 Previous chapters described motion along a straight line › Translational (linear) motion  This chapter we will focus on rotational motion around some fixed axis.  Motion can be more fully described by both translational and rotational motion › Rolling

Rolling Motion

Two views of rolling motion: 1) Pure rotation around the instantaneous axis or 2) rotation and translation.

 Need similar concepts for objects moving in circle (CD, merry-go-round, etc.)  As before: › need a fixed reference system (line) › use polar coordinate system

Previously, we defined a displacement in translational motion as Δx = x – x 0  In order to define displacement for rotation we will use an angular measure called the radian › Θ = 1 radian ≈ 57°

 The radian can be defined as the arc length s along a circle divided by the radius r  Comparing degrees and radians  Converting from degrees to radians

 Every point on the object undergoes circular motion about the point O  Angles generally need to be measured in radians  Note: length of arc radius

 Axis of rotation is the center of the disk › Fixed origin O  Need a fixed reference line › Usually, x-axis  During time t, the reference line moves through angle θ

 The angular displacement is defined as the angle the object rotates through during some time interval  Every point on the disc undergoes the same angular displacement in any given time interval

 Rigid Body › Every point on the object undergoes circular motion about the point O › All parts of the object of the body rotate through the same angle during the same time › The object is considered to be a rigid body  This means that each part of the body is fixed in position relative to all other parts of the body

 Translation Motion Δx = x f – x i  Rotational Motion Δθ = θ f - θ i  The unit of angular displacement is the radian  Each point on the object undergoes the same angular displacement

 The average angular velocity (speed), ω, of a rotating rigid object is the ratio of the angular displacement to the time interval  Analogous to linear motion

 The instantaneous angular speed is defined as the limit of the average speed as the time interval approaches zero  Units of angular speed are radians/sec › rad/s  Speed will be positive if θ is increasing (counterclockwise)  Speed will be negative if θ is decreasing (clockwise)

 What if object is initially at rest and then begins to rotate?  The average angular acceleration, , of an object is defined as the ratio of the change in the angular speed to the time it takes for the object to undergo the change:  Units are rad/s²  Similarly, instant. angular accel.:

 Angular Acceleration  Units of angular acceleration are rad/s²  Every point of the object has the same angular speed and the same angular acceleration  The sign of the acceleration does not have to be the same as the sign of the angular speed  Angular acceleration is positive if an object rotating counterclockwise is speeding up or if an object rotating clockwise is slowing down.  The instantaneous angular acceleration is defined as the limit of the average acceleration as the time interval approaches zero

 Since rotational motion is analogous to translational motion we can apply equations of motion similar to linear motion with constant acceleration  Rotational Kinematics

When a rigid object rotates about a fixed axis, every portion of the object has the same angular speed and the same angular acceleration  i.e. , and  are not dependent upon r, distance form hub or axis of rotation

 Example: A wheel rotates with a constant angular acceleration of 3.50 rad/s 2. If the angular velocity of the wheel is 2.00 rad/s at t=0, a)Through what angle does the wheel rotate between t=0 and 2.00 s? (in radians and revolutions) b)What is the angular velocity of the wheel at t = 2.00s?

 Want to consider how angular quantities relate to linear quantities  Consider an arbitrarily shaped object  Recall s = rθ  Determine change in θ with respect to time

 Starting with, Δθ = Δs / r Divide by Δt, we arrive at v t = rω Where v t is the tangential speed

 Similarly, applying the same procedure as before to the tangential speed, we get a t = rα where a t is the tangential acceleration

Angular quantities vs. Linear quantities on a rotating object  Every point on the rotating object has the same angular motion Same for all r  Every point on the rotating object does NOT have the same linear motion Increases with increasing r

 Displacements  Speeds  Accelerations

A ladybug sits at the outer edge of a merry-go-round, and a gentleman bug sits halfway between her and the axis of rotation. The merry-go-round makes a complete revolution once each second.The gentleman bug’s angular speed is 1. half the ladybug’s. 2. the same as the ladybug’s. 3. twice the ladybug’s. 4. impossible to determine

A ladybug sits at the outer edge of a merry-go-round, and a gentleman bug sits halfway between her and the axis of rotation. The merry-go-round makes a complete revolution once each second.The gentleman bug’s angular speed is 1. half the ladybug’s. 2. the same as the ladybug’s. 3. twice the ladybug’s. 4. impossible to determine Note: both insects have an angular speed of 1 rev/s

 The carnival ride, The Gravitron, spins at 40 revolutions per minute and has a radius of 3 meters. If the distance traveled in one rotation is the circumference of the ride, what is your tangential (linear) speed if you are sitting on the outside edge of this ride?

 As in the linear case, displacement, velocity and acceleration are vectors:  Assign a positive or negative direction  A more complete way is by using the right hand rule › Grasp the axis of rotation with your right hand › Wrap your fingers in the direction of rotation › Your thumb points in the direction of ω

 Angular quantities as vectors › Instinctively, we expect that something be moving along the direction of a vector › This is NOT the case for angular quantities › Instead, something is rotating around the direction of the vector › In the world of rotation, a vector defines an axis of rotation

 Velocity Directions › In a, the disk rotates clockwise, the velocity is into the page › In b, the disk rotates counterclockwise, the velocity is out of the page

 Acceleration Directions › If the angular acceleration and the angular velocity are in the same direction, the angular speed will increase with time › If the angular acceleration and the angular velocity are in opposite directions, the angular speed will decrease with time

 Dynamics of a Rigid Body › Previous chapters discussed whether a force was applied or not, not where it was applied › We can generalize concepts of force to describe rotational dynamics  Generalize Newton’s Laws  Equilibrium › Extend conservation laws  Energy  Momentum

 Consider force required to open door. Is it easier to open the door by pushing/pulling away from hinge or close to hinge? close to hinge away from hinge Farther from from hinge, larger rotational effect!

 Torque  Consider a door that rotates about a hinge › The door is free to rotate about an axis through O › Force is perpendicular › τ = rF › + CCW - CW

 If the applied force is not perpendicular we must take the components of the Force vector  However, only a net torque perpendicular will cause it to rotate

The magnitude of the torque t exerted by the force F is τ = rFsinθ  sin 90 = 1, perpendicular max force  sin 270 = -1, negative dir. max force  sin 0 = sin 180 = 0, no force

 The value of τ depends on the chosen axis of rotation › Once the axis is chosen, apply right hand rule to determine direction 1. Point fingers in direction of r 2. Curl fingers in the direction of F 3. Your thumb points in the direction of the torque Torque is out of the screen

 Newton’s Law Analog 1. The rate of an object does not change, unless acted on by a net torque 2. The angular acceleration of an object is proportional to the net torque  There are three factors that determine the effectiveness of torque: › The magnitude of the force › The position of the application of the force › The angle at which the force is applied

 If the net torque is zero, the object’s rate of rotation doesn’t change  Equilibrium – arbitrary axis › May have convenient location › When solving a problem, you must specify an axis of rotation and maintain it

You are using a wrench and trying to loosen a rusty nut. Which of the arrangements shown is most effective in loosening the nut? List in order of descending efficiency the following arrangements:

2, 1, 4, 3 or 2, 4, 1, 3

 Example:

 The net torque is the sum of all the torques produced by all the forces › Remember to account for the direction of the tendency for rotation  Counterclockwise torques are positive  Clockwise torques are negative

What if two or more different forces act on lever arm?

Given: weights: w 1 = 500 N w 2 = 800 N lever arms: d 1 =4 m d 2 =2 m Find:  = ? 1. Draw all applicable forces 2. Consider CCW rotation to be positive 500 N 800 N 4 m2 m Rotation would be CCW N Determine the net torque:

FFirst Condition of Equilibrium TThe net external force must be zero ›T›This is a necessary, but not sufficient, condition to ensure that an object is in complete mechanical equilibrium ›T›This is a statement of translational equilibrium SSecond Condition of Equilibrium TThe net external torque must be zero TThis is a statement of rotational equilibrium

SSo far we have chosen obvious axis of rotation IIf the object is in equilibrium, it does not matter where you put the axis of rotation for calculating the net torque ›T›The location of the axis of rotation is completely arbitrary ›O›Often the nature of the problem will suggest a convenient location for the axis ›W›When solving a problem, you must specify an axis of rotation OOnce you have chosen an axis, you must maintain that choice consistently throughout the problem

 A zero net torque does not mean the absence of rotational motion › An object that rotates at uniform angular velocity can be under the influence of a zero net torque  This is analogous to the translational situation where a zero net force does not mean the object is not in motion

 Isolate the object to be analyzed  Draw the free body diagram for that object › Include all the external forces acting on the object

Where would the 500 N person have to be relative to fulcrum for zero torque?

Given: weights: w 1 = 500 N w 2 = 800 N lever arms: d 1 =4 m  = 0 Find: d 2 = ? 1. Draw all applicable forces and moment arms 500 N 800 N 2 md 2 m According to our understanding of torque there would be no rotation and no motion! N’ y What does it say about acceleration and force? Thus, according to 2 nd Newton’s law  F=0 and a=0!

So far: net torque was zero. What if it is not?

 Torque, , is the tendency of a force to rotate an object about some axis  Forces cause acceleration a = F/m = Δv / Δt  Torques cause angular accelerations › Angular acceleration about some fixed point O at some length r  Force and torque must be related in some way

WWhen a rigid object is subject to a net torque (≠0), it undergoes an angular acceleration TThe angular acceleration is directly proportional to the net torque ›T›The relationship is analogous to ∑F = ma NNewton’s Second Law

TThe angular acceleration is directly proportional to the net torque TThe angular acceleration is inversely proportional to the moment of inertia of the object TThere is a major difference between moment of inertia and mass: the moment of inertia depends on the quantity of matter and its distribution in the rigid object. TThe moment of inertia also depends upon the location of the axis of rotation

 Moment of Inertia is rotational equivalent of mass  Objects with larger mass (more inertia) are harder to accelerate, objects with larger moments of inertia are harder to rotate › Easier to spin a wheel or rod with mass located at center  Objects moment of inertia depends on not only on the object’s mass but how the mass is distributed

 Moment of Inertia for particles  I = Σmr 2 = m 1 r m 2 r 2 2 +m 3 r 3 2 r2r2 r1r1 r3r3 m1m1 m2m2 m3m3

Moment of Inertia of a Uniform Ring Calculate the moment of inertia Imagine the hoop is divided into a number of small segments, m 1 … These segments are equidistant from the axis

 Other moments of inertia The rotational inertia of an object depends not only on its mass distribution but also the location of the axis of rotation – compare (f) and (g), for example.

a) a) solid aluminum b) hollow gold c) same same mass & radius solid hollow Two spheres have the same radius and equal masses. One is made of solid aluminum, and the other is made from a hollow shell of gold. Which one has the bigger moment of inertia about an axis through its center?

a) a) solid aluminum b) hollow gold c) same same mass & radius solid hollow Two spheres have the same radius and equal masses. One is made of solid aluminum, and the other is made from a hollow shell of gold. Which one has the bigger moment of inertia about an axis through its center? Moment of inertia depends on mass and distance from axis squared. It is bigger for the shell since its mass is located farther from the center.

AAn object rotating about some axis with an angular speed, ω, has rotational kinetic energy ½Iω 2 EEnergy concepts can be useful for simplifying the analysis of rotational motion CConservation of Mechanical Energy ›R›Remember, this is for conservative forces, no dissipative forces such as friction can be present ›R›Rolling race!

Rotational Kinetic Energy When using conservation of energy, both rotational and translational kinetic energy must be taken into account. All these objects have the same potential energy at the top, but the time it takes them to get down the incline depends on how much rotational inertia they have.

 Other moments of inertia The rotational inertia of an object depends not only on its mass distribution but also the location of the axis of rotation – compare (f) and (g), for example.

Work-Energy in a Rotating System  In the case where there are dissipative forces such as friction, use the generalized Work-Energy Theorem instead of Conservation of Energy  W nc =  KE t +  KE R +  PE

 The same basic techniques that were used in linear motion can be applied to rotational motion. › Analogies: F becomes, m becomes I and a becomes, v becomes ω and x becomes θ  Techniques for conservation of energy are the same as for linear systems, as long as you include the rotational kinetic energy

 Example: Problem Use conservation of energy to determine the angular speed of the spool after the bucket has fallen 4.00m starting from rest. The light string attached to the bucket is wrapped around the spool and does not slip as it unwinds.

 Similarly to the relationship between force and momentum in a linear system, we can show the relationship between torque and angular momentum  Angular momentum is defined as L = I ω › and

SSimilarly to the relationship between force and momentum in a linear system, we can show the relationship between torque and angular momentum AAngular momentum is defined as L = I ω IIf the net torque is zero, the angular momentum remains constant CConservation of Linear Momentum states: The angular momentum of a system is conserved when the net external torque acting on the systems is zero. ›T›That is, when (compare to )

 In an isolated system, the following quantities are conserved: › Mechanical energy › Linear momentum › Angular momentum

 With hands and feet drawn closer to the body, the skater’s angular speed increases › L is conserved, I decreases,  increases › Ice skater Ice skater

 Example: