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Physics 218 Lecture 17 Dr. David Toback Physics 218, Lecture XVII 1 Before we begin • Chapter 8HW was due Monday • Ch. 10 reading assignment: due today • Already covered most of chapter 9 in lecture, but I’m changing the Chapter 9 HW due date to next Wednesday • Ch. 10HW still due the following Monday Physics 218, Lecture XVII 2 Rotational Motion Chapters 9 and 10 in four combined lectures • This is the 2nd of the 4 lectures • Concentrate on the relationship between linear and angular variables • Last time started kinematics… Move to dynamics just like earlier this semester Physics 218, Lecture XVII 3 Angular Quantities Last time: • Position Angle q • Velocity Angular Velocity w • Acceleration Angular Acceleration a This time we’ll start by discussing the vector nature of the variables and then move forward on the others: – Force – Mass – Momentum – Energy Physics 218, Lecture XVII 4 Physics 218, Lecture XVII 5 Angular Velocity and Acceleration Are w and a vectors? w and a clearly have magnitude Do they have direction? Physics 218, Lecture XVII 6 Right-Hand Rule Yes! Define the direction to point along the axis of rotation Right-hand Rule This is true for Q, w and a Physics 218, Lecture XVII 7 Angular Quantities • Position Angle q • Velocity Angular Velocity w • Acceleration Angular Acceleration a Moving forward: – Force – Mass – Momentum – Energy Physics 218, Lecture XVII 8 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, Lecture XVII 9 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, Lecture XVII 10 Torque Continued • What if we change the angle at which the Force is applied? • What is the “Effective Radius?” Physics 218, Lecture XVII 11 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, Lecture XVII 12 Torque • Torque is our “slamming” ability • Write Torque as t | t || r || F | sin q t r F • To find the direction of the torque, wrap your fingers in the direction the torque makes the object twist Physics 218, Lecture XVII 13 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, Lecture XVII 14 Example: Composite Wheel Two forces, F1 and F2, act on different radii of a wheel, R1 and R2, at different angles Q1 and Q2. Q1 is a right angle. If the axis is fixed, what is the net torque on the wheel? Q2 F2 Q1 F1 Physics 218, Lecture XVII 15 Angular Quantities • Position Angle q • Velocity Angular Velocity w • Acceleration Angular Acceleration a Moving forward: – Force Torque t – Mass – Momentum – Energy Physics 218, Lecture XVII 16 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 t=Ia Physics 218, Lecture XVII 17 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, Lecture XVII 18 Moment of Inertia • To find the mass of an object, just add up all the little pieces of mass To find the moment of inertia around a point, just add up all the little moments I mr 2 or I r dm Physics 218, Lecture XVII 2 19 Torque and Moment of Inertia • Force vs. Torque F=ma t = Ia • Mass vs. Moment of Inertia m I mr 2 or I r dm 2 Physics 218, Lecture XVII 20 Pulley and Bucket A heavy pulley, with radius R, and known moment of inertia I starts at rest. We attach it to a bucket with mass m. The friction torque is tfric. Find the angular acceleration a Physics 218, Lecture XVII 21 Spherical Heavy Pulley A heavy pulley, with radius R, starts at rest. We pull on an attached rope with a constant force FT. It accelerates to an angular speed of w in time t. What is the moment of inertia of the pulley? Physics 218, Lecture XVII R 22 Less Spherical Heavy Pulley A heavy pulley, with radius R, starts at rest. We pull on an attached rope with constant force FT. It accelerates to final angular speed w 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 tfric. What is this better estimate of the moment of Inertia? Physics 218, Lecture XVII R 23 Next Time • The rest of Chapter 10 – More on angular “Stuff” – Angular Momentum – Energy • Get caught up on your homework!!! • Chapter 8 was due Monday, Chapter 9 homework due Wednesday, Chapter 10 is due the Monday after that Physics 218, Lecture XVII 24 Physics 218, Lecture XVII 25 Example of Cross Product The location of a body is length r from the origin and at an angle q from the x-axis. A force F acts on the body purely in the y direction. What is the Torque on the body? z y q x Physics 218, Lecture XVII 26 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 points? For example a wheel Physics 218, Lecture XVII 27 Rotating Rod A uniform rod of mass m, length l, and moment of inertia I = ml2/3 rotates around a pivot. It is held horizontally and released. Find the angular acceleration a and the linear acceleration a at the end. Where, along the rod, is a = g? Physics 218, Lecture XVII 28 Two weights on a bar Find the moment of inertia for the two different Axes middle Physics 218, Lecture XVII 29 Schedule Changes Please see the handout for schedule changes New Exam 3 Date: Exam 3 Tuesday Nov. 26th Physics 218, Lecture XVII 30 Moments of Inertia Physics 218, Lecture XVII 31