MECHANICS 2

Rotational Motion

Teaching Ideas• Outside – object rotating about a fixed

position (line of students link arms, one end stays fixed and the rest sweep around in a circle)– Can you keep the line straight?– Mix up where you are in the line to

experience the rotation from a different position

– Is it easier to rotate about the person in the centre of the line?

Difference between Rotational motion and circular motion

New vocab!• What are the 3 typical ways we can

describe the LINEAR motion of an object?

AccelerationAccelerationVelocityVelocityDisplacementDisplacement

Rotational motion words

d = displacement(m)

θ= angular displacement (rad)v = velocity (ms-

1)ω= angular velocity (rads-1)

α = angular acceleration (rads-2) a = acceleration (ms-2)

θ

v = ∆d = ∆θ ∆t ∆t

a=∆v α=∆ω ∆t ∆t

Relationship between distance and angular displacement

when θ in radians

d=r θ

d

v=r a=rα

Teaching Idea• Re-write equations of motion with the

new terminology for rotational motion– Build up the four EoM using the jigsaw

pieces– Replace the three (/four) variables with

our new variables

Constant Accelerationvf = vi + at ωf=ωi

+αtd = vi t + ½ at2

vf2 = vi

2 + 2ad

ωf2=ωi

2+2αθ2

i αt21tωθ

2)tv(vd fi 2

)tω(ωθ fi

Combined motionCycloid motion• http://www.youtube.com/watch?

v=vkahXgCaHho• http://www.upscale.utoronto.ca/

GeneralInterest/Harrison/Flash/ClassMechanics/RollingDisc/RollingDisc.html

In order to get something to accelerate we need an unbalanced

FORCE

In order to get something to angular accelerate we need an unbalanced

TORQUE

τ=FdForce

Perpendicular distance to the pivot

Pull at various angles on a roll of wire and predict which way the roll of wire will roll.

Note the helpful dotted lines - they might provide a clue.

Mass and Rotational inertiaMASS (m) is a measure of how

hard it is to accelerate of an object with a force

ROTATIONAL INERTIA (I) is a measure of how hard it is to angular accelerate an object using a torque

Newton’s 2nd law: F=ma. Mass is measured in kg

Newton’s 2nd law τ=Iα Rotational inertia is measured in kg m2

Rotational inertia depends on the mass

of the object and on the distribution of mass around the axis of rotation

MORE ROTATIONAL MOTION QUANTITIESTRANSLATIONAL ROTATIONALF Forcem massF=ma

MORE ROTATIONAL MOTION QUANTITIESTRANSLATIONAL ROTATIONALF Force (N) τ Torque (Nm)m mass (kg) I rotational inertia

(kgm2)F=ma τ=Iα

Rotational inertia for masses moving in a circle

I=mr2

radius

Rotational inertia for masses moving in a circle

I=m1r12+ m2r2

2

Calculating rotational inertiaEither: Measure τ and α to find I Or: use an equation based on the distribution of mass

2

52mrI

For a mass moving in a circle:

For hollow cylinders or hoops

For solid cylinders or disks

For hollow spheres

For solid spheres

2

32mrI

2

21 mrI

2mrI

2mrI

Conservation of angular momentum

The MOMENTUM (p=mv) of a system objects doesn’t change unless there is an an external FORCE. For a single object m can’t change, and v can’t change

Conservation of angular momentum

The MOMENTUM (p=mv) of a system objects doesn’t change unless there is an an external FORCE. For a single object m can’t change, and v can’t change

The ANGULAR MOMENTUM (L=Iω) of a system of objects doesn’t change unless there is an external TORQUE. For a single object I CAN change, so ω CAN change!

Helicopters have two rotors. One big one on the top, and one small one on the tail.

If it there wasn’t a tail rotor what would happen when the helicopter slowed down its main rotor in mid air?

http://www.youtube.com/watch?v=-GTCvyPWzMkhttp://www.youtube.com/watch?v=Ug6W7_tafnc

Angular momentum conservation

Li = Lf

Iiωi=Ifωf

Minimum rotational inertiaMaximum angular velocity

Maximum rotational inertiaMinimum angular velocity

Divers

• http://www.fandome.com/video/102554/Best-of-Athens-2004-Olympic-Diving/

Cat lands on its feet

• http://nz.youtube.com/watch?v=lqsWz9TAwBs

• http://www.upscale.utoronto.ca/GeneralInterest/Harrison/Flash/ClassMechanics/CatOnItsFeet/CatOnItsFeet.swf

Tornado• http://www.youtube.com/watch?

v=xCI1u05KD_s• Tornado tube

Explain how divers can start their dive spinning slowly and start spinning quickly in mid-air.

R- RespondThe divers curl up to spin faster

R – Rule: Conservation of angular momentumAngular momentum is a measure of how fast matter is spinning. It is defined as the rotational inertia multiplied by angular velocity L=Iω. Rotational inertia depends on the mass and the distribution of mass. In the absence of external torques angular momentum is conserved

R – Relate Divers start their dive with their body stretched out. Their mass is spread out so they have a high rotational inertia. They curl their bodies into a ball which decreases their rotational inertia as their mass is closer to their axis of rotation. Since L=Iω is conserved, when they decrease their rotational inertia their angular velocity has to increase to conserve angular momentum.

R- Reread

Angular momentum with a bicycle wheel

• http://www.youtube.com/watch?v=dVwKE9yDqVo

Relationship between angular momentum and translational

momentumL=pr=mvr

L = Angular momentum v = velocityr = perpendicular distance between the direction of motion and the centre of rotation.

r

Merry go round• http://physics.weber.edu/amiri/

director/dcrfiles/momentum/merryGoRoundS.dcr

TYPES OF ENERGY

Kinetic energyLinear kinetic energyEKlin=½ mv2

Rotational kinetic energyEKrot=½ I ω2

Describe the energy transformations involved in

1. A ball rolling down a hill2. Our rotation contraption3. A trampolinist doing a flip

Proving the equation for a ring

EK(LIN) = ½mv2

= ½m(r)2 (as v =r)

=½mr22 EK(ROT) = ½I2 (as I=mr2)

YO- YO PROBLEMWhen a yo-yo is released it accelerates downwardat a constant rate until the string is all unwound.

1. Using forces explain why it accelerates down 2. Using torques explain why it rotates as it falls3. What happens when the string is all unwound?

Explain using conservation of angular momentum.

4. Before the yo-yo is released it has only potential energy. Explain what happens to this energy after the yo-yo is released.

2005 yoyo contest• http://www.youtube.com/watch?

v=XvG3IK-hzRs

Rotational inertia and energyA student started spinning slowly on the stool with her arms out. Her rotational inertia started off as 1.8 kg m2. She pulled her arms in. Her rotational inertia was then 1.2 kg m2. Her initial angular velocity was 1.6 rad/s. Calculate her final angular velocity.Calculate her kinetic energy before and after she moves her arms in. Are they the same? If not explain where the change in energy comes from or goes to.

Two cylinders with the same mass and radius. Why does one roll faster than the

other?Both cylinders transform gravitational potential energy into kinetic energy

mgh =½Iω2+½mv2

The cylinder with the mass concentrated closer to the centre has a smaller rotational inertia so more of its energy goes into linear kinetic energy and therefore it goes faster.

Why does a car wheel roll slower than a car?

Both objects transform gravitational potential energy into kinetic energy

mgh =½Iω2+½mv2

For the car, most of the mass is simply translating, and not rotating, so most of its potential energy turns into the translational kinetic energy giving it a larger speed. The whole of the car wheel has to rotate so a larger proportion of its energy goes into linear kinetic energy and therefore it goes faster.

Rotational Kinetic Energy problems

• Examples page 140• Q2-4 page 142

A student started spinning slowly on the stool with her arms out. Her rotational inertia started off as 1.8 kg m2. She pulled her arms in. Her rotational inertia was then 1.2 kg m2. Her initial angular velocity was 1.6 rad/s. Calculate her final angular velocity.Calculate her kinetic energy before and after she moves her arms in. Are they the same? If not explain where the change in energy comes from or goes to.

•

Rolling racesYou have several things that can rollWhat determines how fast a thing rolls down a slope? Is it

– mass?- radius?- rotational inertia?- shape?- whether it is hollow or filled with stuff?- colour?- how aesthetically pleasing it is?- something else?

Do some informal experimentation and write down your conclusion on the whiteboard

Rotational inertia and energyA student started spinning slowly on the stool with her arms out. Her rotational inertia started off as 1.8 kg m2. She pulled her arms in. Her rotational inertia was then 1.2 kg m2. Her initial angular velocity was 1.6 rad/s, and her final angular velocity was 2.4 rad/s.Calculate her kinetic energy before and after she moves her arms in. Are they the same? If not explain where the change in energy comes from or goes to.

Review quiz1. What does rotational inertia

depend on?

2. A solid wheel of mass 83kg and radius 0.63m is rotated by a torque of 4.3 Nm. Calculate the rotational inertia of the disk and calculate its angular acceleration.

3. Draw and label the forces acting on this plane moving in a circle. Make the size of the arrows in proportion to each other. In a different colour draw in the total force.

Review quiz1. What does rotational inertia depend

on? Mass and the distribution of mass about the axis of rotation

2. A solid wheel of mass 83kg and radius 0.63m is rotated by a torque of 4.3 Nm. Calculate the rotational inertia of the disk and calculate its angular acceleration.I=½mr2=½x83x0.632=16.5 kgm2.α=τ/I=4.3/32.9=0.26rad s-2

3. Draw and label the forces acting on this plane moving in a circle. Make the size of the arrows in proportion to each other. In a different colour draw in the total force.

Lift

TotalWeight/ gravity