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1J-29 Overhanging Blocks How far out from the table can a stack of bricks be balanced ? Length = L, CG = L/2 Length = 3L/2, CG = 3L/4 Length = 7L/4, CG = 25L/24 How do the blocks stay balanced when the top block extends beyond the bottom block ? Δx Blocks can over hang but the Center of Gravity of a block must be inside the block below For 6 blocks max extension Δx : Δx = L/2 + L/4 + L/6 + L/8 + L/10 + L/12 = 1.22(L) IN REALITY EACH BLOCK HAS TO BE MOVED SLIGHTLY BACK TO AVOID TIPPING, SO THE TOTAL EXTENSION WILL BE A LITTLE LESS. 2/25/2011 Rotational Inertia and Newton’s Second Law • In linear motion, net force and mass determine the acceleration of an object. • For rotational motion, torque determines the rotational acceleration. • The rotational counterpart to mass is rotational inertia or moment of inertia. – Just as mass represents the resistance to a change in linear motion, rotational inertia is the resistance of an object to change in its rotational motion. – Rotational inertia is related to the mass of the object. – It also depends on how the mass is distributed about the axis of rotation. Simplest example: a mass at the end of a light rod • A force is applied to the mass in a direction perpendicular to the rod. • The rod and mass will begin to rotate about the fixed axis at the other end of the rod. • The farther the mass is from the axis, the faster it moves for a given rotational velocity. Simplest example: a mass at the end of a light rod • To produce the same rotational acceleration, a mass at the end of the rod must receive a larger linear acceleration than one nearer the axis. • F = ma – It is harder to get the system rotating when the mass is at the end of the rod than when it is nearer to the axis. – I case the distance are equal, it’s harder to move a heavier mass. Rotational Inertia and Newton’s Second Law • Newton’s second law for linear motion: Fnet = ma • Newton’s second law for rotational motion: ∙R=m∙ ∆ ∆ ∙ = ∙ ∙R=m∙ ∙ net = I – The rotational acceleration produced is equal to the torque divided by the rotational inertia. Rotational Inertia and Newton’s Second Law • For an object with its mass concentrated at a point: – Rotational inertia = mass x square of distance from axis – I = mr2 • The total rotational inertia of an object like a merry-goround can be found by adding the contributions of all the different parts of the object. Two 0.2-kg masses are located at either end of a 1m long, very light and rigid rod as shown. What is the rotational inertia of this system about an axis through the center of the rod? a) b) c) d) 0.02 kg·m2 0.05 kg·m2 0.10 kg·m2 0.40 kg·m2 I = mr2 = (0.2 kg)(0.5m)2 x 2 = 0.10 kg·m2 Rotational inertias for more complex shapes: Angular Momentum • Linear momentum is mass (inertia) times linear velocity: p = mv • Angular momentum is rotational inertia times rotational velocity: L = I – Angular momentum may also be called rotational momentum. – A bowling ball spinning slowly might have the same angular momentum as a baseball spinning much more rapidly, because of the larger rotational inertia I of the bowling ball. Conservation of Angular Momentum net = I = ∙ ∆ ∆ = ∆ ∆ = ∆ ∆ i.e. the direction of the angular momentum change is the same as that of the net toque. When net = ∆ 0, ∆ = 0, i.e. L = const. Conservation of Angular Momentum Inertia m : Fnet ma If Fnet 0, p mv p constant net I L I 0, L constant Inertia I : If net Kinetic Energy = 1 = 2 = Conservation of Angular Momentum 1Q- 23 Conservation of angular momentum Changing the moment of inertia of a skater How does conservation of angular momentum manifest itself ? I = 2mR2 = 2/25/2011 2/25/2011 Physics 214 Physics 214 Fall 2009 Fall 2009 14 14 Kepler’s Second Law • Kepler’s second law says that the radius line from the sun to the planet sweeps out equal areas in equal times. • The planet moves faster in its elliptical orbit when it is nearer to the sun than when it is farther from the sun. Kepler’s Second Law • This is due to conservation of angular momentum. • The gravitational force acting on the planet produces no torque about an axis through the sun because the lever arm is zero: the force’s line of action passes through the sun. Kepler’s Second Law • When the planet moves nearer to the sun, its rotational inertia about the sun decreases. • To conserve angular momentum, the rotational velocity of the planet about the sun must increase. Angular momentum is a vector • The direction of the rotational-velocity vector is given by the right-hand rule. • The direction of the angular-momentum vector is the same as the rotational velocity. Inertia I, rotational velocity Angular momentum : L I 1Q-32 Stability Under Rotation Example of Gyroscopic Stability: Swinging a spinning Record Why does the Record not “flop around” once it is set spinning ? L L The dominant physical law is conservation of angular momentum. With no torque the vector L, perpendicular to the plane of rotation always points in the same direction. SINCE THERE IS NO TORQUE ABOUT THE CENTER OF ROTATION OF THE RECORD, THE ANGULAR MOMENTUM VECTOR CANNOT CHANGE. THIS IS GYROSCOPIC STABILITY. THIS IS A VERY SIMPLE GYROSCOPE AND SOPHISTICATED GYROSCOPES ARE USED TO STEER AIRCRAFT AND ORIENT THE HUBBLE TELESCOPE 2/25/2011 2/25/2011 Physics 214 Physics 214 Fall 2009 Fall 2009 19 19 1Q-21 Conservation of angular momentum Conservation of angular momentum using a spinning wheel What happens when the wheel is inverted ? The dominant physical law is conservation of angular momentum. To change the angular momentum of the wheel requires an external torque. So although we can change the direction of the angular momentum of the wheel the force we use is internal to the wheel/stool system so the the stool rotates to keep the net angular momentum the same To turn the wheel requires significant force and work is needed. The energy of the final system is greater than the initial energy by the amount of work that is done. 2/25/2011 2/25/2011 20 20 Angular momentum and bicycles • The wheels have angular momentum when the bicycle is moving. • For straight line motion, the direction of the angularmomentum vector is the same for both wheels and is horizontal. • To tip the bike over, the direction of the vector must change, requiring a torque. Angular momentum and bicycles • If the bike is not perfectly upright, a gravitational torque acts about the line of contact of the tires with the road. • As the bike begins to fall, it acquires a rotational velocity and angular momentum about this axis. • If the bike tilts to the left, the change in angular momentum points straight back. Angular momentum and bicycles • the change in angular momentum caused by the gravitational torque adds to the angular momentum already present from the rotating tires. • This causes a change in the direction of the total-angular momentum vector which can be accommodated by turning the wheel. 1Q-30 Bicycle Wheel Gyroscope Gyroscopic action and precession L What happens to the wheel, does it fall down? F = mg F mg The counterclockwise torque adds to L and produces a precession, providing L is large and the torque is small The torque causes the vector L to precess and changes the direction of the angular momentum vector which is perpendicular to the plane of rotation. This is a very large top. 2/25/2011 24 24 A student sits on a stool holding a bicycle wheel with a rotational velocity of 5 rad./s about a vertical axis. The rotational inertia of the wheel is 2 kg·m2 about its center and the rotational inertia of the student and wheel and platform about the rotational axis of the platform is 6 kg·m2. What is the initial angular momentum of the system? a) b) c) d) 10 kg·m2/s upward 25 kg·m2/s downward 25 kg·m2/s upward 50 kg·m2/s downward L = I = (2 kg·m2)(5 rad./s) = 10 kg·m2/s upward from plane of wheel Quiz: If the student flips the axis of the wheel, reversing the direction of its angular-momentum vector, what is the rotational velocity of the student and stool about their axis after the wheel is flipped? a) b) c) d) 1.67 rad/s 3.33 rad/s 60 rad/s 120 rad/s = L / I = (20 kg·m2/s) / (6 kg·m2) = 3.33 rad/s in direction of original angular velocity