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Physics Review – Circular Motion 1. What are the two dimensions of circular motion? a. Horizontal and vertical b. Speed and direction c. Space and time 2. In our physics class, circular motion is always calculated assuming a constant speed. Yet velocity ( a word we use often instead of speed) is not constant. Why is velocity not constant? a. b. c. d. Speed is not a component of velocity Speed is constant but direction is changing Velocity is a vector that involves more than just speed. Both b and c are correct. 3. You want to make a bowling ball travel in a circular counter-clockwise path around you (this means YOU are the centerpoint). After you set the ball in motion “to the left”, in what direction will force need to be exerted to keep the ball in a circular path? a. You need to constantly push the ball away from you b. You need to constantly push the ball toward you. c. You need to constantly push the ball to the left 4. When referring to circular motion, a period is a. The distance traveled in one revolution b. The time required to travel one revolution c. The distance that you can travel in a second 5. When walking in a circle counterclockwise you are a. Pushing toward the center of your path with your right foot b. Pushing toward the center of your path with your left foot c. Exerting equal force with both feet. 6. Besides gravity (which pushes down on you) and the normal force (pushing up on you), what force is being used to keep you on a circular path? a. Inertia b. Friction c. Centrifugal force 7. We use the formula 2r for distance that is because this formula will give us a. The distance around the circle b. The diameter of the circle c. The volume of the circle. 8. Velocity is defined as a. Circumference divided by frequency (Circumference/frequency) b. Radius multiplied by Period (Radius*Period). c. Circumference divided by Period (Circumference/Period) 9. If you travel around a circle that has a radius of 2 meters in 2 seconds, the distance that you traveled is a. About 12.5 meters 1 b. 0 meters c. 25 meters d. 6.25 meters 10. If you are traveling in a circle at constant speed, are you accelerating? a. Yes, because you are changing your speed b. Yes, because you are changing direction c. No, because you are traveling in a circular path (which is really a straight line). 11. Acceleration which points to the center of a circular path is called a. Centrifugal force b. Inertia c. Centripetal force 12. Consider the following equation Assume that the Period is one second. If this is the case then average speed (v) per second = the circumference of circle. So here is my question …Suppose you have two circles, one is twice as wide as the other. If you want to “walk around each circle” in the same amount of time, the speed at which you travel needs to be a. faster on the larger circle b. faster on the smaller circle 13. Consider the following picture. If you traveled at the same average speed around each of the circles, the period would be a. shorter for the larger circle b. shorter for the smaller circle? The direction of a velocity vector at every instant is tangential to the circle. This means that unless some force is applied to the object, it will not continue to travel in a circle. It wants to “go straight” due to inertia. When we measure velocity, we are really estimating linear velocity at the “edge” of the circle. As the object turns, the direction of linear speed changes, thus we have changing speed but not direction. So based on what you know…answer the question on the next page: 2 14. A tube is been placed upon the table and shaped into a three-quarters circle. A golf ball is pushed into the tube at one end at high speed. The ball rolls through the tube and exits at the opposite end. Describe the path of the golf ball as it exits the tube. 14. What two things MUST be true for velocity to be constant? a. Speed and acceleration must be constant b. Speed and direction must be constant c. Speed must be constant and acceleration must be zero d. Either c or b is correct The acceleration of an object is often measured using a device known as an accelerometer. A simple accelerometer consists of an object immersed in a fluid such as water. Consider a sealed jar which is filled with water. A cork attached to the lid by a string can serve as an accelerometer. To test the direction of acceleration for an object moving in a circle, the jar can be inverted and attached to the end of a short section of a wooden 2x4. A second accelerometer constructed in the same manner can be attached to the opposite end of the 2x4. If the 2x4 and accelerometers are clamped to a rotating platform and spun in a circle, the direction of the acceleration can be clearly seen by the direction of lean of the corks. 15. Which direction will the corks lean? a. They will lean inward toward the center of the circle b. They will lean outward away from the circle c. They will remain “straight up and down 16. The direction in which the corks move demonstrates a. Centripetal force b. Centrifugal force c. Constant velocity and a net force of zero. 3 Another simple homemade accelerometer involves a lit candle centered vertically in the middle of an open-air glass. If the glass is held level and at rest (such that there is no acceleration), then the candle flame extends in an upward direction. However, if you hold the glass-candle system with an outstretched arm and spin in a circle at a constant rate (such that the flame experiences an acceleration), then the candle flame will no longer extend vertically upwards. Instead the flame deflects from its upright position. This signifies that there is an acceleration when the flame moves in a circular path at constant speed. The deflection of the flame will be in the direction of the acceleration. Why? This can be explained by asserting that the hot gases of the flame are less massive (on a per mL basis) and thus have less inertia than the cooler gases which surround. Subsequently, the hotter and lighter gases of the flame experience the greater acceleration and will lean in the direction of the acceleration. 17. Using the example for problem 15, draw a picture of what the flames would look like. 18. Answer the following: e. Acceleration: Yes or No? Explain. If there is an acceleration, then what direction is it? 19. Dizzy Smith and Hector Vector are still discussing # 18. Dizzy says that the ball is not accelerating because its velocity is not changing. Hector says that since the ball has changed its direction, there is an acceleration. Who do you agree with? Argue a position by explaining the discrepancy in the other student's argument. 4 20. Look at each of the pictures below and explain the force that is acting to maintain circular motion. 21. Why does gravity not affect example b. in # 20? Draw a free body diagram to explain this. 22. An object is moving in a clockwise direction around a circle at constant speed. Use your understanding of the concepts of velocity, acceleration and force to answer the next five questions. Use the diagram shown at the right. Click the button to check your answers. a Which vector below represents the direction of the force vector when the object is located at point A on the circle? b Which vector below represents the direction of the force vector when the object is located at point C on the circle? c Which vector below represents the direction of the velocity vector when the object is located at point B on the circle? d. Which vector below represents the direction of the velocity vector when the object is located at point C on the circle? e. Which vector below represents the direction of the acceleration vector when the object is located at point B on the circle? 5 23. A 900-kg car moving at 10 m/s takes a turn around a circle with a radius of 25.0 m. Determine the acceleration and the net force acting upon the car. Known Information: Requested Information: m = 900 kg a = ???? v = 10.0 m/s Fnet = ???? R = 25.0 m a = v2 / R Fnet = m • a Solve the problem. 6 A 95-kg halfback makes a turn on the football field. The halfback sweeps out a path which is a portion of a circle with a radius of 12-meters. The halfback makes a quarter of a turn around the circle in 2.1 seconds. Determine the speed, acceleration and net force acting upon the halfback.(remember that speed uses the letter “v”). Known Information: Requested Information: m = 95.0 kg v = ???? R = 12.0 m a = ???? Traveled 1/4-th of the circumference in 2.1 s Fnet = ???? 24. What equation will you use to solve for v? Write the equation, then solve the problem. 25. What equation will you use to solve for a? Write the equation, then solve the problem. 26. What equation will you use to solve for Fnet? Write the equation then solve the problem. 7 This set of circular motion equations can be used in two ways: as a "recipe" for algebraic problem-solving in order to solve for an unknown quantity. as a guide to thinking about how an alteration in one quantity would affect a second quantity. Equations as a Guide to Thinking An equation expresses a mathematical relationship between the quantities present in that equation. For instance, the equation for Newton's second law identifies how acceleration is related to the net force and the mass of an object. If Fnet = ma, we can rearrange this to state that The relationship expressed by the equation is that the acceleration of an object is directly proportional to the net force acting upon it. In other words, the bigger the net force value is, the bigger that the acceleration value will be. As net force increases, the acceleration increases. In fact, if the net force were increased by a factor of 2, the equation would predict that the acceleration would increase by a factor of 2. Similarly, if the net force were decreased by a factor of 2, the equation would predict that the acceleration would decrease by a factor of 2. Newton's second law equation also reveals the relationship between acceleration and mass. 27. According to the equation shown above, the acceleration of an object is a. inversely proportional b. proportional to the mass of the object. 28. In other words, the bigger the mass value is, Bigger the acceleration Smaller the acceleration 29. In fact, if the mass were increased by a factor of 2, the equation would predict that the acceleration would Increase by a factor of 2 decrease by a factor of 2 30. Similarly, if the mass were decreased by a factor of 2 the equation would predict that the acceleration would increase by a factor of 2 decrease by a factor of two. 8 The predictive ability of an equation becomes more complicated when one of the quantities included in the equation is raised to a power. For instance, consider the following equation relating the net force (Fnet) to the speed (v) of an object moving in uniform circular motion. This equation shows that the net force required for an object to move in a circle is directly proportional to the square of the speed of the object. For a constant mass and radius, the Fnet is proportional to the speed2. The factor by which the net force is altered is the square of the factor by which the speed is altered. Subsequently, if the speed of the object is doubled, the net force required for that object's circular motion is quadrupled. And if the speed of the object is halved (decreased by a factor of 2), the net force required is decreased by a factor of 4 31. Anna Litical is practicing a centripetal force demonstration at home. She fills a bucket with water, ties it to a strong rope, and spins it in a circle. Anna spins the bucket when it is half-full of water and when it is quarter-full of water. In which case is more force required to spin the bucket in a circle? Explain using an equation as a "guide to thinking." 32. A Lincoln Continental and a Yugo are making a turn. The Lincoln is four times more massive than the Yugo. If they make the turn at the same speed, then how do the centripetal forces acting upon the two cars compare. Explain. 33. The Cajun Cliffhanger at Great America is a ride in which occupants line the perimeter of a cylinder and spin in a circle at a high rate of turning. When the cylinder begins spinning very rapidly, the floor is removed from under the riders' feet. What affect does a doubling in speed have upon the centripetal force? Explain. 34. Determine the centripetal force acting upon a 40-kg child who makes 10 revolutions around the Cliffhanger in 29.3 seconds. The radius of the barrel is 2.90 meters. 9 Answers to questions 27-30. According to the equation, the acceleration of an object is inversely proportional to mass of the object. In other words, the bigger the mass value is, the smaller that the acceleration value will be. As mass increases, the acceleration decreases. In fact, if the mass were increased by a factor of 2, the equation would predict that the acceleration would decrease by a factor of 2. Similarly, if the mass were decreased by a factor of 2, the equation would predict that the acceleration would increase by a factor of 2. Answers to questions 31-34 31. It will require more force to accelerate a full bucket of water compared to a half-full bucket. According to the equation Fnet = m•v2 / R, force and mass are directly proportional. So the greater the mass, the greater the force. 32. The centripetal force on the Continental is four times greater than that of a Yugo. According to the equation Fnet=(m•v2) / R, force and mass are directly proportional. So 4 times the mass means 4 times the force. 33. Doubling the speed of the ride will cause the force to be four times greater than the original force. According to the equation Fnet=(m•v2) / R, force and speed2 are directly proportional. So 2X the speed means 4X the force (that's from 22). 34. Answer: Fnet = 533 N Given: m = 40 kg; R = 2.90 m; T = 2.93 s (since 10 cycles takes 29.3 s). First, find speed using speed=(2 • pi • R) / T = 6.22 m/s. Then find the acceleration using a = v2 / R = = (6.22 m/s)2 / (2.90 m) = 13.3 m/s/s Now use Fnet = m • a to find that Fnet = 533 N. 10