Rotational Motion

Rotational Motion NCEA AS 3.4 Text Chapter: 4

The Sun • Use the information you have absorbed this year to estimate the size of the sun. • Need a hint?

CD Study • The CD reads from the inside to the outside.They used to read 4.3 mega bytes per second. • They require a constant linear speed of 1.4ms-1. • This disc needs to rotate at 500rpm at the start and 200rpm at the finish. • a) Convert 500rpm to rads-1 • b) a CD can reach the correct ω in one revolution. What is a? • c) What is the radius of the disc at the start? • A particular CD (Bee Gees) has a playing time of 72 minutes. • d) Convert 200rpm to rads-1.

e) Calculate the angular acceleration as the disc plays from start to finish. • f) Calculate the angle the disc moves through in this time. • g) Convert the angle to revolutions. • h) Calculate the radius of this disc (not Mo’s Rosa) at the finish.

Types • Pure Translation –force acts through the centre of mass, C.o.m moves. • Pure Rotation –2 equal & opposite forces act at a perpendicular distance from the c.o.m (force couple) C.o.m remains stationary, object spins around it • Mixture – single force acts, NOT through c.o.m, object moves and rotates around c.o.m

A B q B A Angular Displacement • Although both points A & B have turned through the same angle, A has travelled a greater distance than B • A must have had the greater linear speed

s r q r Angular Displacement • Symbol q • Measured in radians (rad) • Angular displacement is related to linear distance by:

r Angular Displacement • Remember from Maths: • How to put your calculator into radian mode? • How many radians are in a full circle?

Angular Velocity • Symbol w • Measured in radians per second (rads-1.) • Average angular velocity calculated by:

To put it another way: So angular velocity is related to linear velocity by: s r q r Angular Velocity

Angular Acceleration • Changing angular velocity • Symbol: a • Measured in radians per second squared (rads-2.) • Calculated by:

Angular Acceleration • Angular acceleration and linear acceleration are linked by:

Summary

Gradient = angular velocity w Graphs

Graphs Area under graph = angular displacement q Gradient = angular acceleration a

Kinematic Equations • Recognise these??: • Use them the same way you did last year.

F r Torque • Torque is the turning effect of a force. • Symbol: t • Measured in Newton metres (Nm) • Acts clockwise or anticlockwise • Force and distance from pivot must be perpendicular

r m Example • A mass of 0.1kg is used to accelerate a fly-wheel of radius 0.2m. The mass accelerateds downwards at 1ms-2.

Example FT m FR FW

Torque • Just as force causes linear acceleration, torque causes angular acceleration. • So what is this “I” thing anyway….

Rotational Inertia • Symbol: I • Measured in kgm2 • Rotational inertia is a measure of how hard it is to get an object spinning. • It depends on: • Mass • How the mass is distributed about the axis of rotation

Examples of Inertia’s Solid Cylinder Hollow Cylinder Solid Sphere

Assuming no linear motion of boat – not likely!

Problem • Cameron was pushing his friends on a roundabout (radius 1.5m) at the local park with a steady force of 120N. After 25s it has reached a speed of 0.60rads-1. • What is the torque he is applying? • What is the angular acceleration of the roundabout? • What is the rotational inertia of the roundabout+friends? 180Nm 0.024rads-2. 7500kgm2

Problem • Now it’s Lewis’ turn to push. Cameron and Chris decide to climb into the centre of the roundabout instead of sitting on the seats at the outside. This reduces the inertia of the roundabout + friends to 7000kgm2. • If Lewis pushes with the same force of 120N for 25s, what will the final angular speed of the roundabout be? 0.64rads-1

Problem • Jacob comes along and decides to try and find out what angular speed he would need to spin the roundabout at to make everyone fall off. Assume a 70kg person can hold on with a force equal to their body weight. • Hint: What speed would give you a centripetal force = weight force??

Angular Momentum • Any rotating object has angular momentum, much the same as any object moving in a straight line has linear momentum. • Angular momentum depends on: • The angular velocity w • The rotational inertia I • Symbol: L • Measured in kgm2s-1

Angular momentum • Angular momentum is conserved as long as….. • There are no external torques acting.

Problem: • Lachie is listening to some records one Sunday afternoon. His turntable (I=0.10kgm2) is spinning freely (ie no motor) with an angular velocity of 4rads-1, when he drops a Dire Straits record (I=0.02kgm2) onto it from directly above. What is the angular speed now? 3.33rads-1

Examples: • Helicopters: The blades spin one way so the helicopter body tries to spin the other way – not much use! So we have to supply an external torque (from tail rotor) to keep the body still.

Examples: • Motorbikes (doing wheelies!) – As power goes to the back wheel suddenly to make it spin one way, the bike tries to spin the other way. The weight of the rider and bike body supplies an external torque to keep the front end of bike on the road.

Examples: • Figure Skating – Ice-skaters go into a spin with arms outstretched and a fixed amount of L dependent on the torque used to get themselves spinning. (Once spinning, no external torque) If they then draw in their arms, their inertia decreases, so their angular speed increases in order to keep the total momentum conserved.

Examples: • Balancing on a bicycle – If a stationary bike wheel is supported on one side of the axle, it tips over. If the bike wheel is spinning, it will balance easily when supported on only one side. A large external torque is required to change the direction of the angular momentum.

r v m Angular Momentum • Linear momentum can be converted to angular momentum

Example • A satellite in orbit needs to be turned around. This is done by firing two small “retro-rockets” attached to the side of the satellite. These rockets fire 0.2kg of gas each at 100ms-1. • The satellite has an inertia of 1200kgm2 and the rockets are positioned at a radius of 1.5m • What speed will the satellite turn at?

Solution:

Extra • How would you stop the satellite from rotating once it was in the correct position?? Fire an equal burst of gas from the rockets in the opposite direction to the original.

Problem: • Jethro was at a theme park and wanted a go on the bumper boats. He runs at 4ms-1 and jumps onto a floating boat of radius 1m, landing 40cm from the centre. If the boat + Jethro have an inertia of 100kgm2, what angular speed will they spin at? • What assumption are we making here? (Hint: what kind of motion will be produced?) 1.12rads-1 Assuming no linear motion of boat – not likely!

Assuming no linear motion of boat – not likely!

Rotational Kinetic Energy • The energy of rotating objects Ek(rot)

Example • How much kinetic energy does a 20kgms-1 gear cog have if spinning at 4rads-1?

Conversion of Energy - Example • A toy car operates using a flywheel.The car is pushed across the floor a few times to start the wheel spinning and when let go the rotational kinetic energy is converted to linear kinetic energy as the car moves forward.

Problem • A 65kg trampolinist named Daniel is bouncing on his trampoline so that at the instant he leaves the mat he is travelling at 9ms-1. As he moes upward he curls nto a ball and does a 360° front flip. • How much linear kinetic energy does he have as he leaves the mat? • What happens to this energy? Converted to gravitational and rotational kinetic energy

Problem • If he reaches a maximum height of 3m, how much gravitational energy has he gained? • At the top, he is spinning at 6rads-1. What is his inertia?

Rolling Down Slopes • Which will reach the bottom first?

Rolling Downhill • The ball. • Why? • All have the same Ep to begin with. • The hollow cylinder has the largest I so gains the most Ek(rot) and the least Ek(lin). • It will have the smallest acceleration of rolling – ie will be rolling downhill slower than the others at any given time.

Rotational Motion