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Transcript
Uniform Circular Motion Part I http://en.wikipedia.org/wiki/G-force http://www.youtube.com/watch?v=EXQKl-28zn0 Uniform Circular Motion • Uniform circular motion: motion in a circle of constant radius at constant speed • Instantaneous velocity is always tangent to circle (perpendicular to the radius) Note that the magnitude of v is constant but not the direction v1 v2 v f is the frequency v1 v2 T is the period 2r v r 2fr T is angular velocity v1 v1 v v v3 v2 v2 v v2 v1 v2 (v1 ) v v2 v1 Looking at the change in velocity in the limit that the time interval becomes infinitesimally small, we see that v1 s ac ac | v | s v r vs | v | r | v | vs t r.t v2 ac r v2 v v1 v2 Centripetal Acceleration • an object that moves on a circular path of radius r with constant speed v has an acceleration a. • The direction of the acceleration vector always points towards the center of rotation C (thus the name centripetal) Its magnitude is constant Centripetal Force • If we apply Newton’s law to analyze uniform circular motion we conclude that the net force in the direction that points towards C must have magnitude: mv 2 Fc r • This force is known as “centripetal force” • Centripetal force is not a new kind of force. It is simply the net force that points from the rotating body to the rotation center C. Depending on the situation the centripetal force can be friction, the normal force or gravity or combination of forces. What about Centrifugal Force? • 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. Problem Solving Strategies • Draw the force diagram for the object • Choose one of the coordinate axes (the yaxis in this diagram) to point towards the orbit center C along the radius. • Determine Fnet • Set Fnet mv 2 r Example • A hockey puck moves around a circle at constant speed v on a horizontal ice surface. The puck is tied to a string looped around a peg at point C. In this case the net force along the y-axis is the tension T of the string. Tension T is the centripetal force. Example Example • The Rotor is a large hollow cylinder of radius R that is rotated rapidly around its central axis with a speed v. A rider of mass m stands on the Rotor floor with his/her back against the Rotor wall. Cylinder and rider begin to turn. When the speed v reaches some predetermined value, the Rotor floor abruptly falls away. The rider does not fall but instead remains pinned against the Rotor wall. The coefficient of static friction μs between the Rotor wall and the rider is given. We draw a free body diagram for the rider using the axes shown in the figure. The normal reaction FN is the centripetal force. Fx ,net Fy ,net mv 2 FN ma = (eqs.1) , R f s mg 0 , f s s FN mg s FN (eqs.2) mv 2 Rg If we combine eqs.1 and eqs.2 we get: mg s v2 vmin R s Rg s Example • In a 1901 circus performance Allo Diavolo introduced the stunt of riding a bicycle in a looping-the-loop. The loop sis a circle of radius R. What is the minimum speed v that Diavolo should have at the top of the loop and not fall We draw a free body diagram for Diavolo when he is at the top of the loop. Two forces are acting: Gravitational force Fg and the normal reaction FN from the loop. When Diavolo has the minimum speed v he has just lost contact with the loop and thus FN = 0. The only force acting on Diavolo is Fg The gravitational force Fg is the centripetal force. The skid marks show straight lines. Why? Banked Roads • 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: 1. The kinetic frictional force is smaller than the static. 2. 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. Nonuniform Circular Motion If an object is moving in a circular path but at varying speeds, it must have a tangential component to its acceleration as well as the radial one. Nonuniform Circular Motion This concept can be used for an object moving along any curved path, as a small segment of the path will be approximately circular. A child on a sled comes flying over the crest of a small hill. His sled does not leave the ground, but he feels the normal force between his chest and the sled decreases as he goes over the hill. Explain? A 0.150 kg ball on the end of a 1.10 m long cord is swung in a vertical circle. (a) determine the minimum speed the ball must have at the top of its arc so that the ball continues moving in a circle (b) Calculate the tension in the cord at the bottom of the arc, assuming the ball is moving twice the speed of part(a) What is the apparent weight at the top and the bottom of the track? What must be the rotational speed of a space station in zero gravity so that the occupants experience an artificial g? • https://en.wikipedia.org/wiki/Artificial_gravity