Prof. Rupak Mahapatra. Physics 218, Chapter 15 & 16

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1 Physics 218 Chap 14 & 15 Prof. Rupak Mahapatra Physics 218, Chapter 15 & 16 1

2 Angular Quantities Position Angle θ Velocity Angular Velocity ω Acceleration Angular Acceleration α Moving forward: Force Mass Momentum Energy Physics 218, Chapter 15 & 16 2

3 Torque Torque is the analogue of Force Take into account the perpendicular distance from axis Same force further from the axis leads to more Torque Physics 218, Chapter 15 & 16 3

4 Slamming a door We know this from experience: If we want to slam a door really hard, we grab it at the end If we try to push in the middle, we aren t able to make it slam nearly as hard Physics 218, Chapter 15 & 16 4

5 Torque Continued What if we change the angle at which the Force is applied? What is the Effective Radius? Physics 218, Chapter 15 & 16 5

6 Slamming a door We also know this from experience: If we want to slam a door really hard, we grab it at the end and throw perpendicular to the hinges If we try to pushing towards the hinges, the door won t even close Physics 218, Chapter 15 & 16 6

7 Torque Torque is our slamming ability Need some new math to do Torque Write Torque as τ τ = r F sinθ r τ r r = r F To find the direction of the torque, wrap your fingers in the direction the torque makes the object twist Physics 218, Chapter 15 & 16 7

8 Vector Cross Product C r r r r = A B Θ C = A B Sin Θ This is the last way of multiplying py vectors we will see Direction from the iht right-hand h rule Swing from A into B! Physics 218, Chapter 15 & 16 8

9 Vector Cross Product Cont Multiply py out, but use the Sinθ to give the magnitude, and RHR to give the direction ˆi ˆi = 0 (sin θ = 0 ) îi ĵj = kˆ (i (sin θ = 1 ) îi kˆ = ĵj (i (sin θ = 1 ) Physics 218, Chapter 15 & 16 9

10 Cross Product Example r A = A X ˆî + A Y ˆĵ r B = B îi + = + X r A B using B Y r ĵj What is A B i Unit Vector notation? Physics 218, Chapter 15 & 16 10

11 Torque and Force Torque problems are like Force problems 1. Draw a force diagram 2. Then, sum up all the torques to find the total torque Is torque a vector? Physics 218, Chapter 15 & 16 11

12 Example: Composite Wheel Two forces, F 1 and F 2, act on different radii of a wheel, R 1 and R 2, at different angles Θ 1 and Θ 2. Θ 1 is a right angle. If the axis is fixed, what is the net torque on the wheel? Θ 1 F 1 Θ 2 Physics 218, Chapter 15 & F 2

13 Angular Quantities Position Angle θ Velocity Angular Velocity ω Acceleration Angular Acceleration α Moving forward: Force Torque τ Mass Momentum Energy Physics 218, Chapter 15 & 16 13

14 Analogue of Mass The analogue of Mass is called Moment of Inertia Example: A ball of mass m moving in a circle of radius R around a point has a moment of inertia F=ma τ=ια Physics 218, Chapter 15 & 16 14

15 Calculate Moment of Inertia Calculate the moment of inertia for a ball of mass m relative to the center of the circle R Physics 218, Chapter 15 & 16 15

16 Moment of Inertia To find the mass of an object, just add up all the little l pieces of mass To find the moment m of inertia around a point, just add up all the little moments I mr 2 or = = I r dm 2 Physics 218, Chapter 15 & 16 16

17 Torque and Moment of Inertia Force vs. Torque F=ma τ = Iα Mass vs. Moment of Inertia m I = mr 2 or I 2 = r dm Physics 218, Chapter 15 & 16 17

18 Pulley and Bucket A heavy ypulley, with radius R, and known moment of inertia I starts at rest. We attach it to a bucket with mass m. The fiti friction torque is τ fric. Find the angular acceleration α Physics 218, Chapter 15 & 16 18

19 Spherical Heavy Pulley A heavy pulley, with radius R, starts at rest. We pull on an attached rope with a constant force F T. It accelerates to an angular speed of ω in time t. What is the moment m of inertia of the pulley? R Physics 218, Chapter 15 & 16 19

20 Less Spherical Heavy Pulley A heavy pulley, with radius R R, R starts at rest. We pull on an attached rope with constant force F T. It accelerates to final angular speed ω in time t. A better estimate takes into account that there is friction in the system. This gives a torque (due to the axel) we ll call this τ fric. What is this better estimate of the moment of Inertia? Physics 218, Chapter 15 & 16 20

21 Example of Cross Product The location of a body is length r from the origin and at an angle θ from the x- axis. A force F acts on the body purely in the y direction. What is the Torque on the body? x θ z y Physics 218, Chapter 15 & 16 21

22 Calculate Moment of Inertia 1.Calculate the moment of inertia for a ball of mass m relative to the center of the circle R 2.What about lots of fpoints? For example a wheel Physics 218, Chapter 15 & 16 22

23 Rotating Rod A uniform rod of mass m, length l, and moment of inertia I = ml 2 /3 rotates around a pivot. It is held horizontally and released. Find the angular acceleration α and the linear acceleration a at the end. Where, along the rod, is a = g? g Physics 218, Chapter 15 & 16 23

24 Two weights on a bar Find dthe middle moment of inertia for the two different Axes Physics 218, Chapter 15 & 16 24

25 Moments of Inertia Physics 218, Chapter 15 & 16 25

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