Midterm 3 Thursday April 13th

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1 Welcome back to Physics 215 Today s agenda: Angular momentum Rolling without slipping Midterm Review Physics 215 Spring 2017 Lecture Midterm 3 Thursday April 13th Material covered: Ch 9 Ch 12 Lectures through this Thursday Also draws on material from Ch 1-8 HW, lectures, demos, recitations No office hours after exam on Thursday April 13 th or Friday April 14 th. Extra office hours will be posted online Physics 215 Spring 2017 Lecture

2 Women in Physics organizational meeting Friday (tomorrow) 3-4 pm in 233 snacks and coffee goal is to provide mentorship and support to women undergrads and grads who are interested in physics all are welcome! Physics 215 Spring 2017 Lecture Angular Momentum 1. q r p O Point particle: L = r p sin(q ) = m r v sin(q ) vector form à L = r x p direction of L given by right hand rule (into paper here) L = mvr if v is at 90 0 to r for single particle Physics 215 Spring 2017 Lecture

3 Angular Momentum 2. w o rigid body: * L = I w (fixed axis of rotation/fixed axle) * direction along axis into paper here Physics 215 Spring 2017 Lecture Rotational Dynamics t = I a D L/D t = t These are equivalent statements If no net external torque: t = 0 à * L is constant in time * Conservation of Angular Momentum * Internal forces/torques do not contribute to external torque. Physics 215 Spring 2017 Lecture

4 Demo - wheels Physics 215 Spring 2017 Lecture Linear and rotational motion Force Acceleration F net = F = ma Momentum p = mv Kinetic energy K = 1 2 mv 2 Torque Angular acceleration τ net = τ = I α Angular momentum** L = Iω Kinetic energy K = 1 2Iω 2 ** about a fixed axle or axis of symmetry Physics 215 Spring 2017 Lecture

5 SG A hammer is held horizontally and then released. Which way will it fall? Physics 215 Spring 2017 Lecture QuickCheck SG Two buckets spin around in a horizontal circle on frictionless bearings. Suddenly, it starts to rain. As a result, A. The buckets speed up because the potential energy of the rain is transformed into kinetic energy. B. The buckets continue to rotate at constant angular velocity because the rain is falling vertically while the buckets move in a horizontal plane. C. The buckets slow down because the angular momentum of the bucket + rain system is conserved. D. The buckets continue to rotate at constant angular velocity because the total mechanical energy of the bucket + rain system is conserved. E. None of the above. Physics 215 Spring 2017 Lecture

6 General motion of extended objects Net force è acceleration of CM Net torque about CM è angular acceleration (rotation) about CM Resultant motion is superposition of these two motions Total kinetic energy K = K CM + K rot Physics 215 Spring 2017 Lecture SG Three identical rectangular blocks are at rest on a flat, frictionless table. The same force is exerted on each of the three blocks for a very short time interval. The force is exerted at a different point on each block, as shown. After the force has stopped acting on each block, which block will spin the fastest? 1. A. 2. B. 3. C. 4. A and C. Top-view diagram Physics 215 Spring 2017 Lecture

7 SG Three identical rectangular blocks are at rest on a flat, frictionless table. The same force is exerted on each of the three blocks for a very short time interval. The force is exerted at a different point on each block, as shown. After each force has stopped acting, which block s center of mass will have the greatest speed? 1. A. 2. B. Top-view diagram 3. C. 4. A, B, and C have the same C.O.M. speed. Physics 215 Spring 2017 Lecture EXAMPLE Two Interacting Disks A 20-cm diameter, 2.0 kg solid disk is rotating at 200 rpm. A 20-cm-diameter, 1.0 kg circular loop is dropped straight down onto the rotating disk. Friction causes the loop to accelerate until it is riding on the disk. What is the final angular velocity of the combined system? Physics 215 Spring 2017 Lecture

8 Rolling without slipping D x cm = D x cm = v cm = v cm = Physics 215 Spring 2017 Lecture Rolling without slipping translation rotation v cm = a cm = Physics 215 Spring 2017 Lecture

9 Rolling without slipping F N S F = ma CM W q S t = Ia Now a CM = Ra if no slipping So, ma CM and F = Physics 215 Spring 2017 Lecture Kinetic energy of rolling The total kinetic energy of a rolling object is the sum of its rotational and translational kinetic energies: Physics 215 Spring 2017 Lecture

10 5. Two cylinders with the same radius and same total mass roll down a ramp. In cylinder A, a set of 8 point masses are equally spaced in a circle with radius r1 around the cylinder s axis of rotation, while in cylinder B, the 8 point masses are a distance r2 > r1 from the center. Which cylinder reaches the bottom of the ramp first? A. Cylinder A B. Cylinder B C. They both reach the bottom at the same time D. Not enough information to tell Physics 215 Spring 2017 Lecture Example problem: Calculate the acceleration of the center of mass of a cylindrical object that rolls without slipping down a ramp of height h and angle q. The object has mass M and radius R, with a moment of inertia I = cmr 2 about its center of mass/axis of rotation. Physics 215 Spring 2017 Lecture

11 midterm review Physics 215 Spring 2017 Lecture A toy race car of mass 20 g is traveling on a frictionless loop-the-loop track. The diagram shows the track from the side. The radius of the loop track is R=0.1 m. The initial speed of the car at the bottom of the loop is 2.5 m/s. What is the speed of the car at the top of the loop? What is the normal force on the car at the top? al) R Physics 215 Spring 2017 Lecture

12 A sled slides along a horizontal surface with a coefficient of kinetic friction of 0.2. Its velocity at point A is 8.0 m/s and its velocity at point B is 5.0 m/s. How long does it take the sled to travel from point A to point B? How far did it slide? Use the work-kinetic energy and the impulse-momentum theorems. Physics 215 Spring 2017 Lecture A block of mass m=2.0kg is set into motion down a rough inclined plane with speed 4.0 m/s as shown in the picture. The coefficient of kinetic friction between the block and the plane is 0.4 (take g=9.8 m/s2). What is the speed of the block after it has moved 1.0 m down the incline? v=2.0m/s 30 0 Physics 215 Spring 2017 Lecture

13 A child sits on a seesaw of length L = 2m with his father and his pet dog. The mass of the seesaw is 20 kg, the mass of the father is 50 kg, the mass of the dog is 5 kg and the mass of the child is 20 kg. The father sits on the right edge of the seesaw, while the child sits on the left edge. The dog is 0.5 m from the father. If the seesaw is in equilibrium, what is the distance from the pivot to the child? Confirm that the net torque about this pivot point is zero. Physics 215 Spring 2017 Lecture

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