Chapter 8 Rotational Kinematics – Angular displacement, velocity, acceleration © 2014 Pearson Education, Inc. Info in red font is not necessary to copy

Rotational motion – the motion of a spinning object Any point on a spinning object undergoes circular motion The motion of a point can be described by the angle the point moves in a certain time

Angular Displacements θ l r r = radius of circle or distance from axis l = arc length or distance travelled by the point θ = the angle travelled in radians θ r l Not on equation sheet

Any angle can be converted from radians to degrees & vice versa θ(rad) = π θ(deg) revolution = 2π rad = 360º Radians & Degrees Not on equation sheet

Angular Kinematics Angular displacement: Δθ = θ 2 – θ 1 Angular Velocity (units of rad/s): © 2014 Pearson Education, Inc. Not on equation sheet

Angular Acceleration (units of rad/s/s): © 2014 Pearson Education, Inc. All points on a rotating object have the same angular speed and angular acceleration regardless of their location. However, their linear (tangential) speed and acceleration are different. The tangential speed of the outermost point of the object is also the translational speed of the whole object Different form on equation sheet v = rω Not on equation sheet You must memorize!! a = rα

Objects farther from the axis of rotation will move faster. © 2014 Pearson Education, Inc.

Centripetal Acceleration © 2014 Pearson Education, Inc. Not on equation sheet – you can derive from what is on the equation sheet and what you must memorize

Angular & Linear Kinematic Equations Angular Equations: ω = ω 0 + αt θ = θ 0 + ω 0 t + ½αt 2 ω 2 = ω α(θ-θ 0 ) Linear Equations: v = v 0 + at x = x 0 + v 0 t + ½ at 2 v 2 = v a(x-x 0 ) Not on equation sheet

Frequency and Period The frequency is the number of complete revolutions per second: f = ω/2π Frequencies are measured in hertz. 1 Hz = 1 s −1 The period is the time one revolution takes: =2 π/ ω © 2014 Pearson Education, Inc. On equation sheet

Rolling Motion (Without Slipping) In (a), a wheel is rolling without slipping. The point P, touching the ground, is instantaneously at rest, and the center moves with velocity v. (Remember static friction for rolling wheels!) In (b) the same wheel is seen from a reference frame where C is at rest. Now point P is moving with velocity –v. Relationship between linear and angular speeds: v = rω © 2014 Pearson Education, Inc.

Practice Problems 8-1. A bike wheel rotates 4.50 revolutions. How many radians has it rotated? © 2014 Pearson Education, Inc.

Practice Problems 8-2. A particular bird’s eye can just distinguish objects that move forming an angle no smaller than about 3x10 -4 rad. A) How many degrees is this? B)How small can the length of the object be so the bird can just distinguish it when flying at a height of 100m? © 2014 Pearson Education, Inc.

Practice Problems 8-3 (Modified). On a rotating carousel or merry-go-round, one child sits on a horse near the outer edge and another child sits on a lion halfway out from the center. A) Which child has the greatest linear velocity? B) Which child has the greater angular velocity? C) Each child drops a piece of popcorn. Which piece of popcorn is most likely to move from where it is dropped? © 2014 Pearson Education, Inc.

Practice Problems 8-4. A carousel is initially at rest. At t=0, it is given a constant angular acceleration of α=0.060rad/s/s which increases its angular velocity for 8.0s. A) What is the angular velocity of the carousel? B) What is the linear velocity of a child that is located 2.5m away from the center? © 2014 Pearson Education, Inc.

Practice Problems 8-6 (Modified). A centrifuge rotor is accelerated from rest for 30s to 20,000rpm (rev/min). A) What is its average angular acceleration? B)Through how many radians worth of revolutions does the rotor turn during this time? How many revolutions is this? © 2014 Pearson Education, Inc.

Practice Problems 8-7. A bike slows down from 8.40m/s to rest over a distance of 115m. Each wheel has a diameter of 68.0cm. A)What is the angular velocity of the wheels initially? B)How many revolutions do the wheels rotate while coming to a stop? C)what is the angular acceleration of the wheels? D)How long does it take to stop? © 2014 Pearson Education, Inc.

Chapter 8 Rotational Dynamics – Torque, Momentum, KE © 2014 Pearson Education, Inc. Info in red font is not necessary to copy

Forces Causing Rotational Motion To make an object start rotating, a force is needed The position and direction of the force matter The perpendicular distance from the axis of rotation to the line along which the force acts is called the lever arm (r) or moment arm. The longer the lever arm, the easier the rotation © 2014 Pearson Education, Inc.

Which force will be more successful? Rank the lever arms: r A >r C >r D =0 So, F A has greatest torque and F D has zero Angular acceleration is proportional to the force and the lever arm © 2014 Pearson Education, Inc.

Torque – the product of force and the lever arm a measure of how much a force acting on an object causes that object to rotate (units mN) © 2014 Pearson Education, Inc. =rFsinθ

© 2014 Pearson Education, Inc. Net torque is the sum of all torques Torque from gravity is negative, torque from person is positive

Rotational equilibrium If an object is in angular equilibrium, then it is either at rest or else it is rotating with a constant angular velocity The net torque is zero © 2014 Pearson Education, Inc.

Torque and Rotational Inertia The amount of torque required to get an object rotating depends on the object’s rotational inertia (moment of inertia) I = Σmr 2 is the rotational inertia of an object. I and mass (inertia) are related to each other. One for rotation and the other for translation. The distribution of mass matters—these two objects have the same mass, but the one on the top has a greater rotational inertia (I), as so much of its mass is far from the axis of rotation. More torque is required for the top one. © 2014 Pearson Education, Inc.

Torque and Rotational Inertia © 2014 Pearson Education, Inc.

Different shapes all have their own unique rotational inertia You are not responsible for knowing how to calculate I for different objects – you must understand the concept and apply it in equations when given to you Know this: I long rod< I sphere < I solid cylinder (disc) < I hoop © 2014 Pearson Education, Inc. 5cyfE

Rotational Kinetic Energy A object that has both translational and rotational motion also has both translational and rotational kinetic energy: Conservation of energy must include both rotational and translational kinetic energy © 2014 Pearson Education, Inc. Not on equation sheet

Solid wins because lower I, so more E goes into translational KE Turns out this is independent of mass and radius Speed at bottom only depends on gravitational PE and shape © 2014 Pearson Education, Inc. Which makes it to the bottom first (same mass)?

© 2014 Pearson Education, Inc. Which makes it to the bottom first (same mass and frictionless ramp)?

Your car sliding on ice and anti-lock brakes What kind of friction stronger? How does kinetic energy also now explain this: Sliding: only K=1/2mv2 Rolling: K=1/2mv2 + 1/2Iω2 So translational motion will be faster (and more out of control) when sliding on the ice. Take your foot of the break! © 2014 Pearson Education, Inc.

Angular Momentum and Its Conservation Angular momentum L: The total torque is the rate of change of angular momentum. No net torque from outside forces means angular momentum is conserved. ΔL = τΔt If the net torque on an object is zero, the total angular momentum is constant. Iω = I 0 ω 0 = constant © 2014 Pearson Education, Inc.

Iω = I 0 ω 0 = constant Systems/objects that can change their rotational inertia through internal forces will also change their rate of rotation: © 2014 Pearson Education, Inc.

Right Hand Rule of Angular Momentum The angular velocity vector points along the axis of rotation; its direction is found using a right hand rule -curl fingers of right hand in direction of rotation, thumb points in direction of vector © 2014 Pearson Education, Inc.

Angular acceleration and angular momentum vectors also point along the axis of rotation. Use the right hand rule – in what direction is the vector of the person’s angular velocity and momentum? How about the platform? © 2014 Pearson Education, Inc.

Angular motion demo © 2014 Pearson Education, Inc. momentum and torque are perpendicular, so momentum is chasing torque like video says and change in momentum is same direction as torque

Practice Problems 8-8. The biceps muscle exerts a vertical force of 700N on the lower arm. Calculate the torque about the axis of rotation through the elbow joint assuming the muscle is attached 5.0cm from the axis of rotation. © 2014 Pearson Education, Inc.

Practice Problems A 15.0N force is applied to a cord wrapped around a pulley of mass 4.00kg and a radius of 33.0cm. The pulley accelerates from rest to an angular speed of 30.0rad/s in 3.00s. A) What is the angular acceleration of the pulley? B) If there is a frictional torque of 1.10 mN at the axle, what is the moment of inertia of the pulley? © 2014 Pearson Education, Inc.

Practice Problems 8-12 Modified. What is the speed of a 1.0kg solid sphere that has an I=0.4kgm 2 that rolls down a ramp from a height of 2.5m with an angular velocity of 5.5rad/s? © 2014 Pearson Education, Inc.

Practice Problems Several objects roll without slipping down an incline with a vertical height, h. They all start from rest at the exact same time. The objects are a thin metal hoop, a marble, a solid D-cell battery, and an empty soup can. Additionally, a greased box slides down without friction. In what order do they reach the bottom of the incline or do they all reach it at the same time? © 2014 Pearson Education, Inc.

Practice Problems 8-14 Modified. A simple clutch consists of 2 cylindrical plates that can be pressed together to connect two sections of an axle. Plate A has an I=1.08kgm 2 and plate B has an I=1.62kgm 2. A) If the plates are initially separated and Plate A is accelerated from rest to an angular velocity of 7.2rad/s in 2.0s, what is the final angular momentum of Plate A? B)What was the torque required to cause this acceleration? C) If Plate B is initially at rest and then placed into firm contact with Plate A, what is the new angular velocity of the two plates? © 2014 Pearson Education, Inc.