Physics 103: Lecture 14 Rotations of Extended Objects Today’s lecture will cover Torque Equilibrium Moment of Inertia Torque and Angular Acceleration 03/10/2003 Physics 103, Spring 2004, U. Wisconsin 1

Physics 103, Spring 2004, U. Wisconsin Torque “Point” objects with mass m are linearly accelerated according to Newton’s 2nd Law. But what if the objects have size? Ignore it (treat the mass as a point) Or, account for it: object is rigid … all points are fixed but the extended object can rotate Torque,t, is the tendency of a force to rotate an object about some axis F d a offset force 03/10/2003 Physics 103, Spring 2004, U. Wisconsin

Physics 103, Spring 2004, U. Wisconsin Torque 03/10/2003 Physics 103, Spring 2004, U. Wisconsin

Physics 103, Spring 2004, U. Wisconsin Lever Arm The lever arm, d, is the perpendicular distance from the axis of rotation to a line drawn from the direction of the force d = L sin f t = FL sin f 03/10/2003 Physics 103, Spring 2004, U. Wisconsin

An Alternative Look at Torque The force could also be resolved into its x- and y-components The x-component, F cos f, produces 0 torque The y-component, F sin f, produces a non-zero torque t = FL sin f L 03/10/2003 Physics 103, Spring 2004, U. Wisconsin

Question 1: Opening the door Torque Force Point of application Distance to center of rotation + or - (ccw or cw) In which case (a) thru (f) is the torque most positive? 03/10/2003 Physics 103, Spring 2004, U. Wisconsin

Physics 103, Spring 2004, U. Wisconsin Equilibrium To ensure mechanical equilibrium, you need to ensure rotational equilibrium as well as translational The First Condition of Equilibrium states The net external force must be zero The Second Condition of Equilibrium states The net external torque must be zero 03/10/2003 Physics 103, Spring 2004, U. Wisconsin

Notes About Equilibrium A zero net torque does not mean the absence of rotational motion An object that rotates at uniform angular velocity can be under the influence of a zero net torque This is analogous to the translational situation where a zero net force does not mean the object is not in motion 03/10/2003 Physics 103, Spring 2004, U. Wisconsin

Physics 103, Spring 2004, U. Wisconsin Center of Gravity The force of gravity acting on an object must be considered in determining equilibrium In finding the torque produced by the force of gravity, all of the weight of the object can be considered to be concentrated at one point The coordinates of the center of gravity can be found from the sum of the torques acting on the individual particles within the object being set equal to the torque produced by the weight of the object 03/10/2003 Physics 103, Spring 2004, U. Wisconsin

Experimentally Determining the Center of Gravity The wrench is hung freely from two different pivots The intersection of the lines indicates the center of gravity A rigid object can be balanced by a single force equal in magnitude to its weight as long as the force is acting upward through the object’s center of gravity 03/10/2003 Physics 103, Spring 2004, U. Wisconsin

Physics 103, Spring 2004, U. Wisconsin Preflight 1 A car is parked on a steep hill facing downwards with tires turned towards the curb as they should be. Is the car in dynamic equilibrium. a) yes b) no Equilibrium means the net force and net torque vanish (= zero). This means constant velocity V = 0 (STATIC) V = constant vector (DYNAMIC) The car is in static equilibrium. 03/10/2003 Physics 103, Spring 2004, U. Wisconsin

Physics 103, Spring 2004, U. Wisconsin Preflight 2 A car is running on cruise control on a country road along rolling hills. Assuming that the cruise control does a perfect job, is the car in dynamic equilibrium. a) yes b) no Force is needed to move over the hills even at constant velocity. (Also needed to move on flat curves.) For example, true in Utah on a straight away on the salt flats. 03/10/2003 Physics 103, Spring 2004, U. Wisconsin

Physics 103, Spring 2004, U. Wisconsin Preflight 3 An old carpenter's trick to keep nails from bending when they are pounded into hard materials is to grip the center of the nail firmly with pliers. Why does this help? Because, the middle of the nail is its center of gravity. Holding it firmly there improves its static equilibrium. Any small angle in the hammer striking the nail produces a torque. If the nail is held right in the middle, the torque will be half the value that would occur without the pliers. Both of the above None of the above 03/10/2003 Physics 103, Spring 2004, U. Wisconsin

Physics 103, Spring 2004, U. Wisconsin Stability An object is in equilibrium: Imagine a small shift (displacement) Stable -- generated torque restores equilibrium 03/10/2003 Physics 103, Spring 2004, U. Wisconsin

Physics 103, Spring 2004, U. Wisconsin Stability (reprise) Unstable -- generated torque destroys equilibrium 03/10/2003 Physics 103, Spring 2004, U. Wisconsin

Solving Equilibrium Problems Draw a diagram of the system Isolate the object being analyzed and draw a free body diagram showing all the external forces acting on the object For systems containing more than one object, draw a separate free body diagram for each object Establish convenient coordinate axes for each object. Apply the First Condition of Equilibrium Choose a convenient rotational axis for calculating the net torque on the object. Apply the Second Condition of Equilibrium Solve the resulting simultaneous equations for all of the unknowns 03/10/2003 Physics 103, Spring 2004, U. Wisconsin

Physics 103, Spring 2004, U. Wisconsin Axis of Rotation If the object is in equilibrium, it does not matter where you put the axis of rotation for calculating the net torque The location of the axis of rotation is completely arbitrary Often the nature of the problem will suggest a convenient location for the axis When solving a problem, you must specify an axis of rotation Once you have chosen an axis, you must maintain that choice consistently throughout the problem 03/10/2003 Physics 103, Spring 2004, U. Wisconsin

Physics 103, Spring 2004, U. Wisconsin Question 2 Considering the figure shown, which of the statements below are true? Only the First Condition of Equilibrium is satisfied Only the Second Condition of Equilibrium is satisfied Both conditions are satisfied Neither conditions are satisfied 03/10/2003 Physics 103, Spring 2004, U. Wisconsin

Physics 103, Spring 2004, U. Wisconsin Example 1 Free Body Diagram F R F L W 03/10/2003 Physics 103, Spring 2004, U. Wisconsin

Physics 103, Spring 2004, U. Wisconsin Question 3 Pick free body diagram corresponding to the picture below assuming that the person is able to hold the pole in equilibrium: W F R L A W F R L C W F R L B 03/10/2003 Physics 103, Spring 2004, U. Wisconsin

Example of Free Body Diagram Isolate the object to be analyzed Draw the free body diagram for that object Include all the external forces acting on the object 03/10/2003 Physics 103, Spring 2004, U. Wisconsin

Example of a Free Body Diagram The free body diagram includes the directions of the forces The weights act through the centers of gravity of their objects 03/10/2003 Physics 103, Spring 2004, U. Wisconsin

Physics 103, Spring 2004, U. Wisconsin High Jump Olympic high jumpers (including all high jumpers) manipulate things so that their body goes over the bar -- but their c.m. goes under!! 03/10/2003 Physics 103, Spring 2004, U. Wisconsin

Physics 103, Spring 2004, U. Wisconsin More High Jumping We know how to find the motion of an object with initial velocity After leaving the ground only gravity acts In order to win, the jumper must wiggle his/her body around the cg (rotation) 03/10/2003 Physics 103, Spring 2004, U. Wisconsin

Physics 103, Spring 2004, U. Wisconsin Tides The tides are caused by the action of the moon on the earth. center of mass of moon-earth system A C B mac = Fg At C there is a balance (ac is centripetal acceleration). At A Gravitation is too weak to provide c.a. At B Gravity is stronger than necessary to provide c.a. 03/10/2003 Physics 103, Spring 2004, U. Wisconsin