Dynamics of Rotational Motion Torque: the rotational analogue of force Torque = force x moment arm = Fl moment arm = perpendicular distance through which the force acts a.k.a. lever arm l l F F l l F F = Fl = F r sin = Ftan r
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.
Torque and Angular Acceleration single particle contribution F1,rad F1,tan F1,z z-axis axis of rotation Torque and Angular Acceleration single particle contribution
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
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.
A mass m2 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 m1 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 the given parameters (m1, m2, I, R). Examine the special case where the pulley can be treated as a thin cylindrical shell of mass M. m1 m2 M
More on Combining Translation and Rotation y-axis cm axis axis of rotation F1,y vi’ vcm ri vi 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 with respect to the horizontal. What is the ball’s acceleration?
Rolling Friction: deforming surfaces Distorted surface; normal force offset from cm, net torque slows rotation Stiff surface; normal force through cm, no torque
Work and Power in Rotational Motion Force applied to a point on a rotating object Ftan s R Ftan
If the power output of an engine is 200 hp at 6000rpm, 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 m2. 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?
for a single particle, defined as L r × p = r × mv Angular momentum for a single particle, defined as L r × p = r × mv depends upon origin! L mvr sin= mvr = mvr r mv mv r L Right hand rule for direction Generalize to composite object In the absence of external torques, angular momentum is conserved
Two disks with moments of inertia Ia and Ib share a common axis of rotation, but have different angular velocities (a and b ). The disks are brought together so that they eventually reach a common final angular velocity. Derive an expression for that final angular velocity . 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.
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 embeds itself in the center of the door.
Gyroscopes and Precession L w