Lec 08: Rotation Rotation: Angles, Speed Centripetal Force and Acceleration Seesaws: Torque and Inertia Angular Momentum

Angles: Radians vs. Degrees 1 Revolution = 360o = 2π Radians Q1. How many radians are 8 revolutions? (8 Revolutions ) x (2π Radians) = 16π = 50.26 Rad Q2. How many degrees is 1 radian? (360 degrees) = 2π = 6.28 Rad (???) (6.28 ) = (360) (1) ( ??? ) = 1 Rad (????) = 57.3 degrees

Rotational (Angular) Speed distance traveled = time it took speed Angular Velocity angle swept = time it took Rotational speed RPM Revolutions per Minute Q3. How many rad/s is 20 rpm? Angular Speed: (20 rpm) x (2π Radians) = 2.09 Rad/s (60 seconds) Q4. What is 10 rad/s rpm? (10 rpm) x (60 seconds) = 95.5 rpm (2π Radians)

Circular Motion Q5. When we move in a circle, do we have an acceleration? The change is towards the center (speed)2 = (radius) Centripetal acceleration Q6. What force is the cause for the centripetal acceleration?

Centripetal vs. Centrifugal Force Centripetal Force: In order for an object to rotate along a circle, there is a force responsible for pulling your that object towards the center of a circular arc called Centripetal force Centrifugal Force: When we experience a centripetal acceleration, our feeling is actually being pushed away from the center.

Puzzles P1. When you turn your car, there is a force responsible for pulling your car towards the center of a circular arc called Centripetal force. It is not a new force, but rather one or more forces combining to cause that acceleration. Which is it? Gravity (b) The force from your hands turning the wheel (c) The friction between your car tires and the road? P2. What is the angular speed of the Earth as it rotates about its axis? P3. When you put water in a kitchen blender, it begins to travel in a 5.0-cm radius at a speed of 1 m/s. How quickly is the water accelerating?

Puzzles P4. Why do we not fall during a loop-the-loop ride on a roller coaster? P5. If Earth pulls the Moon towards itself, why does not the Moon fall toward the Earth? P6. Draw vectors for (a) gravitational force; (b) the force exerted by the seat, and (c) the acceleration felt by the passengers.

Center of Mass (Gravity) Pivot/Fulcrum Center of Mass: A point about which an object naturally spins Note: The Center of Mass may be outside the object Two ways to find the Center of Mass: Balance the object Find about which point it naturally rotates

Torque: Cause for Rotation Torque = (Force ) x (Lever Arm) Q6. Which wrench requires you to apply less force in order to loosen a bolt. A long wrench or a short wrench? (a) The long wrench (b) The short wrench (c) It doesn’t matter

Simple machines: LEVER (FORCE) x (distance) = (force) x (DISTANCE)

SEESAWS (FORCE) x (lever arm) = (force) x (LEVER ARM) Example: Two children with different masses must sit at different distance from the fulcrum

Puzzles P7. Balance the seesaw. If the boy is 10 kg and sits 3 m away from the pivot, wher shall the 15-kg girl sit in order for the seesaw to be balanced?

Puzzles (FORCE)(lever arm) = (force) (LEVER ARM) P8. (A see-saw) A 10-lb objects is placed 1 ft to the left of fulcrum. How far from the fulcrum should a 5-lb force be applied to balance it.

Puzzles (FORCE)(lever arm) = (force) (LEVER ARM) P9. (A wheel barrow) A 10-lb objects is placed 1 ft to the left of fulcrum. How far from the fulcrum should a 5-lb force be applied to balance it.

Puzzles P10. Why is it difficult to unscrew a bottle cap? P11. You are trying to replace a light at your skating rink. As you reach up overhead to unscrew the light you twist it, but you begin to rotate yourself. What caused you to rotate? P12. Why do helicopters have two rotors? Which Law is most involved in the explanation?

ANGULAR MOMENTUM ANGULAR (ROTATIONAL) MOMENTUM : ROTATIONAL INERTIA ( I ): How easy it is to make an object change its rotation ANGULAR (ROTATIONAL) MOMENTUM : (ROT. INERTIA ) x (ROTATIONAL SPEED) TORQUE – MOMENTUM PRINCIPLE (NEWTON’S 1ST LAW) : In order to change the rotational momentum, a torque is necessary

Spinning Ice skaters Example: An ice skater is spinning gracefully about herself at 1 rev/seconds while her hands are extended and her rotational inertia is about 2.25 kg.m2. How fast would she be spinning if she pulls her hands in and in the process decreases her rotational inertia to 1.8 kg.m2? (2.25 kg.m2 ) ( 1 rev/s ) = (1.8 kg.m2 ) ( ??? Rev/s)

Collapsing Neutron Stars A neutron star is formed when an object such as our Sun collapses. Suppose our Sun were to collapse so that its rotational inertia becomes 10 billion times less of its original value. If the star originally completed one revolution in 25 days (approx. 0.00000046 rps), how fast does it revolve after the collapse?