Rotational Motion NCEA AS 3.4 Text Chapter: 9
Types of Motion • 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 (omega) w (double u) • 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:
Try These • An old record player rotates at 33 rpm. • What is the displacement per min? • What distance would a fly sitting 10cm fromthe centre travel in a minute? • How fast is the earth travelling through spaceif it is 0.15Tm from the sun
33x2x=207rads-1 • 207 x 0.1 = 21m • How fast is the earth travelling through spaceif it is 0.15Tm from the sun
Graphs Area under graph = angular displacement q Gradient = angular acceleration a
Kinematic Equations • Recognise these??: • Use them the same way you did last year.
The CD • A CD reads from the inside to outside • They used to read 4.3 megabits per second per channel (probably much higher now) • They require a constant linear velocity of 1.40ms-1 • The disc needs to rotate at 500rpm at the start and 200rpm at the finish. 1) Convert 500rpm into rads-1
2) A CD can reach correct w in 1 revolution what is a? 3) What is the radius of the disc at the start? 2.7cm
21rads-1 4) Convert 200rpm to rads-1 A particular CD has a 72 minute playing time 5) Calculate a as the disc plays from start to finish. 6) Calculate the angle the disc moves through during this time 7) Convert this to revolutions 8) Calculate the radius of the disc at the finish. 5) -7.3x10-3rads-2 6) 158000 approx radians 7) 25000 rev 8) 6.7cm
Answer this by yourself Bill and Ben are riding on a merry go round. Bill rides on a horse at the outer rim, twice as far from the center as Ben, who rides on an inner horse. When the merry go round is rotating at a constant angular speed, is Bill’s tangential speed is • Twice Ben’s • The same • Half of Ben’s • Impossible to determine.
Co-operative Challenge Make an order of magnitude estimate of the number of revolutions through which a typical car tyre turns in a year. State the quantities you measure or estimate and their values.
Answer this by yourself Bill and Ben are riding on a merry go round. Bill rides on a horse at the outer rim, twice as far from the center as Ben, who rides on an inner horse. When the merry go round is rotating at a constant angular speed, Bill’s angular speed is • Twice Ben’s • The same • Half of Ben’s • Impossible to determine.
Solve This • What is the angular speed and acceleration of the second hand?
5.00rev/s = 31.4rads-1 Solve This The tub of a washer goes into a spin cycle, starting from rest and gaining angular speed steadily for 8.00s, at which time it is turning at 5.00rev/s. At this point Eiva opens the lid, and a safety switch turns off the washer. The tub smoothly slows to rest in 12.0s.Through how many revolutions does the tub turn while it is in motion? 50 revs
Try This Estimate or calc. the mass of the sun 2.02x1030kg
Remember this? Satellites • By equating Newton’s Law with centripetal force and cancelling: ie. All satellites must have the same orbital speed for a given radius regardless of their mass.
F r Torque • Torque is the turning effect of a force couple. • Symbol: t • Measured in Newton meters (Nm) • Acts clockwise or anticlockwise • Force and distance from pivot must be perpendicular • Does this cause pure rotation? NO
What is Torque? • Objective to find the relationship between t and a
r m Find the relationship between t and a • Assumption: In this demo it is OKto let FT=Fw for small accelerations. • Normally FR=Fw-FT • FT causes the torque • So what are we going to do? • This is great training for 3.1
Torque • Just as force causes linear acceleration, torque causes angular acceleration. • Compare with F=ma • 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
Your Inertia • Calculate the approximate I value for your body. State any assumptions you make.
r m Example • A mass of 0.10kg is used to accelerate a fly-wheel of radius 0.20m. The mass accelerated downwards at 1.0ms-2. • What is the torque? • g=10ms-2 today
Example FT m FR FW
Problem • Bruce 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 Ivys’ turn to push. Bruce and Ben 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 6000kgm2. • If Ivy pushes with the same force of 120N for 25s, what will the final angular speed of the roundabout be now? 0.75rads-1
Answer these Mass and it’s distribution • Which quantities affect rotational inertia? • What are the units of I? • Where does a force act for an object to be travelling with translational motion only? • Explain why you are able to travel at a higher speed around a corner if it is banked. Kgm2 The horizontal componentof the reaction force provides an additional centripetal force whichallows a vehicle to travelfaster without slipping ata tangent. Through COM
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.
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.
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.
Problem: • Tom is listening to some records one Sunday afternoon. His new turntable (I=0.10kgm2) is spinning freely (ie no motor) with an angular velocity of 4.0rads-1, when he drops a record (I=0.02kgm2) onto it from directly above. What is the angular speed now? w = 3.3 rads-1
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.
Roundabout • 7 Physics students get on a roundabout and get it spinning. They then climb towards the middle. • Discuss what happens. Discuss means lots of detail
Hint: want one? • What speed would give you a centripetal force = weight force?? Problem • Sean 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. g is still 10ms-1
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?