Chapter 9 Winding It Up: Rotational Motion and Torque In This Chapter Finding tangential speed and acceleration Getting angular velocity Calculating torque Working with rotational kinematics Handling rotational equilibrium When things rotate, lots of physics happens, If you have a merry-go-round, for example, and apply a force to its edge, what happens next is explained by physics. This chapter covers tangential speed, tangential acceleration, angular velocity and angular acceleration treated as a vector, and torque. I provide you with plenty of practice problems to help you become a master at handling these types of physics problems. Finding Tangential Speed Tangential speed is the magnitude of tangential velocity. Take a look at Figure 9-1, where you see a ball revolving around the origin. As it sweeps around the origin with linear speed (which keeps changing direction as the ball moves in a circle), the angle @ Increases in time. Figure 9-1: A rotating ball. @John Wiley & Sand, Inc.160 Part III: Being Energetic: Work How do you relate the distance the ball has traveled, s, to the angle ? If you measure @ In radians, that relationship is the following, where r is the radius of the circle: s = ro Also, you know that: That means that you can substitute for s to get: And a = 0/t, which means: So MEMBER In this case, o is called tangential velocity - that is, it's the Instantaneous linear velocity of the ball as it goes around in the circle. Tangential velocity is always perpendicular to the radius of the circle and Is in the direction of travel of the object going around the circle. A ball on a string is going around in'a A. The correct answer is 12 m/s. circle at 6.0 radians/s. What is Its tangen- tial velocity if the radius of the circle is 1. Use the equation o = rox 2.0 m? 2. Plug in the numbers: u = ros = (2.0 m)(6.0 radians/s) = 12 m/sChapter 9: Winding It Up: Rotational Motion and Torque 16 1. If a satellite is orbiting Earth, which has 2. You're flying a toy plane on a string, and an average radius of 3,960 miles, at an It's going around at 20.0 mph, 100.0 feet altitude of 150 miles and an angular from you. What is Its angular speed In speed of 1.17 x 10 3 radians/s, what is radians/s? the satellite's tangential speed in mph? 3. A racing car is going around a circular 4. The tip of an airplane propeller is going at track with a 400.0-ft radius at 50.0 mph. 500.0 mph. If the propeller has a radius of What Is its angular speed in radians/s? 3.0 ft, what is its angular speed?Targeting Tangential Acceleration Besides tangential velocity, you can have tangential acceleration. For Instance, if you start a helicopter's rotors, the tip of any rotor starts with a tangential velocity of zero and increases with time. Because the velocity vector's magnitude increases and its direction changes, there's acceleration, which is expressed like so: a = Au At How can you relate this to angular quantities? Because tangential speed is o = ro, you can plug that into the acceleration equation: a=Au Are) raw At where you can write A(rai) = ram because r is constant. And because Am/At = a, which is the angular acceleration, this equation becomes a =Au A(re) Ar L=ra Which breaks down to What this all means is that the tangential acceleration at radius ris a = ra. O. A set of helicopter blades has a radius of 2. Plug in the numbers: 4.3 m. If a point on the tip of one blade a= Au (400 m/s-0 m/s) starts at 0 m/s and ends up 60 seconds Ar (60 s) L= ra later with a speed of 400 m/s, what was the angular acceleration? 3. Divide both sides by r. (400 m/s-0 m/s) A. The correct answer is 1.6 radians/s'. = a (4.3 m) (60 s) 1. Use this equation: 4. Do the math: = Al - ra (400 m/s-0 m/s) (4.3 m)(60 s) 2=a =1.6 radians/s'Chapter 9: Winding It Up: Rotational Motion and Torque 5. If a point on the edge of a tire with a 6. You're flying a toy plane on a string, and radius of 0.50 m starts at rest and ends it's going around at 20.0 m/s. 10.0 m from up 3.5 minutes later at 88 m/s (about you. If it accelerates to a final velocity of 197 mph), what was the magnitude 30.0 m/s in 80.0 seconds, what is its of Its average angular acceleration? angular acceleration? Angular Velocity as a Vector Angular velocity is really a vector, w. The question is, which way does it point? Think of It this way. If you have a flying disk being tossed back and forth between two players, It's spinning. So which way can a point so it stays constant In magnitude and direction? Take a look at Figure 9-2 for the answer. The a vector points out of the plane of rotation. Figure 9-2: Angular velocity as a vector. D John Wiley & Sons, Inc. You find the direction of the a vector by wrapping your right hand around in the direction of rotation. Your right thumb will point In the direction of the as vector. EL MA A helicopter's blades are rotating in a A. The correct answer is upward. horizontal plane, and they're going counterclockwise when viewed from 1. Curl your right hand in the direction of above. Which way does a point? rotational motion - counterclockwise. 2. Your right thumb points upward, Indicating the direction of the ar vector.Part III: Being Energetic: Work 8. Suppose that you're driving forward. 7. Suppose that you're flying a toy plane on a Which way does a point for the string, and it's going around clockwise as left front tire? viewed from above. Which way does a point? Angular Acceleration as a Vector Like angular velocity, angular acceleration is a vector; It's represented by the symbol a. But unlike angular velocity, angular acceleration need not be perpendicular to the plane of rota- tion. The angular acceleration vector just points in the direction of change of the angular velocity vector. Figure 9-3 shows angular acceleration in the same direction as the angular velocity vector. That means the angular velocity vector will grow in time. Figure 9-3: Angular acceleration as a vector. John Whey & Sang, Inc. Bear in mind that a need not be perpendicular to the rotation - It just points in the direction in which a is changing. For example, if you're turning the wheels of a car, the vector at points in a direction so when it's added to the original angular velocity, @.. you get the new angular velocity, or MAPLE A hellcopter's blades are rotating In a 2. Your right thumb points upward, indi- horizontal plane, and they're going coun- cating the direction of the a vector. terclockwise when viewed from above. As they slow down, which way does a 3. The w vector is decreasing in magni- point? tude with time while staying In the - same direction, which means a points A. The correct answer is downward. in the opposite direction - downward. 1. Curl your right hand in the direction of rotational motion - counterclockwise.Chapter 9: Winding It Up: Rotational Motion and Torque 9. Suppose that you're flying a toy plane on a 10. A flying disc tossed from one player to string, and it's going around clockwise as another spins as it flies and slows down. viewed from above. In time, the plane is If it's spinning counterclockwise when slowing down. Which way does a point? viewed from above, which way does a point? Doing the Twist: Torque Torque, represented by the symbol r, is the rotational analog of force in physics. Torque is much like force but differs in that it's the amount of twist, not push, that occurs. For exam- ple, take a look at the door, viewed from above, in Figure 9-4, where the black dot represents the location of the hinge. Figure 9-4: Torque at work. John Wiley & Sons, Inc. If you push on the hinge, as shown in Figure 9-a, the door doesn't open. That's because there's no torque on the door to turn it. On the other hand, If you push on the midpoint of the door, as shown In Figure 9-4b, the door opens. And if you push on the outer edge of the door, it opens more easily (that's Figure 9-4c). The more force you apply, the more torque there Is; the farther out from the turning point (the hinge) you push, the more torque there is. Following is the equation for torque; Fis the force you're applying, and / is the lever arm - the perpendicular distance from the ands of rotation to the point where you apply the external force