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1 7 Rotational Motion Slide 7-2

2 Slide 7-3

3 Recall from Chapter 6 Angular displacement = θ θ= ω t Angular Velocity = ω (Greek: Omega) ω = 2 π f and ω = θ/ t All points on a rotating object rotate through the same angle in the same time, and have the same frequency. Angular velocity: all points on a rotating object have the same angular velocity, ω, but different speeds, v, and v =ωr. v =ωr

4 ω is positive if object is rotating counterclockwise. (Negative if rotation is clockwise.) Conversion: 1 revolution = 2 π rad

5 Checking Understanding Two coins rotate on a turntable. Coin B is twice as far from the axis as coin A. A. The angular velocity of A is twice that of B. B. The angular velocity of A equals that of B. C. The angular velocity of A is half that of B. Slide 7-13

6 Answer Two coins rotate on a turntable. Coin B is twice as far from the axis as coin A. A. The angular velocity of A is twice that of B. B. The angular velocity of A equals that of B. C. The angular velocity of A is half that of B. All points on the turntable rotate through the same angle in the same time. ω = θ/ t All points have the same period, therefore, all points have the same frequency. ω = 2 π f Slide 7-14

7 Checking Understanding Two coins rotate on a turntable. Coin B is twice as far from the axis as coin A. A. The speed of A is twice that of B. B. The speed of A equals that of B. C. The speed of A is half that of B. Slide 7-15

8 Answer Two coins rotate on a turntable. Coin B is twice as far from the axis as coin A. A. The speed of A is twice that of B. B. The speed of A equals that of B. C. The speed, v, of A is half that of B. v = wr Twice the radius means twice the speed, v. Therefore, B moves with twice the speed of A. Slide 7-16

9 Angular Acceleration, α (Greek: Alpha) Angular acceleration α measures how rapidly the angular velocity is changing: If the turning rate, or angular velocity, ω, is constant, then there is no angular acceleration, α, but if ω is changing, then the object is NOT undergoing UCM and the object has angular acceleration Units for Angular Acceleration are rad/s/s or rad/s 2

10 Sign of the Angular Acceleration careful! +ω, +α =speeding up -ω, +α = slowing down +ω, -α = slowing down -ω, -α = speeding up

11 Do NOT confuse angular acceleration, α = Δω/ Δt, with centripetal acceleration, a c = v 2 /r = ω 2 r. Angular acceleration, α, indicates how rapidly the angular velocity, ω, is changing. (No UCM) Centripetal acceleration, a c, is a vector quantity, directed toward the center of a particle s circular path, and is nonzero even if the angular velocity (turning rate) is constant. (a c exists always whether experiencing UCM or not.)

12 Linear and Circular Motion Compared Slide 7-18

13 Linear and Circular Kinematics Compared ω f = ω i +α t Slide 7-19

14 Linear Motion Graphs related to Rotational Motion Graphs: Displacement vs. Time: Slope = Velocity θ t Angular displacement, θ, vs. Time: Slope = Angular Velocity, ω= Δθ/Δt

15 Linear Motion Graphs related to Rotational Motion Graphs: Velocity vs. Time: Slope = Acceleration ω t Angular velocity vs. Time, Slope = Angular Acceleration, α =Δω/Δt

16 Centripetal and Tangential Acceleration No UCM: Slide 7-22

17 Example Problem A high-speed drill rotating CCW takes 2.5 s to speed up to 2400 rpm. A. What is the drill s angular acceleration? B. How many revolutions does it make as it reaches top speed? Slide 7-21

18 K j k ; K j k ; Answer: Given: ω i = 0, ω f = 2400 rev/min, t = 2.5 seconds ω f = 2400 rev/min (2πrad/1 rev)(1 min/60 sec) = rad/s Find: α =?, θ = # revolutions =? Plan: Find α using ω f = ω i +α t, then find θusing θ=ω i t + 1/2 α t 2. Convert θ to revolutions. Solve: ω f = ω i +α t rad/s = 0 + α (2.5 s) α= (251.3 rad/s)/(2.5 s) = rad/s 2 = α Go to next slide for more

19 Δθ=ω i t + 1/2 α t 2 Δθ= 0 + ½ (100.5 rad/s 2 )(2.5 s) 2 Δθ= rad Δθ = rad (1 rev/2πrad) = 50 rev Δθ= 50 revolutions

20

21 Torque rhymes with fork Every time you open a door, turn on a water faucet, or tighten a nut with a wrench, you exert a turning force. This turning force produces a torque. Torque is different from a force. If you want to make an object move, apply a force. Forces tend to make things accelerate. If you want to make an object turn or rotate, apply a torque. Torques produce rotation.

22 Interpreting Torque, ( (Greek: Tau) Torque is due to the component of the force perpendicular to the radial line. t = rf^ = rfsinf Slide 7-25

23 A Second Interpretation of Torque t = r^f = rfsinf Slide 7-26

24 Signs and Strengths of the Torque Slide 7-27

25 Torque: that which causes rotational velocity to change. Variable for torque= (tau) Increasing F causes greater linear acceleration Increasing causes greater rotational acceleration Demonstration: pushing door open When does door open best? A) from outside edge to axis of rotation B) pushing toward hinge C) at an angle D) from center of door to axis of rotation (see next slide)

26 Checking Understanding The four forces shown have the same strength. Which force would be most effective in opening the door? A. Force F 1 B. Force F 2 C. Force F 3 D. Force F 4 E. Either F 1 or F 3 Slide 7-23

27 Answer The four forces shown have the same strength. Which force would be most effective in opening the door? A. Force F 1 B. Force F 2 C. Force F 3 D. Force F 4 E. Either F 1 or F 3 Slide 7-24

28 What does torque depend on? 1. The size of the force (how hard you push) 2. The point where the force is applied relative to the rotation point. 3. The direction of the force.

29 = F r Where F and r have to be to each other! = F r or = F r (Units: N m) F = the force applied r = the moment arm or the lever arm = distance between axis of rotation and line of action so that where the lever arm (r) and line of action meet, a 90 o angle is formed. Pivot point = axis of rotation (in this case, the hinge) Torque is positive when the force tends to produce a counter-clockwise rotation, and negative when force tends to produce clockwise rotation.

30 Even though the magnitude of the applied force is the same in each case, why does the bottom situation produce the greatest amount of torque?

31 For You to Solve: A person pushes on the edge of a 0.90m wide door with a force of 5.0N acting at an angle of 35 0 to the plane of the door. What is the torque of the force acting on the door?

32

33 Another problem for you to solve Revolutionaries attempt to pull down a statue of the Great Leader by pulling on a rope tied to the top of his head. The statue is 17 m tall, and they pull with a force of 4200 N at an angle of 65 to the horizontal. What is the torque they exert on the statue? If they are standing to the right of the statue, is the torque positive or negative? Slide 7-28

34 And the answer is F = F (cosθ) F = 4200N (cos 65 o ) F = 1775 N = F r = (1775 N)(17 m) F =? F = 4200N r = 17 m θ=65 o = 30,175 Nm, clockwise = - 30,175 Nm

35 Which factor does the torque on an object not depend on? A. The magnitude of the applied force. B. The object s angular velocity. C. The angle at which the force is applied. D. The distance from the axis to the point at which the force is applied. Slide 7-7

36 Answer Which factor does the torque on an object not depend on? A. The magnitude of the applied force. B.The object s angular velocity. C. The angle at which the force is applied. D. The distance from the axis to the point at which the force is applied. Slide 7-8

37

38 Slide 7-4

39 A net torque applied to an object causes??? A. a linear acceleration of the object. B. the object to rotate at a constant rate. C. the angular velocity of the object to change. D. the moment of inertia of the object to change. Slide 7-11

40 Answer A net torque applied to an object causes A. a linear acceleration of the object. B. the object to rotate at a constant rate. C.the angular velocity of the object to change. D. the moment of inertia of the object to change. Slide 7-12

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