We define angular displacement, θ, and angular velocity, ω. What's a radian?

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1 We define angular displacement, θ, and angular velocity, ω Units: θ = rad ω = rad/s What's a radian? Radian is the ratio between the length of an arc and its radius note: counterclockwise is + clockwise is - 1

2 Consider purely angular measures for 1 revolution ω = Δ θ Δt recall 2

3 "Suggested " problems for Thursday.... page 101: 34, 37 page 202: 1, 8, 11, 13, 14, 18 Problems if you want extra practice.... page 202: 3, 10, 15, 16, 19, 21 3

4 Rotation of a Rigid Body Translation - the object as a whole moves along a trajectory, but does not rotate. Rotation - the object rotates about a fixed point. Every point on the object moves in a circle. During uniform rotation of a rigid body - every point on the body has the same angular velocity, ω, and the same angular displacement, θ θ= ω t true for both the red and green points v = ωr & v = ωr but v v linear velocities are not the same 4

5 Angular Acceleration - if you push on the edge of a bicycle wheel it begins to rotate, if you continue to push is rotates faster. recall linear acceleration was defined By analogy we define angular acceleration, α, as Linear and Circular Motion Variables standard units: m, m/s, m/s 2 standard units: rad, rad/s, rad/s 2 5

6 Linear and Circular Motion Equations v 2 - v o 2 = 2a(x-x o ) ω 2 ω ο 2 = 2α(θ θ ο ) 1 The disk in a computer disk drive spins up to 5400 rpm in 2.00 seconds. What is the angular acceleration, α, of the disk? (give answer to 3 sig figs and in standard units) 6

7 2 For the same computer disk, at the end of 2 seconds what angle, θ, has the disk turned through? (The disk in a computer disk drive spins up to 5400 rpm in 2.00 seconds.) (standard units and 3 sig figs) a c = v 2 /r -- the acceleration an object in uniform circular motion undergoes --- and a c is always directed toward the center of the circle. Uniform Circular Motion implies ω = 0, so α = 0. But we have just discussed a case where α what does this mean for a c? 7

8 the acceleration has 2 components: the a c from before and a t, tangential acceleration a t is in the same direction as the velocity and is what caused v to increase. a t = v/ t = ( ω/ t)r a t = αr 8

9 x = rθ v = rω a t = rα a c = 0α = 0 3 A ball on the end of string swings in a horizontal circle once every second. The magnitude of which of the following is zero. Choose all the correct quantities. A B Velocity Angular Velocity C Centripetal Acceleration D E Angular Acceleration Tangential Acceleration 9

10 4 A ball on the end of string swings in a horizontal circle once every second. The magnitude of which of the following is constant, but not zero. Choose all the correct quantities. A B C D E Velocity Angular Velocity Centripetal Acceleration Angular Acceleration Tangential Acceleration Torque -- the rotational analog of force 1. the magnitude of the force the ability of a force to cause rotation depends on 3 factors: 2. the distance, r, from the pivot point that the force is applied 3. the angle, φ, at which the force is applied 10

11 τ = r x F τ = rfsinθ, with the direction given by the RHR i x i = j x j = k x k = 0 i x j = k j x k = i k x i = j j x i = -k k x j = -i i x k = -j 11

12 5 In trying to open a door, Ryan pushes perpendicular to the door's surface with a force of 240N at a distance of 75 cm from the hinges. What torque does Ryan exert on the door? (standard units and 2 sig figs) 6 Lulu uses a 20 cm long wrench to turn a nut. The wrench handle is tilted 30 o above the horizontal and Lulu pulls straight down on the end with a force of 100N. How much torque does Lulu exert on the nut? (standard units, 2 sig figs) 12

13 7 Which has the largest torque? The rods all have the same length and are pivoted at the dot. A B 2 N 2 N C 2 N D 4 N E 45 o 4 N 8 Which has the smallest torque? The rods all have the same length and are pivoted at the dot. A B 2 N 2 N C 2 N D 4 N E 45 o 4 N 13

14 Net Torque r = 10 cm 9 Two forces act on the wheel shown. What third force, acting at point P, will make the net torque on the wheel zero? P Pivot A B C D E 14

15 Page 232: 3, 5, 6, 7, 8, 10, 13, 14 F T m A tangential force, F, exerts a torque on the particle of mass, m, and causes a tangential acceleration, a t the torque is given by: how is this related to a t? 15

16 What if there is more than on particle in motion? 16

17 We define the moment of inertia, I, as I = Σm i r i 2 so that Στ = Iα Στ = Iα I = Σm i r i 2 Recall: ΣF = ma What is the significance of the moment of inertia? 17

18 An object with a large moment of inertia is hard to start rotating (and to stop) and vice versa. Depends not only on how much mass, but how the mass is distributed. The further the mass is from the axis of rotation the larger I is and the more torque is needed to make the object rotate. 18

19 Moments for common shapes (calculated from r 2 dm) 1 Which will roll faster down a hill? A B cylindrical hoop cylindrical disk 19

20 An engine on a small airplane is specified to have a torque of 500N. m. This engine drives a 2.0 m, 40 kg single blade propeller. On start up, how long does it take the propeller to reach 2000 rpm? Conditions of Equilibrium 20

21 Consider a 100 N, 3.0 m long ladder supported by two sawhorse, placed at one end and 1 meter from the other end. What forces do the sawhorses exert on the ladder? What is the minimum value of the coefficient of friction so that a 3.0 m long ladder, inclined at an angle of 60 o will not slip? 21

22 Try this with a soda can. Center of Gravity - the point where the force of gravity exerts no torque 22

23 A 10 kg mass hangs from a pulley on a rope. The pulley is 2.00kg and has a radius of 10.0 cm. Find the tension in the rope and the acceleration of the mass. Angular Momentum for a single mass recall p = mv l = r x p = m r x v for a rigid body L = Iω 23

24 An ice skater spins around on the tips of her blades while holding a 5.0 kg mass in each hand. She begins with her hand outstretched and her hands 140 cm apart. While spinning at 2.0 rev/s she pulls the masses in a holds them 50 cm apart against his shoulders. If we neglect the mass of the skater how fast is she spinning after pulling the masses in? Angular Momentum recall: p = mv by direct analogy L = Iω Conservation of Angular Momentum L is conserved if τ net = 0 24

25 Page 234: 26, 30, 32, 34 Page 291: 30, 32 25

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