Chapter 10 Rotational Kinematics and Energy. Copyright 2010 Pearson Education, Inc.

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1 Chapter 10 Rotational Kinematics and Energy Copyright 010 Pearson Education, Inc.

2 10-1 Angular Position, Velocity, and Acceleration Copyright 010 Pearson Education, Inc.

3 10-1 Angular Position, Velocity, and Acceleration Degrees and revolutions: Copyright 010 Pearson Education, Inc.

4 10-1 Angular Position, Velocity, and Acceleration Arc length s, measured in radians: Copyright 010 Pearson Education, Inc.

5 10-1 Angular Position, Velocity, and Acceleration Copyright 010 Pearson Education, Inc.

6 10-1 Angular Position, Velocity, and Acceleration Copyright 010 Pearson Education, Inc.

7 10-1 Angular Position, Velocity, and Acceleration Copyright 010 Pearson Education, Inc.

8 10-1 Angular Position, Velocity, and Acceleration Copyright 010 Pearson Education, Inc.

9 10- Rotational Kinematics If the angular acceleration is constant: Copyright 010 Pearson Education, Inc.

10 10- Rotational Kinematics Analogies between linear and rotational kinematics: Copyright 010 Pearson Education, Inc.

11 Example: A high speed dental drill is rotating at rads/sec. Through how many degrees does the drill rotate in 1.00 sec? Given: ω = rads/sec; Δt = 1 sec; α = 0 Want Δθ. θ = θ θ = θ 0 0 Δθ = ω Δt 0 + ω Δt 0 + ω Δt 0 = = ( rads/sec)( 1.0 sec) αδt rads = degrees Copyright 010 Pearson Education, Inc.

12 10-3 Connections Between Linear and Rotational Quantities Copyright 010 Pearson Education, Inc.

13 10-3 Connections Between Linear and Rotational Quantities Copyright 010 Pearson Education, Inc.

14 10-3 Connections Between Linear and Rotational Quantities Copyright 010 Pearson Education, Inc.

15 10-3 Connections Between Linear and Rotational Quantities This merry-go-round has both tangential and centripetal acceleration. Copyright 010 Pearson Education, Inc.

16 10-4 Rolling Motion If a round object rolls without slipping, there is a fixed relationship between the translational and rotational speeds: Copyright 010 Pearson Education, Inc.

17 10-4 Rolling Motion We may also consider rolling motion to be a combination of pure rotational and pure translational motion: Copyright 010 Pearson Education, Inc.

18 10-5 Rotational Kinetic Energy and the Moment of Inertia For this mass, Copyright 010 Pearson Education, Inc.

19 10-5 Rotational Kinetic Energy and the Moment of Inertia We can also write the kinetic energy as Where I, the moment of inertia, is given by Copyright 010 Pearson Education, Inc.

20 Example: (a) Find the moment of inertia of the system below. The masses are m 1 and m and they are separated by a distance r. Assume the rod connecting the masses is massless. ω r 1 r m 1 m r 1 and r are the distances between mass 1 and the rotation axis and mass and the rotation axis (the dashed, vertical line) respectively. Copyright 010 Pearson Education, Inc.

21 Example continued: Take m 1 =.00 kg, m = 1.00 kg, r 1 = 0.33 m, and r = 0.67 m. (b) What is the moment of inertia if the axis is moved so that is passes through m 1? = Copyright 010 Pearson Education, Inc. I I = mir = = i= 1 (.00 kg)( 0.33 m) + ( 1.00 kg)( 0.67 m) = mir i= 1 i 0.67 kg m i = = (.00 kg)( 0.00 m) + ( 1.00 kg)( 1.00 m) = 1.00 kg m m r m r m r + m r

22 10-5 Rotational Kinetic Energy and the Moment of Inertia Moments of inertia of various regular objects can be calculated: Copyright 010 Pearson Education, Inc.

23 Example: What is the rotational inertia of a solid iron disk of mass 49.0 kg with a thickness of 5.00 cm and a radius of 0.0 cm, about an axis through its center and perpendicular to it? From the previous slide: I 1 = MR = 1 ( 49.0 kg)( 0. m) = 0.98 kg m Copyright 010 Pearson Education, Inc.

24 10-6 Conservation of Energy The total kinetic energy of a rolling object is the sum of its linear and rotational kinetic energies: The second equation makes it clear that the kinetic energy of a rolling object is a multiple of the kinetic energy of translation. Copyright 010 Pearson Education, Inc.

25 10-6 Conservation of Energy If these two objects, of the same mass and radius, are released simultaneously, the disk will reach the bottom first more of its gravitational potential energy becomes translational kinetic energy, and less rotational. Copyright 010 Pearson Education, Inc.

26 Example: Two objects (a solid disk and a solid sphere) are rolling down a ramp. Both objects start from rest and from the same height. Which object reaches the bottom of the ramp first? h θ The object with the largest linear velocity (v) at the bottom of the ramp will win the race. Copyright 010 Pearson Education, Inc.

27 Copyright 010 Pearson Education, Inc.! " # $ % & + =! " # $ % & + =! " # $ % & + = + + = + + = + = R I m mgh v v R I m mgh R v I mv I mv mgh K U K U E E f f i i f i ω Apply conservation of mechanical energy: Example continued: Solving for v:

28 Example continued: The moments of inertia are: I I disk sphere = = 1 5 mr mr For the disk: For the sphere: v v disk = sphere = 4 3 gh 10 7 gh Since V sphere > V disk the sphere wins the race. Compare these to a box sliding down the ramp. vbox = gh Copyright 010 Pearson Education, Inc.

29 Summary of Chapter 10 Period: Counterclockwise rotations are positive, clockwise negative Linear and angular quantities: Copyright 010 Pearson Education, Inc.

30 Summary of Chapter 10 Linear and angular equations of motion: Tangential speed: Centripetal acceleration: Tangential acceleration: Copyright 010 Pearson Education, Inc.

31 Rolling motion: Summary of Chapter 10 Kinetic energy of rotation: Moment of inertia: Kinetic energy of an object rolling without slipping: When solving problems involving conservation of energy, both the rotational and linear kinetic energy must be taken into account. Copyright 010 Pearson Education, Inc.