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Rotational motion Chapter 9 Rigid objects A rigid object has a perfectly definite and unchanging shape and size. In this class, we will approximate everything as a rigid object Radians In describing rotational motion, we will use angles in radians, not degrees. q in radians s q r 180 rad 90 2 rad q in radians An angle in radians is the ratio of two lengths, so it has no units. We will often write “rad” as the units on such an angle to make it clear that it’s not in degrees But in calculations, “rad” doesn’t factor into unit analysis. Angular velocity of change of q w (omega) is the symbol for angular velocity q wav t Rate q w lim t 0 t Angular velocity At any instant, all points on a rigid object have the same angular velocity. The units of angular velocity are rad/s. Sometimes angular velocity is given in rev/s or rpm. 1 rev is 2 radians Angular speed is the magnitude of angular velocity Angular acceleration Rate of change of angular velocity a (alpha) is the symbol for angular acceleration w a av t w a lim t 0 t Angular acceleration The units for angular acceleration are rad/s2. Comparison x is linear position v is linear velocity a is linear acceleration q is angular position w is angular velocity a is angular acceleration Rotation with constant angular acceleration w w0 at v v0 at 1 2 x x0 v0t at 2 1 2 q q 0 w 0 t at 2 v v 2ax x0 w w 2a q q0 v0 v x x0 t 2 w0 w q q0 t 2 2 2 0 2 2 0 Example A CD rotates from rest to 500 rev/min in 5.5 s. What is its angular acceleration, assuming it is constant? How many revolutions does the disk make in 5.5 s? 9.52 rad/s 22.9 rev Relating linear and angular kinematics We might want to know the linear speed and acceleration of a point on a rotating rigid object. So we need relationships between v and w a and a Speed relationship v rw Note: these are speeds, not velocities Acceleration relationship 2 arad v 2 w r r Change in direction Example Find the required angular speed, in rev/min, of an ultracentrifuge for the radial acceleration of a point 2.50 cm from the axis to equal 400,000 times the acceleration due to gravity. 1.25 x 104 rad/s = 1.19 x 105 rev/min Moment of Inertia Rotating objects have inertia, but is more than just their mass. It depends on how that mass is distributed. Moment of inertia The moment of inertia, I, of an object is found by taking the sum of the mass of each particle in the object times the square of it’s perpendicular distance from the axis of rotation. I m r m r ... mi ri 2 1 1 2 2 2 i 2 Moment of Inertia For continuous distributions of particles, i.e. large objects, the sum becomes an integral. The moments of inertia for several familiar shapes with uniform densities are given on page 215 of your book. Moments of inertia are given in terms of masses and dimensions. Kinetic energy of rotating objects 1 2 K Iw 2 Gravitational potential energy of rotating objects Same as for other objects, but use total mass and position of the center of mass. U MgY Example A uniform thin rod of length L and mass M, pivoted at one end, is held horizontal and then released from rest. Assuming the pivot is frictionless, find The angular velocity of the rod when it reaches its vertical position Sqrt(3g/L) On your own An airplane propeller (I=(1/12)ML2) is 2.08 m in length (from tip to tip) with mass 117 kg. The propeller is rotating at 2400 rev/min about an axis through it’s center. What is its rotational kinetic energy? If it were not rotating, how far would it have to drop in free fall to acquire the same kinetic energy? 1.33 x 106 J 1.16 km Torque The measure of the tendency of a force to change the rotational motion of a object. Torque depends on the perpendicular distance between the force and the axis of rotation Torque magnitude t Fl t (tau) [your book uses G (gamma)] is the magnitude of the torque Where Also called moment F is the magnitude of the force l is the perpendicular distance between the force and the axis of rotation. Also called lever arm or moment arm Torque magnitude Torque magnitude Torque Magnitude Torque sign Counterclockwise rotation is caused by positive torques and clockwise rotation is caused by negative torques. We can use this symbol to indicate which direction is positive torque. + Torque Units The SI-unit of torque is the Newtonmeter. Torque is not work or energy, so it should not be expressed as Joules. Torque Vector Direction t Fl t rF sin Visual aid for torque direction Torque Think The is perpendicular to both r and F. of a normal, right-handed screw. torque vector points in the direction the screw moves. Discussion Question Why are doorknobs located far from the hinges? Example Forces F1 = 8.60 N and F2 = 2.40 N are applied tangentially to a wheel with a radius of 1.50 m, as shown on the next slide. What is the net torque on the wheel if it rotates on an axis perpendicular to the wheel and passing through its center? F1 F2 You try Calculate the torque (magnitude and direction) about point O due to the force shown below. The bar has a length of 4.00 m and the force is 30.0 N. F q = 60° O 2m Torque and angular acceleration t Ia Only valid for rigid objects a must be in rad/s2 for units to work Example A torque of 32.0 N-m on a certain wheel causes an angular acceleration of 25.0 rad/s2. What is the wheel’s moment of inertia? On your own A solid sphere has a radius of 1.90 m. An applied torque of 960 N-m gives the sphere an angular acceleration of 6.20 rad/s2 about an axis through its center. Find The moment of inertia of the sphere The mass of the sphere Example An object of mass m is tied to a light string wound around a wheel that has a moment of inertia I and radius R. The wheel is frictionless, and the string does not slip on the rim. Find the tension in the string and the acceleration of the object. T=(I/(I+mR2)*mg a=(mR2/(I+mR2))g On your own a On your own Two blocks are connected by a string that passes over a pulley of radius R and moment of inertia I. The block of mass m1 slides on a frictionless, horizontal surface; the block of mass m2 is suspended from the string. Find the acceleration a of the blocks and the tensions T1 and T2 assuming that the string does not slip on the pulley. a=(m2/(m1+m2+I/R2))m2g T1=(m1/(m1+m2+I/R2))m2g T2=((m1+I/R2)/(m1+m2+I/R2))m2g Rigid object rotation about a moving axis Combined translation and rotation. Translation of center of mass Rotation about the center of mass There is friction, but only static friction to keep the object from slipping Kinetic Energy The kinetic energy is the sum of translational and rotational kinetic energies. 1 1 2 2 K mV I comw 2 2 Rolling without slipping When something is rolling without slipping, V Rw A Ra On your own A hollow cylindrical shell with mass M and radius R rolls without slipping with speed V on a flat surface. What is its kinetic energy? MV2 Example A solid disk and a hoop with the same mass and radius roll down an incline of height h without slipping. Which one reaches the bottom first? The disk What if they had different masses? Different radii? Dynamics of translating and rotating objects We can use both Newton’s 2nd law and its rotational counterpart F mA t Ia Example A uniform solid ball of mass m and radius R rolls without slipping down a plane inclined at an angle q. A frictional force f is exerted on the ball by the incline. Find the acceleration of the center of mass. (5/7) gsinq Work and Power Work done by a constant torque W tq Work and Kinetic Energy Total work done equals change in K Power Power is the rate of doing work dW dq t dt dt P tw P Fv Example A uniform disk with a mass of 120 kg and a radius of 1.4 m rotates initially with an angular speed of 1100 rev/min. A constant tangential force is applied at a radial distance of 0.6 m. What work must this force do to stop the wheel? 780 kJ If the wheel is brought to rest in 2.5 min, what torque does the force produce? 90.4 N-m What is the magnitude of the force? 151 N On your own A playground merry-go-round has a radius of 2.40 m and a moment of inertia 2100 kg-m2 about a vertical axle through its center, and turns with negligible friction. A child applies an 18.0-N Force tangentially to the edge of the merry-go-round for 15.0 s. If the merry-go-round is initially at rest, what is its angular speed after this 15.0-s interval? How much work did the child do on the merry-goround? What is the average power supplied by the child? 0.309 rad/s 100 J 6.67 W Angular momentum Relationship between angular momentum and linear momentum is the same as between torque and force. t rF L rp mvr Units The units for angular momentum are kg-m2/s Angular momentum of rigid objects Look at L for one particle of the object Li mi vr mi riw ri mi ri w 2 Sum over all particles for total angular momentum 2 L Li mi ri w i i L Iw Example A woman with mass 50 kg is standing on the rim of a large disk that is rotating at 0.50 rev/s about an axis through its center. The disk has mass 110 kg and a radius of 4.0 m. Calculate the magnitude of the total angular momentum of the woman-plus-disk system. You can treat the woman as a point. 5275 kg-m2/s On your own Find the magnitude of the angular momentum of the sweeping second hand on a clock about an axis through the center of the clock face. The clock hand has a length of 15.0 cm and a mass of 6.00 g. Take the second hand to be a slender rod rotating with constant angular velocity about one end. 4.71 x 10-6 kg-m2/s Conservation of angular momentum If there is no net external torque acting on a system, then the total angular momentum of the system is conserved. Li L f Example A uniform circular disk is rotating with an initial angular speed w1 around a frictionless shaft through its center. Its moment of inertia is I1. It drops onto another disk of moment of inertia I2 that is initially at rest on the same shaft. Because of surface friction between the disks, they eventually attain a common angular speed wf. Find wf.