Rotational Kinematics Physics C: Rotational Motion 4/21/2017 Rotational Kinematics Bertrand

Physics C: Rotational Motion 4/21/2017 Position In translational motion, position is represented by a point, such as x. x = 3 x 5 linear q r p/2 p 3p/2 angular In rotational motion, position is represented by an angle, such as q, and a radius, r. Bertrand

Physics C: Rotational Motion 4/21/2017 Displacement Dx = - 4 Linear displacement is represented by the vector Dx. x 5 linear Angular displacement is represented by Dq, which is not a vector, but behaves like one for small values. The right hand rule determines direction. Dq p/2 p 3p/2 angular Bertrand

Tangential and angular displacement Physics C: Rotational Motion 4/21/2017 Tangential and angular displacement Dq s A particle that rotates through an angle Dq also translates through a distance s, which is the length of the arc defining its path. r This distance s is related to the angular displacement Dq by the equation s = rDq. Bertrand

Physics C: Rotational Motion 4/21/2017 Speed and velocity vT Dq s The instantaneous velocity has magnitude vT = ds/dt and is tangent to the circle. r The same particle rotates with an angular velocity w = dq/dt. The direction of the angular velocity is given by the right hand rule. Tangential and angular speeds are related by the equation v = r w. w is outward according to RHR Bertrand

Physics C: Rotational Motion 4/21/2017 Acceleration w is out of page (+z in this diagram) according to RHR Tangential acceleration is given by aT = dvT/dt. This acceleration is parallel or anti-parallel to the velocity. Angular acceleration of this particle is given by a = dw/dt. Angular acceleration is parallel or anti-parallel to the angular velocity. Tangential and angular accelerations are related by the equation a = r a. Dq s vT r Dq vT Don’t forget centripetal acceleration. Bertrand

Problem: Assume the particle is speeding up. Physics C: Rotational Motion 4/21/2017 Problem: Assume the particle is speeding up. What is the direction of the instantaneous velocity, v? What is the direction of the angular velocity, w? What is the direction of the tangential acceleration, aT? What is the direction of the angular acceleration a? What is the direction of the centripetal acceleration, ac? What is the direction of the overall acceleration, a, of the particle? What changes if the particle is slowing down? Bertrand

First Kinematic Equation Physics C: Rotational Motion 4/21/2017 First Kinematic Equation v = vo + at (linear form) Substitute angular velocity for velocity. Substitute angular acceleration for acceleration.  = o + t (angular form) Bertrand

Second Kinematic Equation Physics C: Rotational Motion 4/21/2017 Second Kinematic Equation x = xo + vot + ½ at2 (linear form) Substitute angle for position. Substitute angular velocity for velocity. Substitute angular acceleration for acceleration. q = qo + ot + ½ t2 (angular form) Bertrand

Third Kinematic Equation Physics C: Rotational Motion 4/21/2017 Third Kinematic Equation v2 = vo2 + 2a(x - xo) Substitute angle for position. Substitute angular velocity for velocity. Substitute angular acceleration for acceleration. 2 = o2 + 2(q - qo) Bertrand

Physics C: Rotational Motion Practice problem 4/21/2017 The Beatle’s White Album is spinning at 33 1/3 rpm when the power is turned off. If it takes 1/2 minute for the album’s rotation to stop, what is the angular acceleration of the phonograph album? Bertrand

Physics C: Rotational Motion 4/21/2017 Rotational Energy Bertrand

Physics C: Rotational Motion Practice problem 4/21/2017 The angular velocity of a flywheel is described by the equation w = (8.00 – 2.00 t 2). Determine the angular displacement when the flywheel reverses its direction. Bertrand

Inertia and Rotational Inertia Physics C: Rotational Motion 4/21/2017 Inertia and Rotational Inertia In linear motion, inertia is equivalent to mass. Rotating systems have “rotational inertia”. I = mr2 (for a system of particles) I: rotational inertia (kg m2) m: mass (kg) r: radius of rotation (m) Solid objects are more complicated; we’ll get to those later. See page 278 for a “cheat sheet”. Bertrand

Physics C: Rotational Motion Sample Problem 4/21/2017 A 2.0-kg mass and a 3.0-kg mass are mounted on opposite ends a 2.0-m long rod of negligible mass. What is the rotational inertia about the center of the rod and about each mass, assuming the axes of rotation are perpendicular to the rod? Bertrand

Physics C: Rotational Motion 4/21/2017 Kinetic Energy Bodies moving in a straight line have translational kinetic energy Ktrans = ½ m v2. Bodies that are rotating have rotational kinetic energy Krot = ½ I w2 It is possible to have both forms at once. Ktot = ½ m v2 + ½ I 2 Bertrand

Physics C: Rotational Motion Practice problem 4/21/2017 A 3.0 m long lightweight rod has a 1.0 kg mass attached to one end, and a 1.5 kg mass attached to the other. If the rod is spinning at 20 rpm about its midpoint around an axis that is perpendicular to the rod, what is the resulting rotational kinetic energy? Ignore the mass of the rod. Bertrand

Physics C: Rotational Motion 4/21/2017 Rotational Inertia Bertrand

Rotational Inertia Calculations Physics C: Rotational Motion 4/21/2017 Rotational Inertia Calculations I = mr2 (for a system of particles) I =  dm r2 (for a solid object) I = Icm + m h2 (parallel axis theorem) I: rotational inertia about center of mass m: mass of body h: distance between axis in question and axis through center of mass Bertrand

Physics C: Rotational Motion 4/21/2017 Practice problem A solid ball of mass 300 grams and diameter 80 cm is thrown at 28 m/s. As it travels through the air, it spins with an angular speed of 110 rad/second. What is its translational kinetic energy? rotational kinetic energy? total kinetic energy? Bertrand

Physics C: Rotational Motion Practice Problem 4/21/2017 Derive the rotational inertia of a long thin rod of length L and mass M about a point 1/3 from one end using integration of I =  r2 dm using the parallel axis theorem and the rotational inertia of a rod about the center. Bertrand

Physics C: Rotational Motion Practice Problem 4/21/2017 Derive the rotational inertia of a ring of mass M and radius R about the center using the formula I =  r2 dm. Bertrand

Physics C: Rotational Motion 4/21/2017 Bertrand

Torque and Angular Acceleration I Physics C: Rotational Motion 4/21/2017 Torque and Angular Acceleration I Bertrand

Physics C: Rotational Motion 4/21/2017 Equilibrium Equilibrium occurs when there is no net force and no net torque on a system. Static equilibrium occurs when nothing in the system is moving or rotating in your reference frame. Dynamic equilibrium occurs when the system is translating at constant velocity and/or rotating at constant rotational velocity. Conditions for equilibrium: St = 0 SF = 0 Bertrand

Physics C: Rotational Motion 4/21/2017 Torque Torque is the rotational analog of force that causes rotation to begin. Consider a force F on the beam that is applied a distance r from the hinge on a beam. (Define r as a vector having its tail on the hinge and its head at the point of application of the force.) A rotation occurs due to the combination of r and F. In this case, the direction is clockwise. Hinge (rotates) r F Direction of rotation What do you think is the direction of the torque? Direction of torque is INTO THE SCREEN. Bertrand

Physics C: Rotational Motion 4/21/2017 Calculating Torque The magnitude of the torque is proportional to that of the force and moment arm, and torque is at right angles to plane established by the force and moment arm vectors. What does that sound like?  = r  F  : torque r: moment arm (from point of rotation to point of application of force) F: force Bertrand

Physics C: Rotational Motion Practice Problem 4/21/2017 What must F be to achieve equilibrium? Assume there is no friction on the pulley axle. F 3 cm 2 cm 2 kg 10 kg Bertrand

Torque and Newton’s 2nd Law Physics C: Rotational Motion 4/21/2017 Torque and Newton’s 2nd Law Rewrite SF = ma for rotating systems Substitute torque for force. Substitute rotational inertia for mass. Substitute angular acceleration for acceleration. S = I  : torque I: rotational inertia : angular acceleration Bertrand

Physics C: Rotational Motion Practice Problem 4/21/2017 A 1.0-kg wheel of 25-cm radius is at rest on a fixed axis. A force of 0.45 N is applied tangent to the rim of the wheel for 5 seconds. After this time, what is the angular velocity of the wheel? Through what angle does the wheel rotate during this 5 second period? Bertrand

Physics C: Rotational Motion 4/21/2017 Sample problem Derive an expression for the acceleration of a flat disk of mass M and radius R that rolls without slipping down a ramp of angle q. Bertrand

Physics C: Rotational Motion Practice problem 4/21/2017 Calculate initial angular acceleration of rod of mass M and length L. Calculate initial acceleration of end of rod. Bertrand

Physics C: Rotational Motion Sample problem 4/21/2017 Calculate acceleration. Assume pulley has mass M, radius R, and is a uniform disk. m2 m1 Bertrand

Rotational Dynamics Lab Physics C: Rotational Motion 4/21/2017 Rotational Dynamics Lab Bertrand

Work and Power in Rotating Systems Physics C: Rotational Motion 4/21/2017 Work and Power in Rotating Systems Bertrand

Physics C: Rotational Motion 4/21/2017 Practice Problem What is the acceleration of this system, and the magnitude of tensions T1 and T2? Assume the surface is frictionless, and pulley has the rotational inertia of a uniform disk. T1 mpulley = 0.45 kg rpulley = 0.25 m T2 m1 = 2.0 kg m2 = 1.5 kg 30o Bertrand

Work in rotating systems Physics C: Rotational Motion 4/21/2017 Work in rotating systems W = F • Dr (translational systems) Substitute torque for force Substitute angular displacement for displacement Wrot = t • Dq Wrot : work done in rotation  : torque Dq: angular displacement Remember that different kinds of work change different kinds of energy. Wnet = DK Wc = -DU Wnc = DE Bertrand

Power in rotating systems Physics C: Rotational Motion 4/21/2017 Power in rotating systems P = dW/dt (in translating or rotating systems) P = F • v (translating systems) Substitute torque for force. Substitute angular velocity for velocity. Prot = t • w (rotating systems) Prot : power expended  : torque w: angular velocity Bertrand

Conservation of Energy Physics C: Rotational Motion 4/21/2017 Conservation of Energy Etot = U + K = Constant (rotating or linear system) For gravitational systems, use the center of mass of the object for calculating U Use rotational and/or translational kinetic energy where necessary. Bertrand

Physics C: Rotational Motion Practice Problem 4/21/2017 A rotating flywheel provides power to a machine. The flywheel is originally rotating at of 2,500 rpm. The flywheel is a solid cylinder of mass 1,250 kg and diameter of 0.75 m. If the machine requires an average power of 12 kW, for how long can the flywheel provide power? Bertrand

Physics C: Rotational Motion Practice Problem 4/21/2017 A uniform rod of mass M and length L rotates around a pin through one end. It is released from rest at the horizontal position. What is the angular speed when it reaches the lowest point? What is the linear speed of the lowest point of the rod at this position? Bertrand

Rolling without Slipping Physics C: Rotational Motion 4/21/2017 Rolling without Slipping Bertrand

Rolling without slipping Physics C: Rotational Motion 4/21/2017 Rolling without slipping Total kinetic energy of a body is the sum of the translational and rotational kinetic energies. K = ½ Mvcm2 + ½ I 2 When a body is rolling without slipping, another equation holds true: vcm =  r Therefore, this equation can be combined with the first one to create the two following equations: K = ½ M vcm2 + ½ Icm v2/R2 K = ½ m 2R2 + ½ Icm 2 Bertrand

Physics C: Rotational Motion Sample Problem 4/21/2017 A solid sphere of mass M and radius R rolls from rest down a ramp of length L and angle q. Use Conservation of Energy to find the linear acceleration and the speed at the bottom of the ramp. Bertrand

Physics C: Rotational Motion Sample Problem 4/21/2017 A solid sphere of mass M and radius R rolls from rest down a ramp of length L and angle q. Use Rotational Dynamics to find the linear acceleration and the speed at the bottom of the ramp. Bertrand

Angular Momentum of Particles Physics C: Rotational Motion 4/21/2017 Angular Momentum of Particles Bertrand

Physics C: Rotational Motion Sample Problem 4/21/2017 A solid sphere of mass M and radius R rolls from rest down a ramp of length L and angle q. Use Conservation of Energy to find the linear acceleration and the speed at the bottom of the ramp. Bertrand

Physics C: Rotational Motion Sample Problem 4/21/2017 A solid sphere of mass M and radius R rolls from rest down a ramp of length L and angle q. Use Rotational Dynamics to find the linear acceleration and the speed at the bottom of the ramp. Bertrand

Physics C: Rotational Motion 4/21/2017 Practice Problem A hollow sphere (mass M, radius R) rolls without slipping down a ramp of length L and angle q. At the bottom of the ramp what is its translational speed? what is its angular speed? Bertrand

Physics C: Rotational Motion 4/21/2017 Angular Momentum Angular momentum is a quantity that tells us how hard it is to change the rotational motion of a particular spinning body. Objects with lots of angular momentum are hard to stop spinning, or to turn. Objects with lots of angular momentum have great orientational stability. Bertrand

Angular Momentum of a particle Physics C: Rotational Motion 4/21/2017 Angular Momentum of a particle For a single particle with known momentum, the angular momentum can be calculated with this relationship: L = r  p L: angular momentum for a single particle r: distance from particle to point of rotation p: linear momentum Bertrand

Physics C: Rotational Motion Practice Problem 4/21/2017 Determine the angular momentum for the revolution of the earth about the sun. the moon about the earth. Bertrand

Physics C: Rotational Motion 4/21/2017 Practice Problem Determine the angular momentum for the 2 kg particle shown about the origin. about x = 2.0. y (m) 5.0 5.0 x (m) -5.0 v = 3.0 m/s Bertrand

Angular Momentum of Solid Objects and Conservation of Angular Momentum Physics C: Rotational Motion 4/21/2017 Angular Momentum of Solid Objects and Conservation of Angular Momentum Bertrand

Angular Momentum - solid object Physics C: Rotational Motion 4/21/2017 Angular Momentum - solid object For a solid object, angular momentum is analogous to linear momentum of a solid object. P = mv (linear momentum) Replace momentum with angular momentum. Replace mass with rotational inertia. Replace velocity with angular velocity. L = I  (angular momentum) L: angular momentum I: rotational inertia w: angular velocity Bertrand

Physics C: Rotational Motion Practice Problem 4/21/2017 Set up the calculation of the angular momentum for the rotation of the earth on its axis. Bertrand

Law of Conservation of Angular Momentum Physics C: Rotational Motion 4/21/2017 Law of Conservation of Angular Momentum The Law of Conservation of Momentum states that the momentum of a system will not change unless an external force is applied. How would you change this statement to create the Law of Conservation of Angular Momentum? Angular momentum of a system will not change unless an external torque is applied to the system. LB = LA (momentum before = momentum after) Bertrand

Physics C: Rotational Motion 4/21/2017 Practice Problem A figure skater is spinning at angular velocity wo. He brings his arms and legs closer to his body and reduces his rotational inertia to ½ its original value. What happens to his angular velocity? Bertrand

Physics C: Rotational Motion Practice Problem 4/21/2017 A planet of mass m revolves around a star of mass M in a highly elliptical orbit. At point A, the planet is 3 times farther away from the star than it is at point B. How does the speed v of the planet at point A compare to the speed at point B? Bertrand

Physics C: Rotational Motion 4/21/2017 Demonstrations Bicycle wheel demonstrations Gyroscope demonstrations Top demonstration Bertrand

Physics C: Rotational Motion 4/21/2017 Precession Bertrand

Physics C: Rotational Motion Practice Problem 4/21/2017 A 50.0 kg child runs toward a 150-kg merry-go-round of radius 1.5 m, and jumps aboard such that the child’s velocity prior to landing is 3.0 m/s directed tangent to the circumference of the merry-go-round. What will be the angular velocity of the merry-go-round if the child lands right on its edge? Bertrand

Angular momentum and torque Physics C: Rotational Motion 4/21/2017 Angular momentum and torque In translational systems, remember that Newton’s 2nd Law can be written in terms of momentum. F = dP/dt Substitute force for torque. Substitute angular momentum for momentum. t = dL/dt t: torque L: angular momentum t: time Bertrand

So how does torque affect angular momentum? Physics C: Rotational Motion 4/21/2017 So how does torque affect angular momentum? If t = dL/dt, then torque changes L with respect to time. Torque increases angular momentum when the two vectors are parallel. Torque decreases angular momentum when the two vectors are anti-parallel. Torque changes the direction of the angular momentum vector in all other situations. This results in what is called the precession of spinning tops. Bertrand

If torque and angular momentum are parallel… Physics C: Rotational Motion 4/21/2017 If torque and angular momentum are parallel… Consider a disk rotating as shown. In what direction is the angular momentum? F r Consider a force applied as shown. In what direction is the torque? The torque vector is parallel to the angular momentum vector. Since t = dL/dt, L will increase with time as the rotation speeds. L is out t is out Bertrand

If torque and angular momentum are anti-parallel… Physics C: Rotational Motion 4/21/2017 If torque and angular momentum are anti-parallel… Consider a disk rotating as shown. In what direction is the angular momentum? F r Consider a force applied as shown. In what direction is the torque? The torque vector is anti-parallel to the angular momentum vector. Since t = dL/dt, L will decrease with time as the rotation slows. L is in t is out Bertrand

If the torque and angular momentum are not aligned… Physics C: Rotational Motion 4/21/2017 If the torque and angular momentum are not aligned… For this spinning top, angular momentum and torque interact in a more complex way. Torque changes the direction of the angular momentum. This causes the characteristic precession of a spinning top. L  = r  Fg r Fg   t = dL/dt  L Bertrand

Physics C: Rotational Motion 4/21/2017 Rotation Review Bertrand

Physics C: Rotational Motion Practice Problem Physics C: Rotational Motion 4/21/2017 A pilot is flying a propeller plane and the propeller appears to be rotating clockwise as the pilot looks at it. The pilot makes a left turn. Does the plane “nose up” or “nose down” as the plane turns left? Bertrand