Circular Motion Imagine you have attached a rubber stopper to the end of a string and are whirling the stopper around your head in a horizontal circle. If both the speed of the stopper and the radius of its path remain constant, the stopper has uniform circular motion. Figure 1 shows the position and velocity vectors at various positions during the motion of the stopper. Uniform circular motion also occurs if only part of the circle (an arc) is covered. Uniform circular motion occurs for the individual parts of any object spinning at a constant rate, for example, electric fans and motors, lawn mower blades, wheels (from the point of view of the centre of the wheel), and rotating rides at amusement parks. Circular or almost circular motion can also occur for objects or particles in orbits around other objects. For example, to analyze the motion of a planet orbiting the Sun, a satellite around Earth, or an electron around a nucleus, we make the assumption that the motion is uniform circular motion even though the paths are often ellipses rather than circles. As you learned in previous Chapter, an object travelling at a constant speed is undergoing acceleration if the direction of the velocity is changing. This is certainly true for uniform circular motion. The type of acceleration that occurs in uniform circular motion is called centripetal acceleration. Centripetal acceleration is an instantaneous acceleration. Recall that the defining equation for instantaneous acceleration is To apply this equation to uniform circular motion, we draw vector diagrams and perform vector subtractions. Figure 2 shows what happens to v as t decreases. As the time interval approaches zero, the direction of the change of velocity v comes closer to pointing toward the centre of the circle. From the defining equation for instantaneous acceleration, you can see that the direction of the acceleration is the same as the direction of the change of velocity. We conclude that the direction of the centripetal acceleration is toward the centre of the circle. Notice that the centripetal acceleration and the instantaneous velocity are perpendicular to each other.
Figure 2 The Magnitude of Centripetal Acceleration We can derive an equation for the magnitude of the centripetal acceleration in terms of the instantaneous speed and the radius of the circle. Figure 3 shows a particle in uniform circular motion as it moves from an initial position r1 to a subsequent position r2; its corresponding velocities are v1 and v2.
Example 1) A child on a merry-go-round is 4.4 m from the centre of the ride, travelling at a constant speed of 1.8 m/s. Determine the magnitude of the child s centripetal acceleration. For objects undergoing uniform circular motion, often the speed is not known, but the radius and the period (the time for one complete trip around the circle) are known. To determine the centripetal acceleration from this information, we know that the speed is constant and equals the distance travelled (2πr) divided by the period of revolution; T. Example 2) Find the magnitude and direction of the centripetal acceleration of a piece of lettuce on the inside of a rotating salad spinner. The spinner has a diameter of 19.4 cm and is rotating at 780 rpm (revolutions/minute). The rotation is clockwise as viewed from above. At the instant of inspection, the lettuce is moving eastward.
Example 3) Determine the frequency and period of rotation of an electric fan if a spot at the end of one fan blade is 15 cm from the centre and has a centripetal acceleration of magnitude 2.37 x 10 3 m/s 2. Understanding Concepts 1. How can a car moving at a constant speed be accelerating at the same time? 2. Calculate the magnitude of the centripetal acceleration a particle undergoing uniform circular motion at a speed of 4.0 m/s. 3. At a distance of 25 km from the eye of a hurricane, the wind is moving at 180 km/h in a circle. What is the magnitude of the centripetal acceleration, in metres per second squared, of the particles that make up the wind?
4. Calculate the magnitude of the centripetal acceleration in the following situations: (a) An electron is moving around a nucleus with 2.1 8 x 10 6 m/s. The diameter of the electron s orbit is 1.06 x 10 10 m. (b) A cowhand is about to lasso a calf with a rope that is undergoing uniform circular motion. The time for one complete revolution of the rope is 1.2 s. The end of the rope is 4.3 m from the centre of the circle. (c) A coin is placed flat on a vinyl record, turning at 33 3 1 rpm. The coin is 13 cm from the centre of the record 5. A ball on a string, moving in a horizontal circle of radius 2.0 m, undergoes a centripetal acceleration of magnitude 15 m/s 2. What is the speed of the ball? 6. Mercury orbits the Sun in an approximately circular path, at an average distance of 5.79 x 10 10 m, with a centripetal acceleration of magnitude 4.0 x 10 2 m/s 2. What is its period of revolution around the Sun, in seconds? in Earth days?
Main Formulae Linking angular velocity, angle turned and time: Linking angular velocity and Period : Linking frequency and Period: Linking angular velocity, linear velocity and radius: 2π T = ω v = rω Linking linear velocity, radius and inwards acceleration: θ ω = t 1 f = T v a = r 2 Key to symbols used F = Centripetal Force (in N) m = mass (in kg) v = linear velocity (in m/s) ω = angular velocity (in radians/sec) r = radius (in m) T = Period (in secs) f = frequency (in Hertz) Linking Centripetal Force, linear velocity and radius: Linking Centripetal Force, angular velocity and radius: mv F = r 2 2 F = mrω θ = angle turned in radians a = inwards acceleration (in m/s 2 ) t = time (in secs)
Exercise 1. Zach drives his car around a circular section of a road having a radius of 50 meters at a constant speed of 20 m/s. a. What is the centripetal acceleration of Zach's car? b. What is actual force that turns the car? 2. Eva Destruction twirls a 0.3 kg stone attached to a string 2.0 meters long in a horizontal circle. a. At what speed must the stone move for its centripetal acceleration to be equal to the acceleration due to gravity (g = 9.81m/s2)? b. How long does it take the stone to travel once around? c. What is the stone s frequency? d. What is the tension in the string?
3. A toy electric train moving at constant speed on a circular track that has a radius of 1.2 meters goes around the track every 8 seconds. a. What is the speed of the train? b. What is the centripetal acceleration of the train? c. What force turns the train? 4. Noah Formula is in an airplane which is flying at constant speed in a circular course of radius 5000 meters, circling O'Hare Airport over Chicago prior to landing. Noah observes that the plane completes each round trip in 400 seconds. a. What is the speed of the airplane? b. What is its centripetal acceleration? c. What is the force that turns the plane?
5. An electron moves in a circular path with a radius of 0.10 m at a constant speed of 2.0 X10 6 m/s. a. What is the period of the electron s motion? b. What is the centripetal acceleration of the electron? 6. Curious George is whirling a 2.0 kg bunch of bananas on a smooth floor in a circular path having a radius of 0.50 m. What force must he apply to keep the motion constant so that the bananas complete one revolution every 4.0 seconds? 7. Pocky, who has a mass of 60 kg, sits 4.0 meters from the center of a rotating platform that is turning with a period of 10 seconds. a. What is the centripetal force acting on Tommy? b. What is the force of friction acting on Tommy?
8. Baby wants to swing a 0.10-kg Twinkie tied to one end of a cord in a horizontal circle of radius 0.50 m with a tangential speed of 0.20 m/s. What centripetal force must she exert on the mass through the cord? 9. Kim applies a force of 10 N to one end of a cord to keep a bag of rubber frogs tied to the other end moving at a speed of 2.0 m/s in a horizontal circle of radius 3.0 m. What is the mass of the bag of frogs? 10. Heather, who has a mass of 60 kg, drives her Super-Beetle around a turn on a banked circular track which has a radius of 200 m. The Super-Beetle and Heather have a combined mass of 1500 kg. Heather maintains a speed of 20 m/s around the turn. a. What centripetal force does the track exert on the Super-- Beetle? b. Friction, the force of gravity, and the normal force of the track on the Super-Beetle are the three forces acting on the Super-Beetle as it travels around the track. What is the value of the vector sum of these three forces?