Chapter 8 – Rotational Motion Circular Motion Rotational Inertia Torque Center of Mass and Center of Gravity Locating Center of Gravity Stability Centripetal Force Centrifugal Force Centrifugal Force in a Rotating Reference Frame Simulated Gravity Angular Momentum Conservation of Angular Momentum Physics 1100 – Spring 2012

Circular Motion Linear speed - the distance moved per unit time. Also called simply speed. Rotational speed - the number of rotations or revolutions per unit time. Rotational speed is often measured in revolutions per minute (RPM). Physics 1100 – Spring 2012

Rotation Physics 1100 – Spring 2012

Circular Motion The linear speed is directly proportional to both rotational speed and radial distance. v = w r What are two ways that you can increase your linear speed on a rotating platform? Physics 1100 – Spring 2012

Circular Motion The linear speed is directly proportional to both rotational speed and radial distance. v = w r What are two ways that you can increase your linear speed on a rotating platform? Answers: Move away from the rotation axis. Have the platform spin faster. Physics 1100 – Spring 2012

Example Question 1) 1 m/s 2) 2 m/s 3) 3m/s 4) 4m/s You sit on a rotating platform halfway between the rotating axis and the outer edge. You have a rotational speed of 20 RPM and a tangential speed of 2 m/s. What will be the linear speed of your friend who sit at the outer edge? 1) 1 m/s 2) 2 m/s 3) 3m/s 4) 4m/s What will be his rotational speed? 1) 10 RPM 2) 20 RPM 3) 30 RPM 4) 40 RPM Physics 1100 – Spring 2012

Example Question Answer: 4 m/s Answer: 20 RPM You sit on a rotating platform halfway between the rotating axis and the outer edge. You have a rotational speed of 20 RPM and a tangential speed of 2 m/s. What will be the linear speed of your friend who sit at the outer edge? Answer: 4 m/s What will be his rotational speed? Answer: 20 RPM Physics 1100 – Spring 2012

Rotational Inertia An object rotating about an axis tends to remain rotating unless interfered with by some external influence. This influence is called torque. Rotation adds stability to linear motion. Examples: spinning football bicycle tires Frisbee Physics 1100 – Spring 2012

Rotational Inertia The greater the distance between the bulk of an object's mass and its axis of rotation, the greater the rotational inertia. Physics 1100 – Spring 2012

Rotational Inertia Physics 1100 – Spring 2012

Rotational Inertia Physics 1100 – Spring 2012

Torque Torque is the product of the force and lever-arm distance, which tends to produce rotation. Torque = force ´ lever arm Examples: wrenches see-saws Newton’s First Law for Rotation St = 0, no rotational motion !!! Physics 1100 – Spring 2012

Class Problem James finds it difficult to muster enough torque to turn the stubborn bolt with the wrench. He wishes he had a pipe handy to effectively lengthen the wrench handle, but doesn't. He does, however, have a piece of rope. Will torque be increased if he pulls as hard on the rope as shown? Physics 1100 – Spring 2012

Class Problem No, the torque will be the same because the lever-arm distance is the same in both cases. The lever arm is not the distance between axis of turning and the point of application of the force, but the distance from the turning axis to the "line of action" of the applied force. Note the line of action, and hence the lever arm is the same in both cases. The pipe that extends the length of the wrench handle puts the line of action farther from the turning axis—the rope does not. Physics 1100 – Spring 2012

Torque Physics 1100 – Spring 2012

Center of Mass The center of mass of an object is the average position of mass. Physics 1100 – Spring 2012

Center of Mass Objects tend to rotate about their center of mass. Physics 1100 – Spring 2012

Class Problem The broom balances at its center of gravity. If you saw the broom into two parts through the center of gravity and then weigh each part on a scale, which part will weigh more? Handle side Brush Side Both weigh the same Physics 1100 – Spring 2012

Class Problem The short broom part is heavier. It balances the long handle just as kids of unequal weights can balance on a seesaw when the heavier kid sits closer to the fulcrum. Both the balanced broom and seesaw are evidence of equal and opposite torques—not weights. Physics 1100 – Spring 2012

Class Problem The 40-kg woman stands at the end of a 4-meter long uniform plank. If the maximum overhang for balance is 1 meter, estimate the mass of the plank. 10 kg 20 kg 40 kg 80 kg 160 kg Physics 1100 – Spring 2012

Class Problem The mass of the plank is about 40kg. The plank tends to rotate like a seesaw about a pivot point at the end of the building. Her weight multiplied by 1 meter produces a torque that tends to rotate the system clockwise. The counterbalancing torque is produced by the weight of the plank multiplied by the distance from the pivot point to the plank's center of gravity. Note that this distance is also 1 meter. So both the woman and the plank weigh the same. Their masses are equal. Physics 1100 – Spring 2012

Stability For stability center of gravity must be over area of support. Examples: Tower of Pisa Touching toes with legs against wall Meter stick over the edge Physics 1100 – Spring 2012

Angular Momentum Another conserved quantity is angular momentum, relating to rotational inertia: Spinning wheel wants to keep on spinning, stationary wheel wants to keep still (unless acted upon by an external rotational force, or torque) Newton’s laws for linear (straight-line) motion have direct analogs in rotational motion Physics 1100 – Spring 2012

Conservation of Angular Momentum angular momentum = rotational inertia x rotational velocity L = I w Newton's first law for rotating systems: “A body will maintain its state of angular momentum unless acted upon by an unbalanced external torque.” Physics 1100 – Spring 2012

Angular Momentum Angular momentum is proportional to rotation speed times rotational inertia Rotational inertia characterized by (mass)(radius)2 distribution in object Physics 1100 – Spring 2012

Angular Momentum Conservation Speed up rotation by tucking in Slow down rotation by stretching out Seen in diving all the time Figure skaters demonstrate impressively Effect amplified by moving large masses to vastly different radii Physics 1100 – Spring 2012

Centripetal Force Centrifugal Force …is applied by some object. Centripetal means "center seeking". Centrifugal Force …results from a natural tendency. Centrifugal means "center fleeing". Physics 1100 – Spring 2012

Rotational Forces Direction of Motion Centripetal Force Centrifugal Physics 1100 – Spring 2012

What kind of motions do we feel? Aside from vibrations, don’t feel constant velocity But we can feel acceleration Physics 1100 – Spring 2012

The forces real and perceived Rotating Drum Ride Vertical drum rotates, you’re pressed against wall Friction force against wall matches gravity Seem to stick to wall, feel very heavy The forces real and perceived Real Forces: Friction; up Centripetal; inwards Gravity (weight); down Perceived Forces: Centrifugal; outwards Gravity (weight); down Perceived weight; down and out Physics 1100 – Spring 2012

Simulated Gravity Physics 1100 – Spring 2012

Clicker Problems If you replace the wheels and tires of your car with new ones having greater diameters, all of your speedometer readings thereafter will A) increase B) decrease. C) remain the same. Suppose the circumference of a bicycle wheel is 2 meters. If it rotates at 1 revolution per second when you are riding the bicycle, then your speed will be A) 3.14 m/s. B) 2 m/s. C) 6.28 m/s. D) 1 m/s. E) 3 m/s. Physics 1100 – Spring 2012

Clicker Problems An upright broom is easier to balance when the heavier end is A) highest, farthest from your hand. B) nearest your hand. C) same either way. Put a pipe over the end of a wrench when trying to turn a stubborn nut on a bolt, to effectively make the wrench handle twice as long, you'll multiply the torque by A) four. B) eight. C) two. Physics 1100 – Spring 2012

Clicker Questions The famous Leaning Tower of Pisa doesn't topple over because its center of gravity is A) displaced from its center. B) in the same place as its center of mass. C) stabilized by its structure. D) relatively low for such a tall building. E) above a place of support. The chef at the infamous Fattening Tower of Pizza tosses a spinning disk of uncooked pizza dough into the air. The disk's diameter increases during the flight, while its rotational speed A) increases. B) decreases. C) remains constant. Physics 1100 – Spring 2012

Clicker Questions To turn a stubborn screw, it is best to use a screwdriver that has a A) long handle. B) wide handle. C) smooth handle. D) none of these To kick a football so it won't topple end over end, kick it so the force of impact extends A) below its center of gravity. B) through its center of gravity. C) above its center of gravity. Physics 1100 – Spring 2012

Clicker Questions A car travels in a circle with constant speed. The net force on the car is A) directed forward, in the direction of travel. B) zero because the car is not accelerating. C) directed towards the center of the curve. D) none of these If the polar icecaps melted, the resulting water would spread over the entire earth. This new mass distribution would tend to make the length of a day A) longer. B) shorter at first, then longer. C) shorter. D) longer at first, then shorter. E) stay the same. Physics 1100 – Spring 2012