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## Rotational Motion

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**Rotational Motion**Rotation of rigid objects- object with definite shape**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? • What is the difference between rotation and revolution?**Speed**• Rotating objects have 2 speeds: • Linear speed (also known as tangential) • Rotational speed**Linear Speed**• Imagine yourself on a merry-go-round. At any moment, describe the direction of your linear speed • Who goes faster- A or B?**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”**Angular Velocity**• Angular velocity rate of change of angular position • measured in revolutions/time Thus RPM= revolutions per minute**Angular Velocity**• All points in rigid object rotate with same angular velocity (move through same angle in same amount of time) • Related to linear- if you speed up the rotation, both linear and angular velocity increases**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**So how are they related?**• The farther out you are, the faster your linear speed • So linear velocity increases with your radius • The faster your angular speed, the faster your linear speed • So linear velocity increases with angular velocity**Centripetal Acceleration**• If object is moving in a circle, its direction is constantly changing towards the center so the acceleration must be in that direction • Then why when you turn a corner in a car do you feel pushed out, not in?**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 • http://www.glenbrook.k12.il.us/gbssci/phys/mmedia/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 • Towards axis of rotation • Centripetal force • Towards axis of rotation**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**Rolling**• Rolling= rotation + translation • Static friction between rolling object and ground (point of contact is momentarily at rest so static)**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 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 InertiaUnlike translational motion, distribution**of mass is important in rotational motion.**Rotational inertia- baseball**• A long bat that you hold at the end has a lot of rotational inertia- mass is far away from the axis of rotation • Thus it is hard to get moving • Younger players “choke up” on the bat by moving their hands towards the middle- this makes the bat have less rotational inertia- it’s easier to swing • Try the rotating sticks!**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**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 =rF**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**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**Center of mass**• The average position of all the mass of an object • If object is symmetrical- center of mass is at the center of the object • Where is the center of mass of a meter stick? • A donut? • How could you find the center of mass of an object?**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