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?AccelerationVelocityDisplacement

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=r α

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 Acceleration v f = v i + at ωf=ωi +αtωf=ωi +αt d = v i t + ½ at 2 v f 2 = v i 2 + 2ad ω f 2 =ω i 2 +2 αθ

Combined motion Cycloid motion XgCaHhohttp:// XgCaHho Interest/Harrison/Flash/ClassMechanic s/RollingDisc/RollingDisc.htmlhttp:// Interest/Harrison/Flash/ClassMechanic s/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 =Fd Force 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 inertia MASS (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 2 nd law: F=ma. Mass is measured in kg Newton’s 2 nd law τ =I α Rotational inertia is measured in kg m 2 Rotational inertia depends on the mass of the object and on the distribution of mass around the axis of rotation

MORE ROTATIONAL MOTION QUANTITIES TRANSLATIONALROTATIONAL F Force m mass F=ma

MORE ROTATIONAL MOTION QUANTITIES TRANSLATIONALROTATIONAL F Force (N) τ Torque (Nm) m mass (kg)I rotational inertia (kgm 2 ) F=ma τ =I α

Rotational inertia for masses moving in a circle I=mr 2 radius

Rotational inertia for masses moving in a circle I=m 1 r m 2 r 2 2

Calculating rotational inertia Either: Measure τ and α to find I Or: use an equation based on the distribution of mass For a mass moving in a circle: For hollow cylinders or hoops For solid cylinders or disks For hollow spheres For solid spheres

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?

Angular momentum conservation L i = L f I i ω i =I f ω f

Minimum rotational inertia Maximum angular velocity Maximum rotational inertia Minimum angular velocity

Divers /Best-of-Athens-2004-Olympic-Diving/ /Best-of-Athens-2004-Olympic-Diving/

Cat lands on its feet TAwBshttp://nz.youtube.com/watch?v=lqsWz9 TAwBs Interest/Harrison/Flash/ClassMechanic s/CatOnItsFeet/CatOnItsFeet.swfhttp:// Interest/Harrison/Flash/ClassMechanic s/CatOnItsFeet/CatOnItsFeet.swf

Tornado u05KD_shttp:// u05KD_s Tornado tube

Explain how divers can start their dive spinning slowly and start spinning quickly in mid-air. R- Respond The divers curl up to spin faster R – Rule: Conservation of angular momentum Angular 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 E9yDqVohttp:// E9yDqVo

Relationship between angular momentum and translational momentum L=pr=mvr L = Angular momentum v = velocity r = perpendicular distance between the direction of motion and the centre of rotation. r

Merry go round r/dcrfiles/momentum/merryGoRoundS.d crhttp://physics.weber.edu/amiri/directo r/dcrfiles/momentum/merryGoRoundS.d cr

TYPES OF ENERGY

Kinetic energy Linear kinetic energy E Klin =½ mv 2 Rotational kinetic energy E Krot =½ I ω 2

Describe the energy transformations involved in 1.A ball rolling down a hill 2.Our rotation contraption 3.A trampolinist doing a flip

Proving the equation for a ring E K(LIN) = ½mv 2 = ½m(  r) 2 (as v =  r) =½mr 2  2 E K(ROT) = ½I  2 (as I=mr 2 )

YO- YO PROBLEM When a yo-yo is released it accelerates downward at 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 falls 3.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 IK-hzRshttp:// IK-hzRs

Rotational inertia and energy A student started spinning slowly on the stool with her arms out. Her rotational inertia started off as 1.8 kg m 2. She pulled her arms in. Her rotational inertia was then 1.2 kg m 2. 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 =½Iω 2 +½mv 2 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 =½Iω 2 +½mv 2 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 m 2. She pulled her arms in. Her rotational inertia was then 1.2 kg m 2. 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 races You have several things that can roll What 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 energy A student started spinning slowly on the stool with her arms out. Her rotational inertia started off as 1.8 kg m 2. She pulled her arms in. Her rotational inertia was then 1.2 kg m 2. 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 quiz 1.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 quiz 1.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=½mr 2 =½x83x =16.5 kgm 2. α = τ /I=4.3/32.9=0.26rad s 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 Total Weight/ gravity