Chapter 8 Angular, or Rotational Motion The radian There are 2 types of pure unmixed motion: n Translational - linear motion n Rotational - motion involving a rotation or revolution around a fixed chosen axis (an axis which does not move). We need a system that defines BOTH types of motion working together on a system. Rotational quantities are usually defined with units involving a radian measure. If we take the radius of a circle and LAY IT DOWN on the circumference, it will create an angle whose arc length is equal to R. In other words, one radian angle subtends an arc length Ds equal to the radius of the circle (R) Angular Quantities In purely rotational motion, all points on the object move in circles around the axis of rotation (“O”). The radius of the circle is r. Note that all points on a straight line drawn through the axis move through the same angle in the same time. s The angle θ is 1 radian if the arc length is equal to the radius. Therefore, The radian Half a radian would subtend an arc length, s, equal to half the radius, and 2 radians would subtend an arc length equal to two times the radius. A general Radian Angle (Dq) subtends an arc length (Ds) equal to R. The theta in this case represents ANGULAR DISPLACEMENT. Angular Velocity Since velocity is defined as the rate of change of displacement, ANGULAR VELOCITY is defined as the rate of change of ANGULAR DISPLACEMENT. Angular Acceleration Once again, following the same lines of logic… Since acceleration is defined as the rate of change of velocity, we can say the ANGULAR ACCELERATION is defined as the rate of change of the angular velocity. Combining Motions –Tangential Velocity •First we take our equation for the radian measure and divide BOTH sides by a change in time. •The left side is simply the equation for LINEAR velocity. But in this case, the velocity is TANGENT to the circle (according to Newton’s first law). Therefore we call it TANGENTIAL VELOCITY. Inspecting the right side we discover that the formula for ANGULAR VELOCITY appears. Therefore, substituting the appropriate symbols we have a formula that relates translational velocity to rotational velocity. Tangential Acceleration and Rotational Kinematics vt rw Using the same kind of mathematical vt = rw ® = reasoning we can also define linear Dt Dt tangential acceleration. a = ra t Inspecting each equation we discover that there is a DIRECT relationship between the translational quantities and the rotational quantities. We can therefore RE-WRITE each translational kinematic equation and turn it into a rotational kinematic equation. v = vo + at ® w = wo +at x = x + v t + 1 at 2 ® q =q +w t + 1 at 2 o o 2 o o 2 2 2 2 2 v = vo + 2aDx ® w = wo + 2aDq Radians and Degrees In a complete circle, there are 360o, which must correspond to an arc length equal to the circumference of the circle, s = 2p r. Thus = 2p r /r = 2p radians in a complete circle So 360o = 2p rad = 1 revolution One radian is therefore 1 rad x = 360o / 6.28 = 57.3o rad Angular Displacement Angular displacement, q, is defined as the angle, in radians, through which an object travels in some time period. Unlike linear displacement, if an object returns to its original position, the angular displacement is not zero! Example : An object travels around a circle 10.0 full turns in 2.5 seconds. Calculate (a) the angular displacement, θ in radians, and (b) its average angular speed, ω in (rad/s). Solution: (a) θ = 10.0 turns ( 2p rad / turn ) = 62.8 rad (b) w = 62.8 rad / 2.5 sec = 25.2 rad/sec Angular Displacement n Circular motion about AXIS n Three different measures of angles: 1. Degrees 2. Revolutions (1 rev. = 360 deg.) 3. Radians (2p rad = 360 deg.) Example An automobile wheel has a radius of 42 cm. If a car drives 10 km, through what angle has the wheel rotated? a) In revolutions b) In radians c) In degrees Solution Note distance car moves = distance outside of wheel moves, s a) One revolution = 2 p r , where r = 0.42 m but wheel moved through s = 10,000m 10,000m x = 3789 rev b) Find q in radians q = = 10,000m / 0.42m = 2.38 x 104 rad c) Find q in degrees 10,000m x = 1.36 x 106 deg Angular Speed • Can be given in – Revolutions/sec – Radians/sec – Degrees/sec Example A race car engine can turn at a maximum rate of 12,000 rpm. (revolutions per minute). a) What is the angular velocity in radians per second? b) If helicopter blades were attached to the crankshaft while it turns with this angular velocity, what is the maximum radius of a blade such that the speed of the blade tips stays below the speed of sound? Note: The speed of sound is 343 m/s Solution a) Convert rpm to radians per second = 1256 radians/s Solution a) Convert rpm to radians per second = 1256 radians/s b) If helicopter blades were attached to the crankshaft while it turns with this angular velocity, what is the maximum radius of a blade such that the speed of the blade tips stays below the speed of sound? Note: The speed of sound is 343 m/s Known: v = 343 m/s, w = 1256 rad./s Find r v r = = .27 m w Angular Acceleration n Denoted by a n w must be in radians per sec. n Units of angular acceleration are rad/s² n Every portion of the object has same angular speed and same angular acceleration SUMMARY: Relationship Between Angular and Linear Quantities n n Displacements Every point on the rotating s = rq object has the same angular n Speeds motion n Every point on the rotating object does not n Accelerations have the same linear motion a = ra Angular Speed, cont. n Units of angular speed are radians/sec n rad/s n Angular speed/velocity will be positive if θ is increasing (counterclockwise) n Angular speed/velocity will be negative if θ is decreasing (clockwise) Angular Quantities The frequency is the number of complete revolutions per second. Frequencies are measured in hertz. The period is the time one revolution takes: Equations for Linear Speed (in a circular path) = 2 p R f For example, if something circles with a frequency of 3 Hz, and we wanted to translate that into a speed (rev/s into m/s), we would convert = __ m/s f 2 p r = speed • So v = 2 p f r • And we already know that v = w r • Therefore, w = 2 p f Angular Quantities If the angular velocity of a rotating object changes, it has a tangential acceleration: (ch8) But even if the angular velocity is constant, each point on the object still has a centripetal acceleration: ac or Total Acceleration • The tangential component of the acceleration is due to changing speed • The centripetal component of the acceleration is due to changing direction •Total acceleration can be found from these components ac Quick Quiz A racetrack is constructed such that two arcs of radius 80 m at A and 40 m at B are joined by two stretches of straight track. In a particular trial run, a driver travels at a constant speed of 50 m/s for one complete lap. 1. The ratio of the tangential acceleration at A to that at B is (a) ½ (b) ¼ (c) 2 (d) 4 (e) The tangential acceleration is zero at both points. Quick Quiz 2. The ratio of the centripetal acceleration at A to that at B is (a) ½ (b) ¼ (c) 2 (d) 4 (e) The centripetal acceleration is Zero at both points. 3. The angular speed is greatest at (a) A (b) B (c) It is equal at both A and B Equations of Motion for Constant Angular Acceleration The equations of motion for constant angular acceleration are the same as those for linear motion, with the substitution of the angular quantities for the linear ones. Also…..