Objectives: The student will be able to: identify uniform circular motion. determine the directions of the velocity and acceleration vectors for an object in uniform circular motion. calculate the centripetal acceleration of a point mass in uniform circular motion given the radius of the circle and either the linear speed or the period of the motion. state the relationship between the period of the motion and the frequency of rotation and express this relationship using a mathematical equation.

Uniform Circular Motion: Definition Constant speed, or, constant magnitude of velocity Motion along a circle: Changing direction of velocity April 16, 2017

Uniform Circular Motion Uniform circular motion is motion along a circular path in which there is no change in speed, only a change in direction. Constant velocity tangent to path. v Fc Constant force toward center. Question: Is there an outward force on the ball?

Uniform Circular Motion (Cont.) The question of an outward force can be resolved by asking what happens when the string breaks! Ball moves tangent to path, NOT outward as might be expected. v When central force is removed, ball continues in straight line. Centripetal force is needed to change direction.

Kinematics of Uniform Circular Motion Uniform circular motion: motion in a circle of constant radius at constant speed Instantaneous velocity is always tangent to circle.

Kinematics of Uniform Circular Motion This acceleration is called the centripetal, or radial, acceleration, and it points towards the center of the circle.

Examples of Centripetal Force You are sitting on the seat next to the outside door. What is the direction of the resultant force on you as you turn? Is it away from center or toward center of the turn? Fc Car going around a curve. Force ON you is toward the center.

Car Example Continued F’ Fc The centripetal force is exerted BY the door ON you. (Centrally) F’ Reaction Fc There is an outward force, but it does not act ON you. It is the reaction force exerted BY you ON the door. It affects only the door.

Another Example

Spin Cycle on a Washer How is the water removed from clothes during the spin cycle of a washer? Think carefully before answering . . . Does the centripetal force throw water off the clothes? NO. Actually, it is the LACK of a force that allows the water to leave the clothes through holes in the circular wall of the rotating washer.

Example 1: A 3-kg rock swings in a circle of radius 5 m Example 1: A 3-kg rock swings in a circle of radius 5 m. If its constant speed is 8 m/s, what is the centripetal acceleration? r v m m = 3 kg r = 5 m; v = 8 m/s F = (3 kg)(12.8 m/s2) Fc = 38.4 N

Example: The Effect of Radius on Centripetal Acceleration The bobsled track at the 1994 Olympics in Lillehammer, Norway, contained turns with radii of 33 m and 24 m, as the figure illustrates. Find the centripetal acceleration at each turn for a speed of 34 m/s, a speed that was achieved in the two-man event. Express the answers as multiples of g=9.8m/s2.

From ac=v2/r it follows that Radius=33m Radius=24m

Check your understanding 1 The car in the drawing is moving clockwise around a circular section of road at a constant speed. What are the directions of its velocity and acceleration at (a) position 1 and (b) position 2?

(a) The velocity is due south, and the acceleration is due west. (b) The velocity is due west, and the acceleration is due north.

Uniform Circular Motion Transparency

Period and Frequency A circular motion is described in terms of the period T, which is the time for an object to complete one revolution. v= 2pr/T = 2prf The distance traveled in one revolution is r The frequency, f, counts the number of revolutions per unit time.

Tangential (Linear) Velocity The tangential velocity vector is tangent to the circle at the point of study. v v

Problem 1 A biker travels once around a circular track of radius 20.0m in 3s. Calculate: the average tangential speed the frequency the period Answers: 41.9m/s, f=0.33Hz, T=3s 2.09rad/sec 41.89m/s f=0.33Hz T=3s

Record Player

Problem 2 A coin sits 0.10m from the center of a record player spinning at 45rpm. What is the frequency in Hertz? What is the period? What is the linear speed? Answer: 0.75Hz, 1.33s, 0.47m/s

Uniform Circular Motion: Observations Object moving along a curved path with constant speed Magnitude of velocity: same Direction of velocity: changing Velocity : changing Acceleration is NOT zero! Net force acting on an object is NOT zero “Centripetal force” April 16, 2017

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, so can immediately write the force:

Uniform Circular Motion Newton’s 2nd Law: The net force on a body is equal to the product of the mass of the body and the acceleration of the body. *The centripetal acceleration is caused by a centripetal force that is directed towards the center of the circle.

Problem 3 A child on a merry-go-round sits 1.5m from the center. They spin 3 times in one min. The mass of the child is 40kg. Find the friction(centripetal force) acting on the child. Answer: 5.9N

Dynamics of Uniform Circular Motion We can see that the force must be inward by thinking about a ball on a string: Demo – penny and hanger

Dynamics of Uniform Circular Motion There is no centrifugal force pointing outward; what happens is that the natural tendency of the object to move in a straight line must be overcome. If the centripetal force vanishes, the object flies off tangent to the circle. Period 3 start here.

Circular Motion Notes “Centrifugal Force” “centrifugal force” is a fictitious force - it is not an interaction between 2 objects, and therefore not a real force. Nothing pulls an object away from the center of the circle.

Circular Motion Notes “Centrifugal Force” What is erroneously attributed to “centrifugal force” is actually the action of the object’s inertia - whatever velocity it has (speed + direction) it wants to keep.

What provides Centripetal Force ? Centripetal force is not a new kind of force Centripetal force refers to any force that keeps an object following a circular path Centripetal force is a combination of Gravitational force mg: downward to the ground Normal force N: perpendicular to the surface Tension force T: along the cord and away from object Static friction force: fsmax = µsN April 16, 2017

What provides Centripetal Force ? mg April 16, 2017

Circular Motion Lab Do not use clamp.

Highway Curves, Banked and Unbanked When a car goes around a curve, there must be a net force towards the center of the circle of which the curve is an arc. If the road is flat, that force is supplied by friction.

Highway Curves, Banked and Unbanked If the frictional force is insufficient, the car will tend to move more nearly in a straight line, as the skid marks show.

Highway Curves, Banked and Unbanked As long as the tires do not slip, the friction is static. If the tires do start to slip, the friction is kinetic, which is bad in two ways: The kinetic frictional force is smaller than the static. The static frictional force can point towards the center of the circle, but the kinetic frictional force opposes the direction of motion, making it very difficult to regain control of the car and continue around the curve.

Car Negotiating a Flat Turn v Fc What is the direction of the force ON the car? Ans. Toward Center This central force is exerted BY the road ON the car.

Car Negotiating a Flat Turn v Fc Is there also an outward force acting ON the car? Ans. No, but the car does exert a outward reaction force ON the road.

Car Negotiating a Flat Turn The centripetal force Fc is that of static friction fs: R v m Fc n Fc = fs fs R mg The central force FC and the friction force fs are not two different forces that are equal. There is just one force on the car. The nature of this central force is static friction.

Finding the maximum speed for negotiating a turn without slipping. Fc = fs n mg fs R R v m Fc The car is on the verge of slipping when FC is equal to the maximum force of static friction fs. Fc = mv2 R fs = msmg Fc = fs

Optimum Banking Angle n n n m By banking a curve at the optimum angle, the normal force n can provide the necessary centripetal force without the need for a friction force. R v m Fc n fs = 0 n fs q slow speed q n fs q w w w fast speed optimum

Highway Curves, Banked and Unbanked Banking the curve can help keep cars from skidding. In fact, for every banked curve, there is one speed where the entire centripetal force is supplied by the horizontal component of the normal force, and no friction is required. This occurs when:

Class work / Homework Hand-outs Page 156 12-21

Closure Kahoot Circular Motion