Rotational Motion Rotation of rigid objects- object with definite shape

A brief lesson in Greek  theta  tau  omega  alpha

Rotational Motion All points on object move in circles Center of these circles is a line=axis of rotation What are some examples of rotational motion?

Radians Angular position of object in degrees=ø More useful is radians 1 Radian= angle subtended by arc whose length = radius Ø=l/r

Converting to Radians If l=r then  =1rad Complete circle = 360º so…in a full circle 360  =  =l/r=2πr/r=2πrad So 1 rad=360  /2π=57.3  *** CONVERSIONS*** 1rad=57.3  360  =2πrad

Example: A ferris wheel rotates 5.5 revolutions. How many radians has it rotated? 1 rev=360  =2πrad=6.28rad 5.5rev=(5.5rev)(2πrad/rev)= 34.5rad

Example: Earth makes 1 complete revolution (or 2  rad) in a day. Through what angle does earth rotate in 6hours? 6 hours is 1/4 of a day  =2  rad/4=  rad/2

Practice What is the angular displacement of each of the following hands of a clock in 1hr? –Second hand –Minute hand –Hour hand

Hands of a Clock Second: -377rad Minute: -6.28rad Hour: rad

Velocity and Acceleration Velocity is tangential to circle- in direction of motion Acceleration is towards center and axis of rotation

Angular Velocity Angular velocity  = rate of change of angular position As object rotates its angular displacement is ∆  =  2 -  1 So angular velocity is  =∆  / ∆t measured in rad/sec

Angular Velocity All points in rigid object rotate with same angular velocity (move through same angle in same amount of time) Direction: –clockwise is - –counterclockwise is +

Velocity:Linear vs Angular Each point on rotating object also has linear velocity and acceleration Direction of linear velocity is tangent to circle at that point “the hammer throw”

Velocity:Linear vs Angular Even though angular velocity is same for any point, linear velocity depends on how far away from axis of rotation Think of a merry-go- round

Velocity:Linear vs Angular v=  l/  t=r  /  t v=r 

Angular Acceleration If angular velocity is changing, object would undergo angular acceleration  = angular acceleration  =  /  t Rad/s 2 Since  is same for all points on rotating object, so is  so radius does not matter

Angular and Linear Acceleration Linear acceleration has 2 components: tangential and centripetal Total acceleration is vector sum of 2 components a=a tangential +a centripetal

Linear and Angular Measures QuantityLinearAngular Relationship Displacement d(m) Velocity v(m/s) Acceleration a(m/s 2 )

Linear and Angular Measures QuantityLinearAngular Relationship Displacement d(m)  (rad)d=r  Velocity v(m/s)  (rad/s)v=r  Acceleration a(m/s 2 )  (rad/s 2) a=r 

Practice If a truck has a linear acceleration of 1.85m/s 2 and the wheels have an angular acceleration of 5.23rad/s 2, what is the diameter of the truck’s wheels?

Truck Diameter=0.707m Now say the truck is towing a trailer with wheels that have a diameter of 46cm How does linear acceleration of trailer compare with that of the truck? How does angular acceleration of trailer wheels compare with the truck wheels?

Truck Linear acceleration is the same Angular acceleration is increased because the radius of the wheel is smaller

Frequency Frequency= f= revolutions per second (Hz) Period=T=time to make one complete revolution T= 1/f

Frequency and Period example After closing a deal with a client, Kent leans back in his swivel chair and spins around with a frequency of 0.5Hz. What is Kent’s period of spin? T=1/f=1/0.5Hz=2s

Period and Frequency relate to linear and angular acceleration Angle of 1 revolution=2  rad Related to angular velocity:  =2  f Since one revolution = 2  r and the time it takes for one revolution = T Then v= 2  r /T

Try it… Joe’s favorite ride at the 50th State Fair is the “Rotor.” The ride has a radius of 4.0m and takes 2.0s to make one full revolution. What is Joe’s linear velocity on the ride? V= 2  r /T= 2  (4.0m)/2.0s=13m/s Now put it together with centripetal acceleration: what is Joe’s centripetal acceleration?

And the answer is… A=v 2 /r=(13m/s 2) /4.0m=42m/s 2

Centripetal Acceleration acceleration= change in velocity (speed and direction) in circular motion you are always changing direction- acceleration is towards the axis of rotation The farther away you are from the axis of rotation, the greater the centripetal acceleration Demo- crack the whip media/circmot/ucm.gif

Centripetal examples Wet towel Bucket of water Beware….inertia is often misinterpreted as a force.

The “f” word When you turn quickly- say in a car or roller coaster- you experience that feeling of leaning outward You’ve heard it described before as centrifugal force Arghh……the “f” word When you are in circular motion, the force is inward- towards the axis= centripetal So why does it feel like you are pushed out??? INERTIA

Centripetal acceleration and force Centripetal acceleration=v 2 /r –Towards axis of rotation Centripetal force=ma centripetal

Rolling

Rolling= rotation + translation Static friction between rolling object and ground (point of contact is momentarily at rest so static) v=r 

Example p. 202 A bike slows down uniformly from v=8.40m/s to rest over a distance of 115m. Wheel diameter = 68.0cm. Determine (a)angular velocity of wheels at t=0 (b)total revolutions of each wheel before coming to rest (c) angular acceleration of wheel (d) time it took to stop

Torque

How do you make an object start to rotate? Pick an object in the room and list all the ways you can think of to make it start rotating.

Torque Let’s say we want to spin a can on the table. A force is required. One way to start rotation is to wind a string around outer edge of can and then pull. Where is the force acting? In which direction is the force acting?

Torque Force acting on outside of can. Where string leaves the can, pulling tangent.

Torque Where you apply the force is important. Think of trying to open a heavy door- if you push right next to the hinges (axis of rotation) it is very hard to move. If you push far from the hinges it is easier to move. Distance from axis of rotation = lever arm or moment arm

Torque Which string will open the door the easiest? In which direction do you need to pull the string to make the door open easiest?

Torque

 tau = torque (mN) If force is perpendicular,  =rF If force is not perpendicular, need to find the perpendicular component of F  =rFsin  Where  = angle btwn F and r

Torque example (perpendicular) Ned tightens a bolt in his car engine by exerting 12N of force on his wrench at a distance of 0.40m from the fulcrum. How much torque must he produce to turn the bolt? (force is applied perpendicular to rotation) Torque=  =rF=(12N)(0.4m)=4.8mN

Torque- Example glencoe p. 202 A bolt on a car engine needs to be tightened with a torque of 35 mN. You use a 25cm long wrench and pull on the end of the wrench at an angle of 60.0  from perpendicular. How long is the lever arm and how much force do you have to exert? Sketch the problem before solving

More than one Torque When  1 torque acting, angular acceleration  is proportional to net torque If forces acting to rotate object in same direction net torque=sum of torques If forces acting to rotate object in opposite directions net torque=difference of torques Counterclockwise + Clockwise -

Multiple Torque experiment Tape a penny to each side of your pencil and then balance pencil on your finger. Each penny exerts a torque that is equal to its weight (force of gravity) times the distance r from the balance point on your finger. Torques are equal but opposite in direction so net torque=0 If you placed 2 pennies on one side, where could you place the single penny on the other side to balance the torques?

Torque and center of mass Stand with your heels against the wall and try to touch your toes. If there is no base of support under your center of mass you will topple over

Torque and football If you kick the ball at the center of mass it will not spin If you kick the ball above or below the center of mass it will spin

Inertia Remember our friend, Newton? F=ma In circular motion: – torque takes the place of force –Angular acceleration takes the place of acceleration

Rotational Inertia=LAZINESS Moment of inertia for a point object I = Resistance to rotation I =mr 2  = I  I plays the same role for rotational motion as mass does for translational motion I depends on distribution of mass with respect to axis of rotation When mass is concentrated close to axis of rotation, I is lower so easier to start and stop rotation

Rotational Inertia Unlike translational motion, distribution of mass is important in rotational motion.

Changing rotational inertia When you change your rotational inertia you can drastically change your velocity So what about conservation of momentum?

Angular momentum Momentum is conserved when no outside forces are acting In rotation- this means if no outside torques are acting A spinning ice skater pulls in her arms (decreasing her radius of spin) and spins faster yet her momentum is conserved

Angular momentum Angular momentum=L=mvr Unit is kgm 2 /s

Examples… Hickory Dickory Dock… A 20.0g mouse ran up a clock and took turns riding the second hand (0.20m), minute hand (0.20m), and the hour hand (0.10m). What was the angular momentum of the mouse on each of the 3 hands? Try as a group.