PY1008 / PY1009 Physics Rotational motion

Transcription

1 PY1008 / PY1009 Physics Rotational motion M.P. Vaughan

2 Learning Objecties Circular motion Rotational displacement Velocity Angular frequency, frequency, time period Acceleration Rotational dynamics Centripetal force Centrifugal effect Examples: blood centrifuge, planetary orbits Rotational energy Moment of inertia

3 Circular motion

4 Uniform circular motion We shall assume uniform circular motion. This means The radial distance from the centre of rotation does not change An object rotates at a constant rate

5 Circular motion r

6 Rotational displacement r r Rotational displacement: r

7 Velocity r Rotational displacement: r Velocity r t

8 Angular frequency Rotational displacement: r Velocity r t Angular frequency is defined as. t So, r.

9 Time period The time period T is the time it takes to complete a full rotation. Since the circumference of a circle is equal the elocity times the time period But so, T 2r. r, rt 2r, 2 giing T. 2r, this must

10 Frequency Very often, we wish to know the frequency, which is the number of rotations per unit time. Denoting frequency by n, this is related to angular frequency by and to the time period by n n 2 The SI unit of frequency is the hertz 1 Hz = 1 s T.

11 Acceleration Although the magnitude of the elocity (the speed) does not change, the direction does. This means that the object is accelerated.

12 Acceleration r

13 Acceleration Change in elocity:

14 Acceleration We know that acceleration is the rate of change of elocity. Since we hae a t,. t But so... r, t

15 Acceleration Inserting this into gies t a a r., t r 2.

16 Acceleration Alternatiely, in terms of angular frequency we hae Since a, t t r,. we hae 2 a r.

17 Acceleration direction Direction of acceleration

18 Acceleration direction r Direction of acceleration The direction of acceleration is towards the centre of rotation.

19 Uniform circular motion summary Velocity Angular frequency r t Time period T 2 Acceleration (directed towards centre of rotation) a r 2 or 2 a r

20 Rotational dynamics

21 Centripetal force Circular motion implies and acceleration a r 2 or acting towards the centre of rotation. By Newton s Second Law F ma, 2 a r a body of mass m therefore requires a force 2 F mr acting towards the centre of rotation to maintain circular motion.

22 Centripetal force This force F m r 2 or 2 F mr acting towards the centre of rotation, is known as a centripetal force. Do not confuse this with the so-called centrifugal force. This is not a true force! Rather, it is a fictitious force that feels like a force (acting away from the centre of rotation) that should be referred to as the centrifugal effect.

23 Centripetal force F m r 2 F m or t 2 F mr Note: t

24 Centrifugal effect Why do we experience the centrifugal effect? Firstly, recall Newton s First Law that If no net force acts on an object, then it remains in a state of rest or constant elocity Now an object undergoing uniform circular motion is not moing with a constant elocity, since elocity inoles a direction as well as a magnitude (speed).

25 Centrifugal effect Imagine two occupants in a car about to round a corner. This turning motion requires a force to act on the car* acting towards the centre of rotation. Top iew Rear iew *coming from the frictional forces of the tyres in contact with the road.

26 Centrifugal effect Initially, howeer, the occupants of the car will not be acted on by the force directly. At first, these occupants will then continue to moe in a straight line (i.e. with constant elocity). Top iew Rear iew F

27 Centrifugal effect As the car turns, the frame of reference it defines will start to moe towards the centre of rotation. F Relatie to the car s frame of reference, the occupants, will experience this as being propelled towards the outer side of the car. Top iew Rear iew

28 Centrifugal effect The car body will then start to transmit the turning force to the occupants. F The occupants are then turned around the corner along with the car due to this force. Top iew Rear iew