path because of a resultant force that is at right angles to its motion and directed towards the centre of the curve. This resultant, the centripetal force, does no work and does not change the speed.
- The centripetal force is not an extra force. It is provided by real forces such as tension, friction, a normal force or gravity.
Common mistakes
- Using degrees in s = rθ, or f in place of ω in v = rω.
- Saying a body moving in a circle at constant speed has no acceleration.
- Drawing “centripetal force” as an extra arrow on a free-body diagram.
- Saying a body flies outwards along the radius when the string breaks; it moves off along the tangent.
- Using the same tension equation at the top and the bottom of a vertical circle.