2 Anatomy of a CircleThe direction from the edge of the circle towards the center of the circle is called the radial directionSign convention is that the +r direction is toward the center of the circleThe direction perpendicular to the radial direction is called the tangential directiontangentialradial
3 Circular Velocity & Acceleration Because the direction that the object is moving in is constantly changes, velocity constantly changesAn object’s instantaneous velocity is directed in the tangential directionBecause velocity constantly changes, there is a constant accelerationThe acceleration is always directed radially inward (in the negative direction)
4 Uniform Circular Motion Uniform Circular Motion (UCM) occurs when an object follows a circular path at a constant speedThe object experiences an acceleration toward the center of the path called centripetal accelerationThe object must be kept in circular motion by at least one force, the total of which we call the centripetal forceComes from the Latin “centrum” = center and “petus” = seeking
5 Periodic MotionAny repeated cycle, such as UCM, can be described in terms of its period and its frequencyPeriod (T) is the time that it takes for one cycle to be completed, usually measured in secondsFrequency (f) is the number of cycles completed per unit time, usually measured in Hz1 Hz = 1/s or s-1
6 Converting units for fOur standard units for frequency is Hz, but another common unit is revolutions/minute or rpmExample: Convert 45 rpm to Hz
7 Centripetal Acceleration Always directed radially inwardv is the object’s speed measured in m/s and r is the radius measured in mBecause the is constantly changing, the direction of ac is always changing
8 Example 11-1To test the effects of high accelerations on the human body, NASA has constructed a large centrifuge at the Manned Spacecraft Center in Houston. In this device, astronauts are place in a capsule that moves in a circular path with a radius of 15 m. If the astronauts in this centrifuge experience 9.0 g of acceleration, what is the linear speed of the capsule?
9 Example 11-2The moon’s nearly circular orbit about the Earth has a radius of about 384,000 km and a period of 27.3 days. Determine the acceleration of the Moon toward the Earth
10 Since v is equal to the distance traveled (the Since v is equal to the distance traveled (the circumference of the circle) divided by the time that it takes to travel around the circle (the period), we haveand so
11 Two different ways to express ac ac as a function of Tac as a function of f
12 Centripetal ForceThe condition necessary for an object to move in UCM is that the net force in the radial direction must be equivalent to the centripetal force whereIf this condition is NOT met, then the motion of the object will not be a circle (it’ll become straight line motion, a parabola, etc.
13 The “Centrifugal” Force An extremely common misconception is that there is a force, the so called centrifugal (“center-fleeing”) force that acts radially outward on an object moving in a circular path – this is incorrect, there is NO outward force!!!!This perceived force is due to the object’s tendency to follow a straight-line path (inertia)Example: ball on a string… why does it pull your hand outward?
14 Example 11-2A puck attached to a string undergoes circular motion on an air table. If the string breaks at the point indicated in the picture, which path will the puck take? Why does it take this path?
15 Example 11-3 (conceptual) A car is driven with constant speed around a circular track. a) Is the car’s velocity constant? b) Is its speed constant? c) Is the magnitude of its acceleration constant? d) Is the direction of its acceleration constant?
16 Horizontal UCM Problems Draw a free body diagram (LABEL YOUR DIRECTIONS! – especially the +r direction)Common radial forces: Tension, friction, normal forceWrite an equation that satisfies the condition for UCM (ΣFr = macp)Many times you’ll have to apply some other condition (usually that f ≤ fs,max)
17 Example 11-4A 1200 kg car rounds a corner of radius r = 45 m. If the coefficient of static friction between the tires and the road is μs = 0.82, what is the greatest speed the car can have in the corner without skidding?
18 Example 11-5When you take your 1300 kg car out for a spin, you go around a corner of radius 59 m with a speed of 16 m/s. You have a 110 g fuzzy dice hanging from your rearview mirror. You notice that it hangs at an angle to the vertical. a) Assuming your car doesn’t skid, what is the force exerted on it by static friction? b) Calculate the angle that your dice make with the vertical. c) Calculate the tension in the string holding up the dice. d) You notice that your keys (which are heavier than your dice) are also hanging off to the side. Compare the angle of you keys to the angle of your dice.
19 Example 11-6You may find it surprising that the rotation of the Earth in fact makes us feel lighter. To illustrate this fact calculate what a bathroom scale would report for the weight of a 100 kg person (a) at the North Pole (where there is no rotation) and (b) at the equator. Earth’s equatorial radius is 6,378.1 km. (c) Does the weight your scale measures increase or decrease as your latitude increases? WHY?
20 Example 11-7You place a coin 15 cm from the center of a record on a turntable. You can adjust the “speed” of the turntable, measured in rpm. a) What is the largest value for the “speed” so that the coin does not slip off the record? b) Calculate the linear speed of coin when the record spins at this rate.
21 Example 11-8A puck of mass m = 1.5 kg slides in a circle of radius r = 20 cm of a frictionless table while attached to a hanging cylinder of mass M = 2.5 kg by a cord through a hole in the table. What speed does the puck need to slide to keep the cylinder at rest?
22 Example 11-9The Rotor, an amusement park ride, is essentially a large hollow cylinder that is rotated rapidly through its central axis. The rider enters through a door, leans up against a canvas-covered wall and the cylinder begins to spin. When the cylinder’s frequency reaches a predetermined value, the floor will fall away. If you are the designer of this ride, what minimum frequency could you safely have the the floor fall away so that the riders remain pinned the the wall and not fall? The coefficient of static friction between a rider’s clothes and canvas is 0.40 and the cylinder’s radius is 2.1 m.
23 Vertical UCM ProblemsDraw a free body diagram (LABEL YOUR DIRECTIONS! – especially the +r direction)Common radial forces: Tension, normal force, gravityVERY IMPORTANT!! The +r direction will change based on where the object is in its motionWrite an equation that satisfies the condition for UCM (ΣFr = macp)Many times you’ll have to solve for some min or max value (remember that T and FN cannot be negative!)
24 Example 11-10As part of a circus act, a person drives a motorcycle with a constant speed v around the inside of a vertical track of radius r = 5 m. The combined mass of the rider and motorcycle is m = 350 kg. a) Calculate the minimum value for v so that the motorcycle’s wheels maintain contact with the track. b) Calculate the normal force exerted on the motorcycle at points A, B, and C.
25 Example 11-11As you ride on a Ferris wheel, you notice that your apparent weight is different at the top than at the bottom. Consider a rider of mass 55 kg riding a Ferris wheel of radius 7.2 m that completes one revolution every 28 s. a) What is the rider’s apparent weight at the top of the Ferris wheel? b) What is the rider’s apparent weight at the bottom of the Ferris wheel? c) What is the minimum period such that a rider wouldn’t lose contact with his/her seat?