Dynamics of Rotational Motion Torque the rotational analogue

Dynamics of Rotational Motion Torque: the rotational analogue of force Torque = force x moment arm t = Fl moment arm = perpendicular distance through which the force acts a. k. a. lever arm l l F F t = Fl = F r sin f = Ftan r Phys 211 C 10 p 1

Example: To loosen a difficult nut, a wrench with a piece of pipe for additional leverage is used. If the distance from the center of the nut to the end of the pipe is 0. 80 m, and the wrench and pipe make an angle of 20º with respect to the horizontal, determine the magnitude and direction of the torque applied by a 900 N man standing on the edge of the pipe. Phys 211 C 10 p 2

Torque and Angular Acceleration single particle contribution y-axis of rotation F 1, y r 1 F 1, tan F 1, rad r Phys 211 C 10 p 3

Problem solving tactics: • Draw a diagram • For each body, draw a free body diagram • Choose coordinates and directions (including for rotations) • Relate torques to angular accelerations and forces to linear accelerations • for more than one body, repeat and note any geometrical relation between motions of bodies Phys 211 C 10 p 4

A cord is wrapped around a solid 50 kg cylinder which has a diameter of 0. 120 m, and which rotates (frictionlessly) about an axis through its center. A 9. 00 N force is applied to the end of the cable, causing the cable to unwind and the drum (initially at rest) to rotate. What is the angular acceleration of the drum? 9. 00 N A mass m is suspended by a string wrapped around a pulley of radius R and moment of inertia I. The mass and pulley are initially at rest. Determine the acceleration of the disk and of the suspended mass in terms of the given parameters (m, I, R). Examine the special case where the pulley is a uniform disk of mass M. Phys 211 C 10 p 5

A mass m 2 is suspended by a string wrapped around a pulley of radius R and moment of inertia I. The string is then connected to a mass m 1 which slides on a horizontal frictionless surface. The masses and pulley are initially at rest. Determine the accelerations of the masses and the pulley in terms of the given parameters (m 1, m 2, I, R). Examine the special case where the pulley can be treated as a thin cylindrical shell of mass M. M m 1 m 2 Phys 211 C 10 p 6

More on Combining Translation and Rotation y-axis cm axis of rotation ri F 1, y v i’ vi vcm Phys 211 C 10 p 7 vcm

example: A primitive yo-yo is made of a solid cylinder of mass M with string wrapped around its radius R. What is the speed of the cylinder after it has dropped a distance h? What is acceleration of the cylinder and the tension in the string? example: A solid bowling bowl rolls without slipping down a ramp inclined at an angle b with respect to the horizontal. What is the ball’s acceleration? Phys 211 C 10 p 8

Rolling Friction: deforming surfaces Stiff surface; normal force through cm, no torque Distorted surface; normal force offset from cm, net torque slows rotation Phys 211 C 10 p 9

Work and Power in Rotational Motion Force applied to a point on a rotating object Ftan s R q Ftan Phys 211 C 10 p 10

Phys 211 C 10 p 11

If the power output of an engine is 200 hp at 6000 rpm, what is the corresponding torque? A motor produces a torque of 10 N-m with disk mounted to its shaft. The disk has a moment of inertia of 2. 0 kg m 2. If the system starts from rest, determine the work done by the motor in 8. 0 s, and the kinetic energy of the disk at the end of this time. What is the average power delivered during this 8 seconds? Phys 211 C 10 p 12

Angular momentum for a single particle, defined as L r × p = r × mv depends upon origin! L = mvr sinf = mvr = mv r mv f mv r r L w Generalize to composite object In the absence of external torques, angular momentum is conserved Right hand rule for direction Phys 211 C 10 p 13

Two disks with moments of inertia Ia and Ib share a common axis of rotation, but have different angular velocities (wa and wb ). The disks are brought together so that they eventually reach a common final angular velocity. Derive an expression for that final angular velocity w. Suppose the two disks are uniform with masses of 2. 0 kg and 4. 0 kg, radii of. 20 m and. 10 m and initial angular speeds of 50 rad/s and 200 rad/s. Determine the final angular speed and the change in the kinetic energy of the system. Phys 211 C 10 p 14

A bullet (10 g and initial speed 400 m/s) is fired into a door from a direction perpendicular to the door. The door has a mass of 15 kg and is 1. 0 m wide. Determine the angular velocity of the door after the bullet imbeds itself in the center of the door. Phys 211 C 10 p 15

Gyroscopes and Precession t L w w Phys 211 C 10 p 16