Circular Motion Skill Building Problems

PSI AP Physics 1 Uniform Circular Motion Skill Building Problems 1. Determine the minimum angle at which a frictionless road should be banked so that a car traveling at 20.0 m/s can safely negotiate the curve if the radius of the curve is 200.0 m. 2. Determine the velocity that a car should have while traveling around a frictionless curve of radius 100m and that is banked 20 degrees. 3. If the road in number 2 was not frictionless, which way would the frictional force point if the car exceeded this velocity? Which way would it point if the car went slower than this velocity? 4. A curve with a radius of 50 m is banked at an angle of 25˚. The coefficient of static friction between the tires and the roadway is 0.3. a. Find the correct speed of an automobile that does not require any friction force to prevent skidding. b. What is the maximum speed the automobile can have before sliding up the banking? c. What is the minimum speed the automobile can have before sliding down the banking? 5. A ball of mass M attached to a string of length L moves in a circle in a vertical plane as shown above. At the top of the circular path, the tension in the string is twice the weight of the ball. At the bottom, the ball just clears the ground. Air resistance is negligible. Express all answers in terms of M, L, and g. a. Determine the magnitude and direction of the net force on the ball when it is at the top. b. Determine the speed vo of the ball at the top. After a few circles, the string breaks when the ball is at the highest point. c. Determine the time it takes the ball to reach the ground. d. Determine the horizontal distance the ball travels before hitting the ground. 6. A ball of mass M is attached to a string of length R and negligible mass. The ball moves clockwise in a vertical circle, as shown above. When the ball is at point P, the string is horizontal. Point Q is at the bottom of the circle and point Z is at the top of the circle. Air resistance is negligible. Express all algebraic answers in terms of the given quantities and fundamental constants. a. On the figures below, draw and label all the forces exerted on the ball when it is at points P and Q, respectively. b. Derive an expression for vmin the minimum speed the ball can have at point Z without leaving the circular path. c. The maximum tension the string can have without breaking is Tmax Derive an expression for vmax, the maximum speed the ball can have at point Q without breaking the string. d. Suppose that the string breaks at the instant the ball is at point P. Describe the motion of the ball immediately after the string breaks. 7. A 0.10-kilogram solid rubber ball is attached to the end of an 0.80-meter length of light thread. The ball is swung in a vertical circle, as shown in the diagram above. Point P, the lowest point of the circle, is 0.20 meter above the floor. The speed of the ball at the top of the circle is 6.0 meters per second, and the total energy of the ball is kept constant. a. Determine the total energy of the ball, using the floor as the zero point for gravitational potential energy. b. Determine the speed of the ball at point P, the lowest point of the circle. c. Determine the tension in the thread at i. the top of the circle; ii. the bottom of the circle. The ball only reaches the top of the circle once before the thread breaks when the ball is at the lowest point of the circle. d. Determine the horizontal distance that the ball travels before hitting the floor. 8. To study circular motion, two students use the handheld device shown above; this consists of a rod on which a spring scale is attached. A polished glass tube attached at the top serves as a guide for a light cord attached the spring scale. A ball of mass 0.200 kg is attached to the other end of the cord. One student swings the teal around at constant speed in a horizontal circle with a radius of 0.500 m. Assume friction and air resistance are negligible. a. Explain how the students, by using a timer and the information given above, can determine the speed of the ball as it is revolving. b. How much work is done by the cord in one revolution? Explain how you arrived at your answer. c. The speed of the ball is determined to be 3.7 m/s. Assuming that the cord is horizontal as it swings, calculate the expected tension in the cord. d. The actual tension in the cord as measured by the spring scale is 5.8 N. What is the percent difference between this measured value of the tension and the value calculated in part c.? e. The students find that, despite their best efforts, they cannot swing the ball so that the cord remains exactly horizontal. i. On the picture of the ball, draw vectors to represent the forces acting on the ball and identify the force that each vector represents. ii. Explain why it is not possible for the ball to swing so that the cord remains exactly horizontal. iii. Calculate the angle that the cord makes with the horizontal. 9. An object of mass M on a string is whirled with increasing speed in a horizontal circle, as shown above. When the string breaks, the object has speed vo and the circular path has radius R and is a height h above the ground. Neglect air friction. a. Determine the following, expressing all answers in terms of h, vo, and g. i. The time required for the object to hit the ground after the string breaks ii. The horizontal distance the object travels from the time the string breaks until it hits the ground iii. The speed of the object just before it hits the ground b. On the figure below, draw and label all the forces acting on the object when it is in the position shown in the diagram above. c. Determine the tension in the string just before the string breaks. Express your answer in terms of M, R, vo, and g. 10. (Short answer) A red car is traveling on a straight road and is speeding up with an acceleration of 5 m/s2. A blue car is traveling on a circular track of radius 20 m with a constant speed of 10 m/s. Which car has a constant acceleration? Justify your answer. Answers 1. ϴ = 11.3O 2. V = 19 m/s 3. If v >19 m/s, then f is inward If v <19 m/s, then f is outward 4. a. v = 15 m/s b. v = 21.1 m/s c. v = 8.5 m/s 5. a. FNET = 3 Mg; FNET is pointed downwards b. vO = √(3Lg) c. t = 2√(L/g) d. x = 2L√3 6. a. FT FT mg b. c. d. 7. Vmin = √(Rg) V = √(R( – g)) Velocity is upwards, acceleration is g down a. E = 3.56 J b. 8.2 m/s c. i. 3.5 N ii. 9.4 N d. 1.6 m 8. a. Get n and t Calculate T = Calculate v = 2 b. none, Δx = 0, also F is perpendicular to v c. T = 5.5 N d. % difference = 5% e. i. FT mg a= mg ii. The net force in the vertical direction must be zero. Tension must support the weight iii. ϴ = 20O 9. a. i. t = √(2h/g) ii. x = vO√(2h/g) iii. v = √(vO2 + 2gh) b. FT a= mg c. T = m√() 10. Acceleration is a vector, which means it has both a magnitude and a direction. For acceleration to be constant, both of these things must be constant. Since the blue care is traveling in a circle, its acceleration is always towards the center of the circle; therefore it is changing direction and is not constant.