Slide 1 / 71 Slide 2 / 71 AP Physics 1 Circular Motion 2015-12-02 www.njctl.org Slide 3 / 71 Slide 4 / 71 Topics of Uniform Circular Motion (UCM) Kinematics of UCM Click on the topic to go to that section Period, Frequency, and Rotational Velocity Dynamics of UCM Kinematics of UCM Vertical UCM Buckets of Water Rollercoasters Cars going over hills and through valleys Horizontal UCM Return to Unbanked Curves Table of Banked Curves Contents Conical Pendulum Slide 5 / 71 Slide 6 / 71 Kinematics of Uniform Circular Motion Kinematics of Uniform Circular Motion This acceleration is called the centripetal, or Uniform circular motion: radial, acceleration, and it points towards the center motion in a circle of of the circle. constant radius at constant speed Instantaneous velocity is always tangent to circle.
Slide 7 / 71 Slide 8 / 71 Kinematics of Uniform Circular Motion Kinematics of Uniform Circular Motion Looking at the change in velocity in the limit that the time If displacement is equal to interval becomes infinitesimally small, we see that we get velocity multiplied by time, two similar triangles. then vt is the displacement During that same time, velocity covered in time t. changed by an amount, Δv. # l A A B v 1 vt B r v 2 v 1 # r r r ## # r v r 2 # v C C ## Slide 9 / 71 Slide 10 / 71 Kinematics of Uniform Circular Motion Kinematics of Uniform Circular Motion These are similar triangles because Transpose v 2 to see the angles are all congruent, so the the vector addition. sides must be in proportion. v 1 v 1 # v vt # v v 2 v 2 Δv = v r vt v 1 # v v 2 = v 2 θ θ t r r r v a = 2 r θ The change in velocity, # v, shows the direction of the acceleration. θ In the picture, you can see # v points towards the center of the circle. That's the magnitude of the acceleration. Slide 11 / 71 Slide 12 / 71 Kinematics of Uniform Circular Motion 1 Is it possible for an object moving with a constant speed to accelerate? Explain. No, if the speed is constant then the A This acceleration is called acceleration is equal to zero. the centripetal, or radial, acceleration. No, an object can accelerate only if there is a B net force acting on it. It direction is towards the center of the circle. Yes, although the speed is constant, the C direction of the velocity can be changing. It's magnitude is given by Yes, if an object is moving it is experiencing D acceleration.
Slide 13 / 71 Slide 14 / 71 2 Consider a particle moving with constant speed such 3 An object moves in a circular path at a constant that its acceleration of constant magnitude is always speed. Compare the direction of the object's velocity and acceleration vectors. perpendicular to its velocity. A It is moving in a straight line. A Both vectors point in the same direction. B It is moving in a circle. B The vectors point in opposite directions. C It is moving in a parabola. C The vectors are perpendicular. None of the above is definitely true all of the The question is meaningless, since the D D time. acceleration is zero. Slide 15 / 71 Slide 16 / 71 4 Two cars go around the same circular track at the same speed. The first car is closer to the inside of the circle. Which of the following is true about their centripetal acceleration? A Both cars have the same centripetal motion since Period, Frequency, and they both have the same speed. B The centripetal acceleration of the first car is Rotational Velocity greater since its radius is smaller. C The centripetal acceleration of the second car is greater since its radius is larger. D The centripetal acceleration of the first car is less since its radius is smaller. Return to Table of Contents Slide 17 / 71 Slide 18 / 71 Frequency Period The number of revolutions that an object completes in The time it takes for an object to c omplete a given amount of time is called the frequency of its one trip around a circular path is called its motion. Period. The symbol for frequency is "f" The symbol for Period is "T" Frequency are measured in units of revolutions per Periods are measured in units of time; we unit time; we will usually use 1/seconds (s -1 ). Another will usually use seconds (s). name for s -1 is Hertz (Hz). Frequency can also be measured in revolutions per minute (rpm), etc. Often we are given the time (t) it takes for an object to make a number of trips (n) around a Often we are given the time (t) it takes for an circular path. In that case, object to make a number of revolutions (n). In that case,
Slide 19 / 71 Slide 20 / 71 Period and Frequency Period and Frequency Frequency Period We can relate them mathematically: Period is the inverse of Frequency These two equations look similar. In fact, they are exactly opposite one another. Frequency is the inverse of Period Another way to say this is that they are inverses. Slide 21 / 71 Slide 22 / 71 Rotational Velocity Rotational Velocity In kinematics, we defined the speed of an object Each trip around a circle, an object travels a length equal as to the circle's circumference. The circumference of a circle is given by For an object moving in a circle, instead of using t (time), we will The time it takes to go around once is the period measure the speed with T (period), the time it takes to travel around a circle. And the object's speed is given by To find the speed, then, we need to know the distance around the circle. So the magnitude of its velocity must be: Another name for the distance around a circle is the circumference. Slide 23 / 71 Slide 24 / 71 Rotational Velocity Rotational Velocity * A velocity must have a magnitude and a direction. Since , we can also determine the velocity of an object The magnitude of an object's instantaneous velocity is in uniform circular motion by the radius and frequency of its motion. its speed. So for an object in uniform circular motion, the magnitude of its velocity is: and so If an object is in uniform circular motion, the Of course the direction of its velocity is still tangent to direction of its velocity is always changing! its circular motion. We say the velocity is tangent to its circular motion.
Slide 25 / 71 Slide 26 / 71 5 A girl whirls a toy at the end of a string around her head. 6 A ball is swung in a circle. The frequency of the ball is The string makes one complete revolution every second. doubled. By what factor does the period change? She keeps the radius constant but increases the speed so that the string makes two complete revolutions per A It remains the same. second. What happens to the centripetal acceleration? B It is cut to one half. A The centripetal acceleration remains the same. C It doubles. B The centripetal acceleration doubles. D It is cut to one fourth. C The centripetal acceleration quadruples. D The centripetal acceleration is cut to half. Slide 27 / 71 Slide 28 / 71 Dynamics of Uniform Circular Motion For an object to be in uniform circular motion, there must be a net force acting on it. We already know the acceleration, Dynamics of UCM so we can write the force: Return to Table of Contents Slide 29 / 71 Slide 30 / 71 Dynamics of Uniform Circular Motion Dynamics of Uniform Circular Motion There is no centrifugal force pointing outward; what happens is that the natural tendency of the object to We can see that the force move in a straight line must be must be inward by thinking Force on ball overcome. exerted by about a ball on a string: string This If the centripetal force vanishes, the Force on hand happens. object flies off tangent to the circle. exerted by string This does NOT happen.
Slide 31 / 71 Slide 32 / 71 Curved Paths Centrifugation This concept can be used for an object moving along any A centrifuge works by curved path , as a spinning very fast. This small segment of the means there must be a path will be very large centripetal approximately force. The object at A circular. would go in a straight line but for this force; as it is, it winds up at B. Slide 33 / 71 Slide 34 / 71 7 When an object experiences uniform circular 8 A boy whirls a toy at the end of a string around his head. motion, the direction of the net force is: The string makes one complete revolution every second. He keeps the radius constant but decreases the speed A in the same direction as the motion of the object. so that the string makes one revolution every two in the opposite direction of the motion of the seconds. What happens to the tension in the string? B object. A The tension remains the same. C is directed toward the center of the circular path. B The tension doubles. is directed away from the center of the circular D C The tension is cut to half. path. D The tension is cut to one fourth. Slide 35 / 71 Slide 36 / 71 9 (Multi-correct Directions: For each of the following, two of the suggested answers will be correct. Select the best two choices to earn credit. No partial credit will be earned if only one correct choice is selected.) An object is moving in uniform circular motion with a mass m, a speed v, and a radius r. Which of the following will quadruple the centripetal force on the Vertical UCM object? A Doubling the speed. B Cutting the speed to one half. C Cutting the radius to one half. D Cutting the radius to one fourth. Return to Table of Contents
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