Rotational Motion and Angular Displacement
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1 Physics 20 AP - Assignment #5 Angular Velocity and Acceleration There are many examples of rotational motion in everyday life (i.e. spinning propeller blades, CD players, tires on a moving car ). In this lesson, we focus on the rotational motion of a rigid body - a body that has a definite shape that doesn t change, so that the particles that make it up stay in fixed positions relative to one another. Rotational Motion and Angular Displacement When a rigid body rotates through a circular path, it must do so around a center known as its axis of rotation. As the rigid body rotates it sweeps out an angle about the fixed axis. This angle is known as the angular displacement. Angular Displacement The angle ( ) swept out by a line passing through any point on the body and intersecting the axis of rotation perpendicularly. Angular displacement is positive (+) if it is counterclockwise and negative (-) if it is clockwise. The units for angular displacement are radians (rad). In rotational motion, all points of the rigid body move in circles about some axis of rotation, O, such as the point P moving counterclockwise below. r P s Reference Line Referring to the diagram above, a point particle P is moving counterclockwise around the circle, making an angle,, with the positive x-axis when it travels the distance, s, measured along the circumference of its circular path. Angles are commonly measured in degrees, but the mathematics of circular motion is much simpler if we use the radian for angular measure. One radian (rad) is the angle swept out by an arc whose length is equal to the radius. In our circle, if point P is a distance, r, from the axis of rotation and it has moved a distance, s, along the arc of the circle; if s = r, then = 1 rad. Generally, any angle is given by: (radians) = s/r = arc length/ radius Then, if in a complete circle, there are 360 and this corresponds to an arc length, s, equal to the circumference of the circle, s = 2 r. If = s/r = 2 r/r = 2 rad in a complete circle, so 360 = 2 rad 1 rad = 57.3
2 Example #1 2 synchronous satellites are put into an orbit whose radius is 4.23 x 10 7 m. The orbit is in the plane of the equator, and the two adjacent satellites have an angular separation of = Find the arc length that separates the satellites. ** Once the angle ( ) is converted into radians, the relation s = r can be used = 2.00 x (2 radians/360 ) = radians s = r = ( rad)(4.23 x 10 7 m) = 1.48 x 10 6 m Angular Velocity and Acceleration To describe rotational motion, we make use of angular quantities such as angular velocity and acceleration. Consider an object already undergoing periodic, uniform circular, motion. The object maintains a constant speed as it revolves around a circle of radius, r, in a period of time, T. Clearly, if the total distance traveled around the circle is its circumference, 2 r, and the time for one complete rotation is its period, T, then the constant average speed is v t = d/t = 2 r/t Also, if the circle maintains a constant radius, the quantity 2 /T is called the angular velocity,, and the tangential velocity is then: v t = r The units of involve units related to arc length. Angular Velocity Angular velocity is defined in analogy with ordinary linear velocity. Instead of distance traveled, we use the angular distance,. Thus, the average angular velocity,, is defined as = - / t-t = / t so, angular velocity is the angle through which the body has rotated in time, t, and its units are radians/s. Now, let us return to the idea of uniform circular motion. If the object is moving around a circle at a constant rate, its linear (tangential) and angular velocities are related by the angle swept out per second and the radius of the circle. Consider two people standing on a moving merry-go-round; one person near the center and one near the outer edge of the merry-go-round. Both people experience the same angular speed, = / t, since they both sweep out equal angles in equal time intervals, but the person on the outer edge has a greater tangential speed, since he is sweeping out a larger arc length in a same amount of time. v t = d/t= s/t = r /t = r
3 Example 2 A gymnast on a highbar swings through two revolutions in a time of 1.9 s. Find the average angular velocity of the gymnast in rad/s. = 2.00 rev x (2 radians / revolution) = 12.6 radians = / t = 12.6 rad / 1.90 s = 6.63 rad / s Angular Acceleration Angular acceleration, in analogy to ordinary linear acceleration, is defined as the change in angular velocity divided by the time required to make this change. The average angular acceleration,, is = - /t-t = / t in units of radians/seconds 2. Since, the angle through which the body is rotated in a set time interval is constant, for an object undergoing uniform circular motion, then = 0, and the angular acceleration,, is 0. Thus, we are still left with the question concerning the nature of the acceleration involved. Suppose you tie a rock to the end of a string and spin it in a circle. The string holding the rock is exerting a force inward toward your hand. This force is called a centripetal force. Since the rock is spinning in a circle, and thus constantly changing direction, it is being accelerated by this centripetal force. This acceleration is directed radially inwards, in the same direction as the force and is known as the centripetal acceleration, a c. The magnitude of this acceleration depends on how fast you spin the rock and how long the string is. a c = v t 2 /r = (r ) 2 /r = r 2 Example 3 A jet waiting to take off rev's his engine. As the engine idles, the fan blades rotate with an angular velocity of +110 rad/s. As the plane takes off, the angular velocity of the blades reaches +330 rad/s in a time of 14 s. Find the angular acceleration of the blades. = / t = (+ 220 rad/s / 14 s) = 16 rad/s 2
4 Rotational Kinematics Our understanding of rotational velocity and acceleration takes into the realm of rotational kinematics. Using our previous kinematic equations we are able to derive equations explaining rotational motion. The mathematical forms of the linear and rotational kinematic equations are identical with the exception of the replacement of the linear variables with rotational variables. Quantity Rotational Motion Linear Motion Displacement d Initial velocity o v o Final velocity f v f Acceleration a Time t t Rotational Motion Linear Motion ( = constant) (a = constant) = o + t v = v o + at = ½ ( o + )t d = ½ (v o + v)t = o t + ½ t 2 d = v o t + ½ at 2 2 = 2 o + 2 v 2 = v 2 o + 2ad Example Problem: The blades of an electric blender are whirling with an angular velocity of +375 rad/s while the puree button is depressed. When the blend button is pressed, the blades accelerate and reach a greater angular velocity in rad (seven revolutions). The angular acceleration has a constant value of rad/s 2. Find the final angular velocity of the blades. 2 = o = (375 rad/s) 2 + 2(1740 rad/s 2 ) (44.0 rad) = 2.94 x 10 5 rad 2 /s 2 = +542 rad/s (The negative root is disregarded since the blades do not reverse direction.)
5 Physics 20 AP Assignment Angular Velocity and Acceleration Questions 1-4 are based on the following: A fly and a mosquito are sitting on a phonograph record which is turning at a constant rate. The fly is near the outer edge of the record and the mosquito is near the center. 1. How do their angular velocities compare? a) The fly has the greater angular velocity. b) The mosquito has the greater angular velocity. c) Both the fly and the mosquito have the same nonzero angular velocity. d) The angular velocity for both is zero. 2. How do their tangential velocities compare? a) The fly has the greater tangential velocity. b) The mosquito has the greater tangential velocity. c) Both the fly and the mosquito have the same nonzero tangential velocity. d) The tangential velocity for both is zero. 3. How do their angular accelerations compare? a) The fly has the greater angular acceleration. b) The mosquito has the greater angular acceleration. c) Both the fly and the mosquito have the same nonzero angular acceleration. d) The angular acceleration for both is zero. 4. How do their centripetal accelerations compare? a) The fly has the greater centripetal acceleration. b) The mosquito has the greater centripetal acceleration. c) Both the fly and the mosquito have the same nonzero centripetal acceleration. d) The centripetal acceleration for both is zero. 5. A particular bird s eye can just distinguish objects which subtend an angle no smaller that about 3.00 X 10-4 rad. How small an object can the bird just distinguish when flying at a height of 100 m? a cm c m b cm d m corresponds to how many radians? a c. 172 b d. 188
6 7. An electric grinder uses a grinding wheel of 12 cm radius. It takes the electric motor 3.0 s to reach its rated speed of 1500 rev/min starting from rest. What is the angular acceleration? a. 8.3 rad/s 2 c. 52 rad/s 2 b. 24 rad/s 2 d. 3.1 x 10 3 rad/s 2 8. A small bug sits on a turntable, 10.0 cm from the center. What is the centripetal acceleration of the bug when the turntable rotates at 45.0 rev/min? a m/s 2 c m/s 2 b m/s 2 d. 203 m/s 2 9. What is the above bug s tangential velocity? a m/s c m/s b m/s d. 450 m/s 10.A space station is constructed in the shape of a doughnut whose outer radius is 480 m. At what rate should the space station rotate so that its occupants will experience an acceleration of magnitude g when they are located at the outer diameter of the station? a rev/min c rev/min b rev/min d rev/min Long Answer 1. An electric fan is set on its HIGH setting. After the LOW push button is depressed, the angular speed of the fan blades decreases to a value of 800 rev/min in 1.75 s. The deceleration is 42.0 rad/s 2. Determine the initial angular speed of the blades in rev/min. (4 marks)
7 2. An airliner arrives at the terminal, and the pilot shuts off the engines. The initial angular velocity of the fan blades is 1800 rad/s, and it takes 120s for them to come to rest. What is the angular displacement of the blades?
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