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Rotational Motion and Angular Momentum. Unit 6. A particle at point P at a fixed distance r from origin is rotating about axis O. Lesson 1 : Angular Position, Velocity, and Acceleration.

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## Rotational Motion and Angular Momentum

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**A particle at point P at a fixed distance r from origin is**rotating about axis O. Lesson 1 : Angular Position, Velocity, and Acceleration When a rigid object rotates about its axis, at any given time different parts of the object have different linear velocities and linear accelerations.**q is measured counterclockwise from some reference line (q =**0) s q = r P is at polar coordinate (r, q) s = rq**One radian is the angle subtended by an arc length equal to**the radius of the arc. 2pr 360o = rad = 2p rad 360o r = 57.3o 1 rad = s q = 2p r q is measured in radians (rad)**p**q (rad) = q (deg) 180o Converting from Degrees to Radians 90o = p/2 rad 60o = p/3 rad 45o = p/4 rad 270o = 3p/2 rad**As particle moves from A to B in time interval Dt, the**reference line of length r sweeps out an angle Dq = qf - qi Angular Displacement (Dq)**qf - qi**Dq w = = tf - ti Dt Average Angular Speed (w) Ratio of the angular displacement of a rigid object to the time interval Dt. The rad/s is the unit for angular speed. w is positive when q increases (counterclockwise motion) w is negative when q decreases (clockwise motion)**wf - wi**Dw a = = tf - ti Dt Average Angular Acceleration (a) Dq dq w = lim = Dt dt Dt 0 Instantaneous Angular Speed (w)**Dw**dw a = lim = Dt dt Dt 0 Instantaneous Angular Acceleration (a) The rad/s2 is the unit for angular speed. a is positive when object rotates counterclockwise and speeds up OR when object rotates clockwise and slows down**The directions of w and a are along the axis of rotation.**Direction of Angular Velocity and Acceleration Vectors w and a are vector quantities with magnitude and direction**Right-Hand Rule**Wrap four fingers of the right hand in the direction of rotation. Thumb will point in the direction of angular velocity vector (w).**Example 1**A rigid object is rotating with an angular speed w < 0. The angular velocity vector w and the angular acceleration vector a are antiparallel. The angular speed of the rigid object is a) clockwise and increasing b) clockwise and decreasing c) counterclockwise and increasing d) counterclockwise and decreasing**Example 2**During a certain period of time, the angular position of a swinging door is described by q = 5.00 + 10.0 t + 2.00 t2, where q is in radians and t is in seconds. Determine the angular position, angular speed, and angular acceleration of the door at a) at t = 0 b) at t = 3.00 s**Lesson 2 : Rotational Kinematics with Constant Angular**Acceleration dw a = dt wf = wi + at (by integrating) dq Since w = dt dq wi + at = dt dw = a dt**qf = qi + wit+ ½ at2 (by integrating)**wf2 = wi2 + 2a(qf – qi) qf = qi + ½ (wi + wf) t Eliminating t from previous two equations, Eliminating a from previous two equations,**q**Position x w Velocity v a Acceleration a**Example 1**A wheel rotates with a constant angular acceleration of 3.50 rad/s2. a) If the angular speed of the wheel is 2.00 rad/s at ti = 0, through what angular displacement does the wheel rotate in 2.00 s ?**b) Through how many revolutions has the wheel turned during**this time interval ? c) What is the angular speed of the wheel at t = 2.00 s ?**Example 2**A rotating wheel requires 3.00 s to rotate through 37.0 revolutions. Its angular speed at the end of the 3.00 s interval is 98.0 rad/s. What is the constant angular acceleration of the wheel ?**Every particle of a rigid rotating object moves in a circle**whose center is the axis of rotation. ds Tangential speed v = dt dq v = r dt v = rw Lesson 3 : Angular and Linear Quantities Since s = rq,**angular speed**perpendicular distance of a point from the axis of rotation Tangential Speed v = rw Tangential speed depends on distance from axis of rotation Angular speed is the same for all points**Tangential Acceleration**dv at = dt dw at = r dt at = ra Since v = rw,**v2**ac = r ac = rw2 Centripetal Acceleration in terms of Angular Speed Since v = rw,**a = at2 + ar2**a = r2a2 + r2w4 a = r a2 + w4 Total Linear Acceleration**In order to keep the tangential speed of the disc surface at**the location of the lens constant, the disc’s angular speed must vary as the lens moves radially along the disc. In a typical compact disc player, the constant speed of the surface at the point of the laser-lens system is 1.3 m/s. Example 1**a) Find the angular speed of the disc in rev/min when**information is being read from the innermost first track (r = 23 mm) and the outermost final track (r = 58 mm).**b) The maximum playing time of a standard music CD is 74**min 33 s. How many revolutions does the disc make during that time ?**c) What total length of track moves past the lens during**this time ?**d) What is the angular acceleration of the CD over the**4,473 s time interval ? [Assume that a is constant.]**The drive train of a bicycle is shown to the right. The**wheels are 67.3 cm in diameter and pedal cranks 17.5 cm long. The cyclist pedals at a steady angular rate of 76.0 rev/min. The chain engages with a front sprocket 15.2 cm in diameter and a rear sprocket 7.00 cm in diameter. Example 2**a) Calculate the speed of a link of the chain relative to**the bicycle frame. b) Calculate the angular speed of the bicycle wheels.**c) Calculate the speed of the bicycle relative to the road.**d) What pieces of data, if any, are not necessary for the calculations ?**Lesson 4 : Rotational Kinetic Energy**The total kinetic energy of a rotating rigid object is the sum of the kinetic energies of its individual particles. KErot = SKEi = S ½mivi2 Since v = rw, KErot = ½ S miri2wi2 Factoring out w2, KErot = ½ (S miri2) w2**Moment of Inertia (I)**I = S miri2 Moment of inertia is a measure of the resistance of an object to changes in its rotational motion. KErot = ½ (S miri2) w2 The kg . m2 is the SI unit for moment of inertia.**Rotational Motion**Linear Motion KErot = ½ Iw2 KErot = ½ Iw2 KE = ½ mv2 Substituting I, I m w v**Example 1**Consider an oxygen molecule (O2) rotating in the x-y plane about the z-axis. The rotation axis passes through the center of the molecule, perpendicular to its length. The mass of each oxygen atom is 2.66 x 10-26 kg, and at room temperature the average separation between the two atoms is d = 1.21 x 10-10 m. (The atoms are modeled as particles.) a) Calculate the moment of inertia of the molecule about the z-axis.**b) If the angular speed of the molecule about the z-axis is**4.60 x 1012 rad/s, what is its rotational kinetic energy ?**ò**Dmi 0 I = lim Sri2Dmi = r2 dm ò I = rr2 dV Lesson 5 : Calculation of Moments of Inertia Moment of inertia of a rigid object is evaluated by dividing the object into many small volume elements, each with mass = Dmi. Since r = m/V, dm = r dV**Find the moment of inertia of a uniform thin hoop of mass M**and radius R about an axis perpendicular to the plane of the hoop and passing through its center. Example 1**Calculate the moment of inertia of a uniform rigid rod of**length L and mass M about an axis perpendicular to the rod (the y-axis) and passing through its center of mass. Example 2**A uniform solid cylinder has a radius R, mass M, and length**L. Calculate its moment of inertia about its central axis (the z-axis). Example 3**The moment of inertia about any axis parallel to and a**distance D away from this axis is I = ICM + MD2 Parallel-Axis Theorem**Consider once again the uniform rigid rod of mass M and**length L. Find the moment of inertia of the rod about an axis perpendicular to the rod through one end (the y’ axis). Example 4**Moment of Inertia of a Long Thin Rod**(Axis Through Center)**Moment of Inertia of a Long Thin Rod**(Axis Through End)

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