Outline. Rolling Without Slipping. Additional Vector Analysis. Vector Products. Energy Conservation or Torque and Acceleration

Transcription

1 Rolling Without Slipping Energy Conservation or Torque and Acceleration hysics 109, Class eriod 10 Experiment Number 8 in the hysics 11 Lab Manual (page 39) 3 October, 007 Outline Additional Vector analysis Former experimental approach Current approach Description of the experiment Questions Additional Vector Analysis Vector roducts We need to define what is meant by the vector dot product and the vector cross product. First, the dot product. It is the product of the projection of the second vector on the first, and is a scalar quantity A. B ABcos Then, the cross product. It has the magnitude of the area between the two vectors, and is a vector that is perpendicular to the plane of the two vectors, using the right-hand rule. A B AB sin C B A B A A.B AB cos A B AB sin C 1

2 Rotation and Linear motion Angular acceleration,, is the analog of linear acceleration, a. Torque,, is the analog of force. Units are Newton.m, but we reserve Joule for energy. Rotational inertia, or moment of inertia, I, is the analog of mass. = Iis the rotational analog of F = ma is angular velocity =/t, with units of radians/sec. Linear speed v =r Tangential acceleration = r Radial acceleration = r Torque Torque is the cross product of the vector r with the vector F. r F rfsin Former Approach Energy Conservation Kinetic energy at any time is given by the sum of the translational kinetic energy of the center of mass plus that due to the rotation about the center of mass. Equations: KE cm 1 m v and Rot 1 I cm Where I = moment of inertia about the center of mass. When rolling on a surface of radius R, v cm R so the energy can be written: KE 1 mv cm 1 I cm mr define I cm mr a constant k where 0 k 1 Energy Conservation (cont) The kinetic energy is then: 1 m v cm 1 k The previous version of this experiment measured the velocity after the object rolled down a ramp. Equating the change in potential energy to the change in kinetic energy allowed calculation of k.

3 New Approach Torque and Acceleration In this experiment we will find the acceleration of the object rolling down a ramp with an angle with respect to the horizontal. The force of gravity causes a torque about point the torque is negative. Torque is: I R F mgrsin Acceleration,, is positive. I is the moment of inertia about So, for this case, the equations are: a cm g sin 1 k giving k gsin a cm 1 Experimental Arrangement Motion Sensor N R Fg x Disk Rolling on an Inner Axle R Ra Fg x Equations:q k ' k R R a and k' g sin a cm 1 The Race of the Rolling Bodies Various round, rigid bodies are started rolling down a plane together. Which will reach the bottom first? Using conservation of energy: Mgh K 1 U 1 K U 0 Mgh 1 Mv cm 1 Mv cm 1 I 1 cmr v cm R (Note that k can be greater than 1) 3

4 The Race of the Rolling Bodies () From the previous chart: Mgh 1 1 c Mv cm So the speed at the bottom of the incline is: v cm gh 1 c 0.5 The Race of the Rolling Bodies (3) So, comparing the multiplier for moment of inertia, we have: Solid sphere, c = /5 Solid cylinder, c = ½ Hollow sphere, c = /3 Hollow cylinder, c = 1 As a result, this will be the order of finish. Note that the radius and mass of the body drops out, and the remaining factor is the multiplier for moment of inertia! Class roblems roblems based on homework A cross product problem Four problems related to the experiment A moment of inertia problem How do we calculate angular speed? A. Divide the angular change by the time C B.. Divide the number of revolutions by the time C. Both D. Neither 4

5 A CD maintains a constant linear rate as it rotates. How does it accomplish this? A. The head moves out faster at the rim. B. It changes RM as the distance from the center changes. C C. Neither of the above. A Torque of 110 N.m is required to turn the revolving door.? 90 N A child can push with a force of 90N. How far from the center must she push? A. 1 m. B. 1.5 m. C. Between these C D. Outside these. Hint: Torque = force x radius. The Space Station roblem A M diameter space station of mass 5 x 10 5 kg, mostly in the rim. Rockets of 100N thrust each. Space Station roblem (cont.) You are asked to calculate the time to reach the required spin rate, and how many revolutions it will require. What is the equation for angular acceleration? A.= torque/mass B. = torque/moment of inertia.c C.= force/mass D. None of the above. How do we calculate the time to reach the rate? A. Angular velocity divided by the linear acceleration B. Linear velocity divided by radial acceleration C. Angular velocity divided by angular acceleration. C 5

6 Space Station roblem, concluded Where do we consider the acceleration of gravity? A. In order to calculate the required angular velocity B. To calculate the time to reach rotation rate C. None of the above. D. Both of the above. C The vector cross product of U and V is: B. z C. V B is correct y x U A. The following factors apply to the ball rolling down the plane: oint has something in common with: A.The angle of the plane B. The force of gravity C. Friction at point D.A and B E. A, B and C Correct A. A block sliding down the plane B. A ball thrown into the air C. C. A skier going down the slope D. None of the above 6

7 The common factor is: We calculate the moment of inertia about point using: A. Rolling motion B. An instant of zero velocity C. C. A normal force D. None of the above A. The moment of inertia about the axis of the cylinder. B. The arallel Axis Theorem C. Both of the above. C. D. Neither of the above Which of the cylinders has the higher moment of inertia about its axis? The masses and diameters are the same A. C. B. Why? A. More of the mass is concentrated at the center. B. More of the mass is concentrated at the rim. C. C. Not enough information to tell. 7