Rotational Motion About a Fixed Axis Chapter 10 Rotational Motion About a Fixed Axis

A body with a definite shape that doesn’t change Rigid body A body with a definite shape that doesn’t change 1. Vibrating or deforming can be ignored 2. Distance between particles does not change Motion of a rigid body can be expressed as Translational motion of its CM + Rotational motion about its CM Pure rotational motion: all move in circles centers of these circles all lie on a line Axis of rotation Fixed axis

Angular coordinate / Angle Angular quantities Angular coordinate / Angle Angular velocity y x o  R s Angular acceleration Radial acceleration Frequency Period

Hard drive Example1: The platter of hard disk rotates 5400rpm. a) Angular velocity; b) Speed of reading head located 3cm from the axis; c) Acceleration at that point; d) How many bits the writing head writes per second at that point if 1 bit needs 0.5µm length. Solution: a) angular velocity b) speed c) acceleration d)

Useful Similarities Notice the similarities in different motion Uniformly accelerated rotational motion Analogous thinking is very helpful

Vector nature of angular quantities Angular velocity and acceleration → vectors  Points along the axis, follows the right-hand rule How does change? Angular acceleration pulley spinning top

What causes acceleration of rotational motion? Rotational dynamics What causes acceleration of rotational motion? Force: magnitude, direction and point of action Push a door Archimedes’ Lever “Give me a fulcrum, and I shall move the world. ” —— Archimedes Lever arm: the perpendicular distance from the axis to the line along which the force acts 7

Torque about fixed axis The effect of force → angular acceleration Torque = force × lever arm R F R⊥: lever arm or moment arm Net torque causes acceleration of rotational motion positive rotational direction Balance of rigid body Play on a seesaw 8

Torque and balance Example2: A 15kg mass locates 20cm from the axis of massless lever. Determine: a) torque on the lever; b) Force required respectively to balance the lever. Solution: a) 15cm 20cm b) 5cm 25cm 150N What about F4 ?

A particle mi in rigid body Rotational theorem A particle mi in rigid body fi Fi mi Ri Rotational inertia Net external torque or moment of inertia Rotational theorem about fixed axis 10

Properties of Rotational theorem 1) Only external torques are effective Sum of the internal torques is 0 from N-3 2) Rotational equivalent of Newton’s second law. 3) Analogy of rotational and translational motion change in motion cause of the change inertia of motion 11

Determining rotational inertia several particles continuous object How mass is distributed with respect to the axis Example3: Rotational inertia of 3 particles fixed on a massless rod about a, b, c axis. Solution: c b a m 2m 3m l 12

Solution: Choose a segment dx Uniform thin rod Example4: Rotational inertia of uniform thin rod with mass m and length l. a) Through center; Solution: Choose a segment dx C dx dm x o b) Through end o Typical result, should be memorized 13

Example5: Uniform thin hoop (mass m, radius R) Uniform circles Example5: Uniform thin hoop (mass m, radius R) Solution: Choose a segment dm R dm Example6: Uniform disk/cylinder (m, R) Solution: Choose a hoop dm r dr Homework: Uniform sphere (m, R) (P246) 14

Parallel-axis theorem If I is the rotational inertia about any axis, and IC is the rotational inertia about an axis through the CM, and parallel to the first but a distance l, then I Ic l C M o Proof: IC is always less than other I of parallel axes 15

Perpendicular-axis theorem The sum of the rotational inertia of a plane about any two perpendicular axes in the plane is equal to the rotational inertia about an axes through the point of intersection ⊥ the plane. x y z 1) Only for plane figures or 2-dimensional bodies 2) x ⊥ y ⊥ z and intersect at one point 3) Try to prove it by yourself 16

Application of two theorems Example7: Rotational inertia of uniform thin disk about the line of diameter Solution: You can choose a dm dm or apply the ⊥-axis theorem Example8: Thin hoop about a tangent line Solution: Perpendicular-axis theorem Parallel-axis theorem 17

Massive Pulley Example9: A box is hanging on a pulley by massless rope, then the system starts to move without slip, determine the angular acceleration and tension. mg  Solution: T = mg ? m M R  Free-body diagram T Rotational theorem: Newton’s second law: No slip motion: Two boxes? 18

Solution: a) Gravity → torque Rotating rod Example10: A uniform rod of mass M and length l can pivot freely about axis o, released horizontally. Determine: a) α; b) aC; c) force acted by the axis. Solution: a) Gravity → torque F o Mg C b) Acceleration of CM a of the end? c) 19

Rotational theorem can be written in terms of L Angular momentum Linear momentum p = m v rotational analog Angular momentum Rotational theorem can be written in terms of L The rate of change in angular momentum of a rigid body is equal to the net torque applied on it. Comparing with Newton’s second law 20

Conservation of angular momentum Torque & angular momentum 1) It is valid even if I changes 2) Valid for a fixed axis or axis through its CM Law of conservation of angular momentum: The total angular momentum of a rotating body remains constant if the net external torque is zero. 1) It holds for inertial frames or frame of CM 2) One of fundamental laws of conservation 21

Examples in sports Figure skating Diving

Helicopter

Falling cat A falling cat can adjust his posture to avoid injury, how to make it? Angular momentum is conserved 1) Bend his body 2) Rotate his upper part to proper position about blue axis, meanwhile the lower part rotates a less angle 3) Rotate his lower part to proper position about the red axis 4) Get the work done

A. increase; B. decrease; C. remain constant Rotating disk Example11: A disk is rotating about its center axis, and two identical bullets hit into it symmetrically. The angular velocity of system will ________ A. increase; B. decrease; C. remain constant Solution: Total angular momentum of the system is conserved  . o Total rotational inertia increases So angular velocity will decrease What if two opposite forces act on the disk? 25

Man on rotating platform Example12: A platform is rotating about its center axis, and a man standing on it (treat as a particle) starts to move. How does  change if he goes: a) to point o; b) along the edge with relative speed v. Solution: a) Conservation of angular momentum  I o R m b) choose a positive direction 26

Solution: Conservation of total L Hits on a rod Example13: A bullet hits into a hanged uniform rod, determine the angular velocity after collision. Solution: Conservation of total L o M, l A r mv Notice: Momentum is not conserved in general! Require: Force acted by axis remains constant Only if the bullet hits on position r = 2l / 3 27

Rotational kinetic energy Kinetic energy in translational motion rotational motion Total Ek is the sum of Ek of all particles Work done on a rotating body: Power 28

Energy in rotational motion Rotational theorem: Work-energy principle: Comparing with Total mechanical energy is conserved in rotational motion if only conservative forces do work. Potential energy of gravity: 29

Solution: Distance oC=l/6 Rotating rod Example14: A uniform rod (M, l ) can pivot freely about axis o (Ao=l/3), and it is released horizontally. Determine ω at the vertical position. Solution: Distance oC=l/6 A o . . C Rotational inertia Conservation of mechanical energy Any position? α = ? 30

For general motion of a rigid body Translational motion of its CM: + Rotational motion about its CM: The total kinetic energy (Proved in P259) Examples: 31

A typical motion: rolling without slipping Rolling motion A typical motion: rolling without slipping To make sure vC Relationship between translational and rotational motion Valid only if no slipping Motion of pulley, tire, … 32

Rolling down an incline Example15: A uniform cylinder (m, R) rolls down an incline without slipping, determine its speed if the CM moves a vertical height H. Solution: Conservation of energy where Which is faster? Comparing with sliding: 33

Forces acting on the cylinder Dynamics in rolling F mg N Forces acting on the cylinder Translational motion of CM Rotational motion about CM We can obtain: aC = 2gsin /3， F = mgsin /3 Static friction causes the rolling motion It also rearranges the kinetic energy

Challenging question A stick (M, l ) stands vertically on a frictionless table, then it falls down. Describe the motion of its CM, and of each end. 35