VIEWS: 22 PAGES: 34 POSTED ON: 3/23/2011
Rotational Kinematics Tipler Ch. 9 Position • In translational motion, x=3 x position is represented by a 0 5 point, such as x. linear p/2 • In rotational motion, position is r represented by an q p 0 angle, such as q, and a radius, r. angular 3p/2 Displacement Dx = - 4 • Linear displacement is represented by the x vector Dx. 0 5 linear p/2 • Angular displacement is represented by Dq, which is not a vector, Dq but behaves like one p 0 for small values. The right hand rule determines direction. angular 3p/2 Tangential and angular displacement • A particle that rotates through an angle Dq s also translates through a distance s, which is r the length of the arc Dq defining its path. • This distance s is related to the angular displacement Dq by the equation s = rDq. Speed and velocity • The instantaneous velocity has magnitude vT = ds/dt s vT and is tangent to the circle. • The same particle rotates r with an angular velocity w = Dq dq/dt. vT w is outward • The direction of the angular according to velocity is given by the right RHR hand rule. • Tangential and angular speeds are related by the equation v = r w. Acceleration • Tangential acceleration is given by aT = dvT/dt. s vT • This acceleration is parallel or anti-parallel to the velocity. Dq r • Angular acceleration of this vT w is outward particle is given by a = according to dw/dt. RHR • Angular acceleration is parallel or anti-parallel to the angular velocity. Don’t forget • Tangential and angular centripetal accelerations are related by acceleration. the equation a = r a. Problem: Assume the particle is speeding up. a) What is the direction of the What changes if instantaneous velocity, v? the particle is b) What is the direction of the angular velocity, w? slowing down? c) What is the direction of the tangential acceleration, aT? d) What is the direction of the angular acceleration a? e) What is the direction of the centripetal acceleration, ac? f) What is the direction of the overall acceleration, a, of the particle? First Kinematic Equation • v = vo + at (linear form) – Substitute angular velocity for velocity. – Substitute angular acceleration for acceleration. • w = wo + at (angular form) 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 + wot + ½ at2 (angular form) Third Kinematic Equation • v2 = vo2 + 2a(x - xo) – Substitute angle for position. – Substitute angular velocity for velocity. – Substitute angular acceleration for acceleration. • w2 = wo2 + 2a(q - qo) 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 a : torque I: rotational inertia a: angular acceleration 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.) Hinge (rotates) A rotation occurs due to the combination of r and F. In r this case, the direction is clockwise. Direction of rotation What do you think is the F Direction of torque is direction of the torque? INTO THE SCREEN. 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? • =rF – : torque – r: moment arm (from point of rotation to point of application of force) – F: force Practice Problem 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. a) After this time, what is the angular velocity of the wheel? b)Through what angle does the wheel rotate during this 5 second period? Rotational Inertia Calculations • I = Smr2 (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 Sample Problem • 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? Practice Problem Derive the rotational inertia of a long thin rod of length L and mass M about the center. Practice Problem Derive the rotational inertia of a long thin rod of length L and mass M about a point 1/3 from one end a) using integration of I = r2 dm b) using the parallel axis theorem and the rotational inertia of a rod about the center. Practice Problem Derive the rotational inertia of a ring of mass M and radius R about the center using the formula I = r2 dm. 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. Angular Momentum of a particle • For a single particle with known momentum, the angular momentum can be calculated with this relationship: • L=rp – L: angular momentum for a single particle – r: distance from particle to point of rotation – p: linear momentum Practice Problem Determine the angular momentum for the revolution of a) the earth about the sun. b) the moon about the earth. Practice Problem Determine the angular momentum for the 2 kg particle shown a) about the origin. y (m) b) about x = 2.0. 5.0 5.0 x (m) -5.0 v = 3.0 m/s 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 w (angular momentum) – L: angular momentum – I: rotational inertia – w: angular velocity Practice Problem Set up the calculation of the angular momentum for the rotation of the earth on its axis. 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) 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? Practice Problem 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? Practice Problem 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? 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. = dL/dt – t: torque – L: angular momentum – t: time So how does torque affect angular momentum? • If = 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. If torque and angular momentum are parallel… Consider a disk rotating as shown. In what direction is the angular momentum? Consider a force applied as shown. In what r direction is the torque? F The torque vector is parallel to the angular momentum vector. L is out Since = dL/dt, L will increase with time as the rotation speeds. is out If torque and angular momentum are anti-parallel… Consider a disk rotating as shown. In what direction is the angular momentum? Consider a force applied as shown. In what r direction is the torque? F The torque vector is anti-parallel to the angular momentum vector. L is in Since = dL/dt, L will decrease with time as the rotation slows. is out If the torque and angular momentum are not aligned… • For this spinning top, angular momentum and torque interact in a more complex way. r L • Torque changes = r Fg the direction of the angular momentum. Fg • This causes the = dL/dt characteristic precession of a DL spinning top.