HW 7 Q 14,20,20,23 P 3,4,8,6,8. Chapter 7. Rotational Motion of the Object. Dr. Armen Kocharian

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1 HW 7 Q 14,20,20,23 P 3,4,8,6,8 Chapter 7 Rotational Motion of the Object Dr. Armen Kocharian

2 Axis of Rotation The radian is a unit of angular measure The radian can be defined as the arc length s along a circle divided by the radius r s θ = r

3 More About Radians Comparing degrees and radians 360 1rad = = π Converting from degrees to radians π θ [ rad] = θ 180 [degrees]

4 Angular Displacement Axis of rotation is the center of the disk Need a fixed reference line During time t, the reference line moves through angle θ

5 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

6 Angular Displacement, cont. The angular displacement is defined as the angle the object rotates through during some time interval Δ = θ θ θ f i The unit of angular displacement is the radian Each point on the object undergoes the same angular displacement

7 Speed of Rotation The average angular speed, ω, of a rotating rigid object is the ratio of the angular displacement to the time interval ω av θ θ Δθ t t Δt f i = = f i

8 Angular Speed, cont. 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)

9 Average Angular Acceleration The average angular acceleration α 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: α av Δ = f i = t t Δt ω ω ω f i

10 Relation between the Period and Frequency Longer the period the lower frequency The period is equal to reciprocal of the frequency Period = 1 Frequency Equation between the frequency and period is T = 1 f = 1 f T

11 Angular Acceleration, final The sign of the acceleration does not have to be the same as the sign of the angular speed The instantaneous angular acceleration is defined as the limit of the average acceleration as the time interval approaches zero

12 Vector Nature of Angular Quantities Angular displacement, velocity and acceleration are all vector quantities Direction can be more completely defined 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

13 Velocity Directions, Example 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

14 Force vs. Torque Forces cause accelerations Torques cause angular accelerations Force and torque are related

15 Torque The door is free to rotate about an axis through O There are three factors that determine the effectiveness of the force in opening the door: The magnitude of the force The position of the application of the force The angle at which the force is applied

16 Torque, cont Torque, τ, is the tendency of a force to rotate an object about some axis τ = r F τ is the torque F is the force symbol is the Greek tau r is the length of the position vector SI unit is N. m

17 Direction of Torque Torque is a vector quantity The direction is perpendicular to the plane determined by the position vector and the force If the turning tendency of the force is counterclockwise, the torque will be positive If the turning tendency is clockwise, the torque will be negative

18 Multiple Torques When two or more torques are acting on an object, the torques are added As vectors If the net torque is zero, the object s rate of rotation doesn t change

19 General Definition of Torque The applied force is not always perpendicular to the position vector The component of the force perpendicular to the object will cause it to rotate

20 Torque, cont When the force is parallel to the position vector, no rotation occurs When the force is at some angle, the perpendicular component causes the rotation

21 Torque, final Taking the angle into account leads to a more general definition of torque: F is the force r is the position vector θ is the angle between the force and the position vector

22 Direction of Rotation Point the fingers in the direction of the position vector Curl the fingers toward the force vector The thumb points in the direction of the torque

23 Net Torque 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

24 Torque and Equilibrium First Condition of Equilibrium The net external force must be zero This is a necessary, but not sufficient, condition to ensure that an object is in complete mechanical equilibrium This is a statement of translational equilibrium

25 Torque and Equilibrium, cont To ensure mechanical equilibrium, you need to ensure rotational equilibrium as well as translational The Second Condition of Equilibrium states The net external torque must be zero r τ = 0

26 Equilibrium (Balance) The woman, mass m, sits on the left end of the see-saw The man, mass M, sits where the seesaw will be balanced Apply the Second Condition of Equilibrium and solve for the unknown distance, x

27 Axis of Rotation If the object is in equilibrium, it does not matter where you put the axis of rotation for calculating the net torque The location of the axis of rotation is completely arbitrary Often the nature of the problem will suggest a convenient location for the axis When solving a problem, you must specify an axis of rotation Once you have chosen an axis, you must maintain that choice consistently throughout the problem

28 Notes About Equilibrium 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

29 Torque and Angular Acceleration When a rigid object is subject to a net torque ( 0), it undergoes an angular acceleration The angular acceleration is directly proportional to the net torque The relationship is analogous to F = ma Newton s Second Law

30 Moment of Inertia The angular acceleration is inversely proportional to the analogy of the mass in a rotating system This mass analog is called the moment of inertia, I, of the object I mr SI units are kg m 2 2

31 Newton s Second Law for a Rotating Object τ = Iα The angular acceleration α is directly proportional to the net torque The angular acceleration is inversely proportional to the moment of inertia of the object

32 More About Moment of Inertia 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. The moment of inertia also depends upon the location of the axis of rotation

33 Rotational Kinetic Energy An object rotating about some axis with an angular speed, ω, has rotational kinetic energy ½Iω 2 Energy concepts can be useful for simplifying the analysis of rotational motion

34 Angular Momentum, cont If the net torque is zero, the angular momentum remains constant Conservation of Angular Momentum states: The angular momentum of a system is conserved when the net external torque acting on the systems is zero. That is, when τ = L = L or Iω = I ω 0, i f i i f f

35 Conservation Rules, Summary In an isolated system, the following quantities are conserved: Mechanical energy Linear momentum Angular momentum

36 Angular Momentum 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 ΔL and τ = Δ t

37 Moment of Inertia of a Uniform Ring Image the hoop is divided into a number of small segments, m 1 These segments are equidistant from the axis I = mr = MR 2 2 i i

38 Other Moments of Inertia

39 Conservation of Angular Momentum, Example With hands and feet drawn closer to the body, the skater s angular speed increases L is conserved, I decreases, ω increases

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