Uniform circular motion

Learning outcomes

Misconceptions Teaching challenges Common sense suggests there is an outward (centrifugal) force. Students often get the impression that ‘centripetal force’ is a new force, when the term simply describes the direction of existing forces. Teaching challenges Convincing students that something travelling at a constant speed is accelerating. Introducing the radian as a measurement unit for angles. Analysis of the motion of moons and planets needs the relationship but this may not have been taught.

Newton’s conceptual leap “The supreme act of imagination in the construction of modern dynamics” - Richard Westfall (1971) Force in Newton’s Physics

Getting a feel for circular motion In threes: Do PP class experiment: Whirling a rubber bung on a string, answering associated questions (on small sheet). To be followed by: PP demo experiment Introducing circular motion

Examples Discuss, in pairs conker on a string clothes in a spin drier blood sample in a centrifuge child on a playground roundabout car, bus or train going round a corner Olympic sport ‘throwing the hammer’ cycle racing on an indoor track Discuss, in pairs In each case, what force keeps the object moving in a circle?

A video clip Bowling ball and mallet Discuss, in pairs What does this video demonstrate about force and motion? How does it relate to this diagram of a planetary orbit, from Newton’s Principia? 7

Vector analysis 1: acceleration of a falling object 8

Straight line motion is easy 9

Vector analysis 2: acceleration & velocity in different directions Projectile motion: horizontal and vertical motions are independent, so analyse these separately. Circular motion: the direction of motion is constantly changing. Use a more complex diagram that shows changes over very short time intervals. Use Advancing Physics diagrams. 10

Centripetal acceleration Magnitude Direction constantly changing but always acts towards the centre of the circle. and so force

Experimental test of F = mv²/r Measure tension, F bung mass m radius r periodic time, T Calculate mv2/r and compare with F. VPLab simulation Circular motion

The Earth and Moon If the force acting on an object is always at right angles to its velocity (momentum), then the object moves with constant speed in a circle. Describe the force that keeps the Moon in orbit round the Earth as a centripetal force if you want, but remember it’s GRAVITY. Be careful you don’t confuse students (or yourself). Real forces can act centripetally. 13

What to measure? When things moving in circles, or parts of circles (arcs), you can often directly measure angle of rotation rate of rotation; you must calculate distance travelled orbital speed.

In radian world s = r θ defines the radian. When θ = 1 radian, s = r.

The radian Another way of measuring angles 2π (~ 6) radians in a circle How many radians in a right angle? 1 rad = 57.2958 degrees 57°17′45″ ′ = minutes of arc (1/60 of a degree) ″ = seconds of arc (1/60 of a minute) 16

Angular velocity,   = /t linear motion rotational s  v  t Another key quantity. The number of radians swept out per second (rad s-1)  = /t linear motion rotational s  v  t Yes, it’s not really a vector - pesudovector 17

Orbital speed from   and periodic time acceleration & force periodic time, T = time for one revolution acceleration & force

D R G ? 2π rad = 360 degrees

Rotation in a vertical circle PP experiment Looping the loop Rotating a bucket of water Draw the free body force diagram. Vector analysis shows how the forces acting (weight, central force such as tension) combine differently as the object circles round.

More free body force diagrams conical pendulum car/bike/train on a ramped curve roller coaster

Artificial satellites PP experiment: Sketching a satellite orbit and predicting its period This represents part of a circular orbit for a satellite at an altitude of 200 km. ‘Apparent weightlessness’: An orbiting spacecraft and its contents are in free fall.

Kepler’s third law If for each planet you take an average radius, R, and time interval the planet takes to go once round its orbit (its year), T, then the ratio R3/T2 is the same for all planets.

Law of gravitation G = universal gravitational constant = 6.67 × 10-11 m3 kg-1 s-2 m1 and m2 = masses of interacting bodies r = distance between their centres

Endpoints NOTE: In circular motion, F is always perpendicular to s. This means that no energy is required e.g. to keep planets in orbit.