Motion, Forces and Energy Lecture 5: Circles and Resistance m FrFr FrFr m FrFr A particle moving with uniform speed v in a circular path of radius r experiences an acceleration which has magnitude a = v 2 /r (see Serway Section 4.4 for derivation). Acceleration is ALWAYS directed towards the centre of the circle, perpendicular to the velocity vector. When string breaks, ball moves along tangent. NB The centripetal force is a familiar force acting in the role of a force that causes a circular motion. r

Tension as a centripetal force If the ball has a mass of 0.5kg and the cord is 1.5m long, we can find the maximum speed of the ball before the cord breaks (for a given tension). We assume the ball remains perfectly horizontal during its motion:. The TENSION in the cord PROVIDES the CENTRIPETAL FORCE: m FrFr FrFr r For a tension of say 100 N (breaking strain), the Maximum speed attainable is (100x1.5/0.5) 1/2 = 17.3 ms -1.

Conical Pendulum T mg L  A conical pendulum consists of a “bob” revolving in a horizontal plane. The bob doesn’t accelerate in the vertical direction: The centripetal force is provided by the tension component: (1) / (2) gives Finally we note that r = L sin  so that:

A car moving round a circular curve The centripetal force here is provided by the frictional force between the car tyres and the road surface: The maximum possible speed corresponds to the maximum frictional force, f smax =  s n. Since here n = mg, f smax =  s mg FrFr So for  s = 0.5 and r = 35 m, v max = 13.1 ms -1.

Non-uniform circular motion Sometimes an object may move in a circular path with varying speed. This corresponds to the presence of two forces: a centripetal force and a tangential force (the latter of which is responsible for the change in speed of the object as a function of time. An example of such motion occurs if we whirl a ball tied to the end of a string around in a VERTICAL circle. Here, the tangential force arises from gravity acting on the ball. m T R  mgcos  mgsin  mg centre v top v bot T top T bot mg We must consider both radial and tangential forces!

Analysis Tangential acceleration At the bottom of the path where  = 0 o : At the bottom of the path where  = 180 o : General equation Therefore, the maximum tension occurs at the bottom of the circle, where the cord is most likely to break.

Motion with resistive forces We’ll take a quick look at fluid resistance (such as air resistance or Resistance due to a liquid). Resistive forces can depend on speed In a complex way, but here we will look at the simplest case: R = b v ie the resistance is proportional to the speed of the moving object. mg R v With fluid (air) resistance, the falling object does not continue to accelerate, but reaches terminal Velocity (see Terminal Velocity Analysis notes).