AP Physics “C” – Unit 4/Chapter 10 & 11 Notes – Yockers Rotational Motion Angular Position, Speed, and Acceleration - rotational motion – motion of a body that spins about an axis - axis of rotation – the line about which rotation occurs - circular motion – occurs when any single point on an object travels in a circle around an axis of rotation →r – radius →s – arc length – distance moved along the circumference of a circle - angles can be measured in radians →radian – an angle whose arc length is equal to its radius arc length 360 →rad = radian(s); 1 rad = /length of radius when s=r; 1 rad 57.3 2 → the radian is a pure number with no dimensions (ratio of lengths) →in general, any angle, , measured in radians, is defined by the relation s or s r r →conversions rad deg 180 - angular displacement, , describes how much an object has rotated →the angle through which a point, line, or body is rotated in a specified direction and about a specified axis → - angular displacement s s s s f 0 f 0 f 0 r r r s r - s is positive (+) when rotation is counterclockwise (ccw) - s is negative (-) when rotation is clockwise (cw) - angular speed, , describes rate of rotation →the rate at which a body rotates about an axis, usually expressed in rad/s (or s-1 because radians are not dimensional) →average angular speed = avg f 0 avg unit is rad/s or revolutions/unit time t t f t0 - instantaneous angular speed d lim t0 t dt - angular acceleration, , occurs when angular speed changes 2 -2 →the time rate of change of angular speed, expressed in rad/s/s or rad/s (or s ) →average angular acceleration = avg f 0 avg t t f t0 - instantaneous angular acceleration d lim t0 t dt - rigid body model - all points on a rotating rigid object have the same angular acceleration and angular speed →if not, the object’s shape would change (it would not be rigid) - similarities of angular and linear quantities linear angular displacement s speed v acceleration a - direction of and →the direction of is along the axis of rotation - if rotation is counterclockwise then the direction of is out of the plane of the diagram - if rotation is clockwise then the direction of is into the plane of the diagram d →the direction of follows from its vector definition of (along the same axis as ) dt - for rotation about a fixed axis →the direction of is the same as if the angular speed (the magnitude of ) is increasing in time →the direction of is antiparallel to if the angular speed is decreasing in time Rotational Kinematics: The Rigid Body Under Constant Angular Acceleration - for rotational motion about a fixed axis, the simplest accelerated motion to analyze is motion under constant angular acceleration →manipulating the equation for instantaneous angular acceleration results in f t d dt f 0 t 0 0 →manipulating the equation for instantaneous angular speed results in t t f 1 2 d dt 0 t dt f 0 0t t 0 0 2 0 →eliminating results in 1 f 0 2 0 f t →and eliminating t results in 2 2 f 0 2 f 0 Tangential and centripetal acceleration - objects in circular motion have a tangential speed - an object farther from the axis of rotation must travel at a higher tangential speed around the circular path, s, to travel the same angular displacement as would an object closer to the axis - tangential speed ds d v r dt dt v r - tangential acceleration is tangent to the circular path →the instantaneous linear acceleration of an object directed along the tangent to the object’s circular path dv d a r t dt dt at r - centripetal acceleration v f v0 →remember a and that velocity can be changed either by magnitude or t f t0 direction; a change in either produces acceleration →for an object moving in a circular path with a constant speed, acceleration is due to a change in direction →centripetal acceleration (ac) – acceleration directed toward the center of a circular path - magnitude of ac is given by v2 a t c r - because tangential speed is related to the angular speed by vt r , the centripetal acceleration can also be found using angular speed 2 ac r - http://www.phy.ntnu.edu.tw/oldjava/circularMotion/circular3D_e.html - http://www.walter-fendt.de/ph14e/carousel.htm - tangential and centripetal accelerations are perpendicular →tangential component of acceleration is due to changing speed →centripetal component of acceleration is due to changing direction - total acceleration is found using the Pythagorean theorem 2 2 2 2 2 4 2 4 a at ac r r r - direction of the acceleration can be found using a tan 1 c at Moment of Inertia and Rotational Kinetic Energy - if a rigid object is considered to be a collection of particles that rotates about a fixed axis with an angular speed →each particle has some kinetic energy determined by its mass and tangential speed →if the mass of the ith particle is mi and its tangential speed is vi, the kinetic energy of the particle is 1 2 Ki 2 mivi →the total kinetic energy KR of the rotating object is 1 2 1 2 2 1 2 2 1 2 KR Ki 2mivi 2 miri 2 miri K 2 mv i i i i - the quantity in parentheses above is the moment of inertia, I, of the rigid object 2 I miri i →moment of inertia is the rotational analog of mass →the greater the moment of inertia of a body, the greater its resistance to change in its angular speed - therefore, the kinetic energy of the rotating rigid object can be expressed as 1 2 KR 2 I - on the storage side of the continuity equation for energy (Unit 3 Notes), the kinetic energy term should now be considered the sum of the changes in both translational and rotational kinetic energy - for an extended continuous object, the moment of inertia can be calculated by dividing the object into many small elements 2 2 I lim r mi r dm m 0 i i - it is usually easier to calculate the moments of inertia in terms of the volume of the elements rather than their mass (especially if is uniform) m V I r 2 dV - examples (moment of inertia problems) Torque - (tau) – SI derived unit is N·m - point mass vs. extended object →extended object – has a definite, finite size and shape →point mass – assuming all of an object’s mass is concentrated at its center of mass - rotational and translational motion can be separated →translational motion – motion along a path →rotational motion – the motion of a body that spins about an axis - net torque produces rotation →torque – a quantity that measures the ability of a force to rotate an object about some axis (torque measures the effectiveness of a given force in producing rotation of a body about a specified axis) →torque is not work: work requires motion and torque does not; work is a dot product (the product of two vectors results in a scalar quantity) whereas torque is a cross product (the product of two vectors results in a third vector quantity) - torque depends on a force and a lever arm →how easily an object rotates depends not only on how much force is applied but also on where the force is applied →lever arm – the ┴ distance from the axis of rotation to a line drawn along the direction of the force - torque also depends on the angle between a force and a lever arm →forces do not have to be ┴ to cause rotation Fr(sin) Fd - two equal but opposite forces can produce a rotational acceleration if they do not act along the same line - if torques are equal and opposite, there will be no rotational acceleration →seesaw – momentary torque produced by pushing with legs Rotational Equilibrium - equilibrium requires zero net force and zero net torque →if the net force on an object is zero, the object is in translational equilibrium →if the net torque on an object is zero, the object is in rotational equilibrium - the dependence of equilibrium on the absence of net torque is called the second condition of equilibrium →the resultant torque acting on an object in rotational equilibrium is independent of where the axis is placed - an unknown force that acts along a line passing through this axis of rotation will not produce any torque - beginning a diagram by arbitrarily setting an axis where a force acts can eliminate an unknown in a problem - conditions for equilibrium Type of Equation Symbolic Eq. Meaning translational Fnet=0 Fnet on an object must be zero rotational net=0 net on an object must be zero - unstable equilibrium – center of mass is above support point - stable equilibrium – center of mass is below support point The Rigid Object Under a Net Torque - only the tangential component of an applied force contributes to the torque of a rotating rigid object - the radial component of an applied force provides no torque because its line of action is through the axis of rotation - the torque on each particle in a rigid rotating object Fti mi ati ri Fti ri mi ati 2 i mi ri i - the sum of the torque on all particles in a rigid rotating object 2 i miri i i i 2 miri i - recognizing moment of inertia in the second equation above I F ma - the net torque on a rigid body is proportional to the body’s angular acceleration Work and Energy in Rotational Motion - the work done by external forces on an object will equal the change in rotational kinetic energy as long as energy is not stored by any other means - work done by a torque dW F ds F sinr d →the radial component of F does no work because it is perpendicular to the displacement of the point of application of the force →the above equation can be written as dW d this is analogous to the work done in translational motion (the product of force and translational