# Rotation, Rolling, Torque, Angular Momentum

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1 Halliday, Resnick & Walker Chapter 10 & 11 Rotation, Rolling, Torque, Angular Momentum Physics 1A PHYS1121 Professor Michael Burton Rotation

2 10-1 Rotational Variables! The motion of rotation! The same laws apply! But new quantities are needed to express them o o Torque Rotational Inertia (or Moment of Inertia)! A rigid body rotates as a unit, locked together! Consider just rotation about a fixed axis! This excludes: o o The Sun, where layers of gas rotate separately A rolling bowling ball, where rotation and translation occur 10-1 Rotational Variables! The fixed axis is called the axis of rotation! The angular position,!, of a reference line is taken relative to a fixed direction, the zero angular position From side From above

3 10-1 Rotational Variables! Unit is radians (rad): dimensionless Eq. (10-1) Eq. (10-2)! Do not reset! to zero after a full rotation! Knowledge of!(t) yields the kinematics of the rotation! The angular displacement is: 10-1 Rotational Variables! Clocks are negative! i.e. their angular displacement is in the negative direction!

4 10-1 Rotational Variables! Average angular velocity: angular displacement during a time interval Eq. (10-5)! Instantaneous angular velocity: limit as "t " 0 Eq. (10-6)! If the body is rigid, these equations hold for all points on the body! Magnitude of angular velocity = angular speed 10-1 Rotational Variables! Average angular acceleration: angular velocity change during a time interval Eq. (10-7)! Instantaneous angular velocity: limit as "t " 0 Eq. (10-8)

5 RMM02VD3: Angular Acceleration 10-1 Rotational Variables! With right-hand rule to determine direction, angular velocity & acceleration can be written as vectors! If the body rotates around the vector, then the vector points along the axis of rotation

6 10-1 Rotational Variables! Angular displacements are not vectors, because the order of rotation matters for rotations around different axes! Does not obey the rule for vector addition! The wheels of a bicycle roll without slipping on a horizontal road. The bicycle is moving due east at a constant velocity. What is the direction of the angular velocity of the wheels? a) down b) west c) east d) north e) south E

7 10-2 Rotation with Constant Angular Acceleration! The same equations hold as for constant linear acceleration.! Simply change linear quantities to angular ones.! Eqns and are the basic equations: all others can be derived from them Relating the Linear and Angular Variables! Linear and angular variables are related by r, the perpendicular distance from the rotational axis! Position (note! must be in radians):!! is the angular position! Speed:! # is the angular velocity (in radians / s)! The Period (in s) is given by: T = 2"r v = 2"r #r = 2" #

8 10-3 Relating the Linear and Angular Variables! Tangential acceleration, a t :! \$ angular acceleration (radians / s 2 ):! Radial acceleration, a r, can be written in terms of angular velocity: An insect rides the rim of a rotating merry-go-round. If the angular speed, #, of the system (i.e. merry-goround + insect) is constant, does the insect have: (a) Radial acceleration? (b) Tangential acceleration? If # is decreasing, does the insect have: (c) Radial acceleration? (d) Tangential acceleration? #

9 10-4 Kinetic Energy of Rotation! Kinetic energy, summing over all the particles: K = % &m i v 2 i! But for rotation each point has a different velocity???! Write velocity in terms of angular velocity, i.e. v i =#r i! I=!m i r i 2 is the rotational inertia, or moment of inertia! A constant for a rigid object & given rotational axis! Caution: the axis for I must always be specified 10-4 Kinetic Energy of Rotation! We write I, the moment of inertia, as:! The axis must always be specified.! And rewrite the kinetic energy, K, as:

10 RMM03VD1: Rotational Kinetic Energy 10-5 Calculating the Moment of Inertia! Integrating over a continuous body: Eq. (10-35)! For many shapes the Moment of Inertia has been calculated for commonly used axes

11 10-5 Calculating the Moment of Inertia 10-5 Calculating the Moment of Inertia! Parallel-axis theorem:! For body of mass M, about an axis distance h from a centre of mass axis, with moment of inertia I com! Note the axes must be parallel, and the first must go through the centre of mass! Does not relate the moment of inertias for two arbitrary axes

12 A flat disk, a solid sphere, and a hollow sphere each have the same mass m and radius r. The three objects are arranged so that an axis of rotation passes through the centre of each object. The rotation axis is perpendicular to the plane of the flat disk. Which of the three objects has the largest rotational inertia? a) The solid sphere and hollow sphere have the same rotational inertia and it is the largest. b) The hollow sphere has the largest rotational inertia. c) The solid sphere has the largest rotational inertia. d) The flat disk has the largest rotational inertia. e) The flat disk and hollow sphere have the same rotational inertia and it is the largest Calculating the Rotational Inertia Example Calculate the moment of inertia about an axis through one particle in 2 ways: o (a) Summing by particle: o (b) Use parallel-axis theorem

13 Catastrophic release of Rotational Kinetic Energy! 10-6 Torque! Force needed to rotate an object depends on! (a) the angle of the force and! (b) where it is applied! Resolve force into components to see how it affects rotation! Only tangential component contributes

14 10-6 Torque! Torque takes these factors into account:! i.e.! Line extended through the applied force is called the line of action of the force! Perpendicular distance from the line of action to the axis is called the moment arm! Unit of torque is the newton-metre, N m! Note that 1 J = 1 N m, but torques are never expressed in joules, torque is not energy 10-6 Torque! Torque is positive if it would cause a counterclockwise rotation, otherwise negative! Net torque or resultant torque is the sum of individual torques

15 The corner of a rectangular piece of wood is attached to a rod that is free to rotate as shown. The length of the longer side of the rectangle is 4.0 m, which is twice the length of the shorter side. Two equal forces are applied to two of the corners with magnitudes of 22 N. What is the magnitude of the net torque and direction of rotation on the block, if any? a) 44 N"m, clockwise b) 44 N"m, counterclockwise c) 88 N"m, clockwise d) 88 N"m, counterclockwise e) zero N"m, no rotation 4m 2m RMM05VD1: Torque 1-30

16 10-7 Newton's Second Law for Rotation! Rewrite F = ma with rotational variables: Eq. (10-42)! Torque causes angular acceleration Figure Work and Rotational Kinetic Energy! The rotational work-kinetic energy theorem states: Eq. (10-52)! The work done in a rotation about a fixed axis can be calculated by: Eq. (10-53)! Which, for a constant torque, reduces to: Eq. (10-54)

17 10-8 Work and Rotational Kinetic Energy! We can relate work to power with the equation: Eq. (10-55) 11-1 Rolling as Translation and Rotation Combined Straight Line Cycloid

18 RMM07VD1: Rolling Motion 1 min 11-1 Rolling as Translation and Rotation Combined! Consider only objects that roll smoothly (no slip)! The centre of mass (c.o.m.) of the object moves in a straight line parallel to the surface! The object rotates around the c.o.m. as it moves! The rotational motion is defined by: Eq. (11-1) # Eq. (11-2) With " = d# dt

19 11-1 Rolling as Translation and Rotation Combined! Rolling is a combination of translation and rotation.! Result depends on position on the wheel. Spokes at top blurred as they are moving faster than those at bottom Rolling as Translation and Rotation Combined The rear wheel on a bicycle is half the radius of that of the front wheel. (a) When moving is the linear speed on the top of the rear wheel greater than, less than, or the same as the top of the front wheel? i.e. > or = or < (a) Is the angular speed of the real wheel greater than, equal or less than the front wheel? i.e. > or = or <

20 11-2 Kinetic Energy of Rolling! Combine translational and rotational kinetic energy: KE of rotation about c.o.m. + KE of translation of c.o.m Forces associated with (smooth) Rolling! If a wheel accelerates, its angular speed changes! Friction must act to prevent slip, f s (static friction) Since v com = "R Then dv com = d" dt dt R i.e. a com = #R where # is the angular acceleration Note: if the wheel does slide then the frictional force is the kinetic friction, f k. The motion is not then smooth and the above equation does not apply.

21 11-2 Forces associated with smooth Rolling (no slip)! For smooth rolling down a ramp: 1. The gravitational force is vertically down 2. The normal force is perpendicular to the ramp 3. The force of friction points up the slope We make use of the angular version of N2L: Torque " net =I#\$ 11-2 Forces associated with smooth Rolling (no slip) Thus, applying N2L along direction of ramp (x - direction) : f s " Mgsin# = Ma com,x Applying rotational form of N2L about an axis through c.o.m.: Rf s = I com " (the moment arm for the friction force is R; for other forces it is 0) Smoothly rolling, so a com,x = "#R The negative sign because # is positive, but a com,x is negative. a Solve for f s : f s = "I com,x com R 2 gsin# And subsitute to yield : a com,x = " 1+ I com / MR 2

22 11-3 The Yo-Yo! As a yo-yo moves down a string, it loses potential energy mgh but gains rotational and translational kinetic energy! Yo-yo accelerating down its string is like a rolling down a ramp: 1. With an angle of Rolling on axle instead of its outer surface 3. Slowed by tension T rather than friction 11-3 The Yo-Yo gsin" So place " = 90 in a com,x = # 1+ I com / MR 2 g to obtain a com = # 1+ I com / MR with I = 1 2 com 2 MR2 Example Calculate the acceleration of the yo-yo: o M = 150 grams, R=2 cm, R 0 = 3 mm, o So I com = MR 2 /2 = 3E-5 kg m 2 o Therefore a com = -9.8 m/s 2 / (1 + 3E-5 / (0.15 * )) = m/s 2

23 11-4 Torque Revisited! Previously, torque was defined only for a rotating body and a fixed axis! Now we redefine it for an individual particle that moves along any path relative to a fixed point! The path need not be a circle; torque is now a vector! Direction determined with right-hand-rule Figure Torque Revisited! The general equation for torque is:! We can also write the magnitude as:! Or, using the perpendicular component of force or the moment arm of F:

24 11-4 Torque Revisited The position vector r of a particle points along the positive z-direction. (i.e. out of the board remember the right hand rule for xyz and a b) What is the direction of the force that is causing the torque if the torque is (a) Zero (b) In the negative x-direction (c) In the negative y-direction 11-5 Angular Momentum! Angular momentum is the angular counterpart to linear momentum! Direction given by right-hand rule! Perpendicular to both r and p! Magnitude! Particle does need not rotate around O to have angular momentum about it! Unit of angular momentum:! kg m 2 /s, or J s

25 11-6 Newton's Second Law in Angular Form! We rewrite Newton's second law as:! i.e. Torque is the rate of change of angular momentum! The torque and the angular momentum must be defined with respect to the same point (usually the origin)! Note the similarity to the linear form: 11-7 Angular Momentum of a Rigid Body! We sum the angular momenta of the particles to find the angular momentum of a system of particles:! The rate of change of the net angular momentum is:! In other words, the net torque is defined by the rate of change of the net angular momentum:! With

26 11-7 Angular Momentum of a Rigid Body 11-7 Angular Momentum of a Rigid Body I = 1 2 MR2 I = MR 2 I = 2 5 MR2

27 11-8 Conservation of Angular Momentum! Since we have a new version of Newton's Second Law, we also have a new conservation law: Eq. (11-32)! The law of conservation of angular momentum states that, for an isolated system, (net initial angular momentum) = (net final angular momentum) Eq. (11-33) 11-8 Conservation of Angular Momentum! Since these are vector equations, they are equivalent to the three corresponding scalar equations! This means we can separate axes and write:! If the distribution of mass changes with no external torque, we have:

28 11-8 Conservation of Angular Momentum Examples! A student spinning on a stool: rotation speeds up when arms are brought in, slows down when arms are extended! A springboard diver: rotational speed is controlled by tucking her arms and legs in, which reduces rotational inertia and increases rotational speed! A long jumper: the angular momentum caused by the torque during the initial jump can be transferred to the rotation of the arms, by windmilling them, keeping the jumper upright

29 11-8 Conservation of Angular Momentum An insect rides the rim of a rotating disk. As it crawls towards the disk s centre, do the following increase, decrease or stay the same? (a) Moment of Inertia (b) Angular Momentum (c) Angular Speed # 10 Summary Angular Position! Measured around a rotation axis, relative to a reference line: Eq. (10-1) Angular Displacement A change in angular position Eq. (10-4) Angular Velocity and Speed Average and instantaneous values: Eq. (10-5) Angular Acceleration Average and instantaneous values: Eq. (10-7) Eq. (10-6) Eq. (10-8)

30 10 Summary Kinematic Equations Given in Table 10-1 for constant acceleration Match the linear case Linear and Angular Variables Related Linear and angular displacement, velocity, and acceleration are related by r Rotational Kinetic Energy and Rotational Inertia Eq. (10-34) The Parallel-Axis Theorem Relate moment of inertia around any parallel axis to value around com axis Eq. (10-36) Eq. (10-33) 10 Summary Torque Force applied at distance from an axis: Moment arm: perpendicular distance to the rotation axis Work and Rotational Kinetic Energy Eq. (10-39) Eq. (10-53) Newton's Second Law in Angular Form Eq. (10-42) Eq. (10-55)

31 11 Summary Rolling Bodies Eq. (11-2) Eq. (11-5) Eq. (11-6) Torque as a Vector! Direction given by the righthand rule Eq. (11-14) Angular Momentum of a Particle Newton's Second Law in Angular Form Eq. (11-18) Eq. (11-23) 11 Summary Angular Momentum of a System of Particles Angular Momentum of a Rigid Body Eq. (11-31) Eq. (11-26) Eq. (11-29) Conservation of Angular Momentum Eq. (11-32) Precession of a Gyroscope Eq. (11-46) Eq. (11-33)

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