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10/28/2014 Angular Position Circles Phy 114 Eyres The qualitative velocity change method for circular motion Circles • Instantaneous velocity is tangent to the displacement for any instant. – In circular motion, the system object travels in a circle and the velocity is tangent to the circle at every instant. • Even if an object is moving at constant speed around a circle, its velocity changes direction. – A change in velocity means there is acceleration! C = 2πr v= 2πr T T is time to go around once © 2014 Pearson Education, Inc. Using the velocity change method to estimate the direction of acceleration • This method is used to estimate the direction of acceleration of any object during a short time interval Δt = tf – ti. Circles ∆s ∆v = r v (∆s )v = (∆v )r r r vi vf r r Δs (∆s )v (∆v)r = ∆t ∆t 2 v = ar a= v2 r © 2014 Pearson Education, Inc. 1 10/28/2014 Radial and Tangential • Always draw a picture where you can see the center of the circle • Use radial and tangential coordinate axes. • Set positive toward the center • Remember that radial acceleration is: Testing Experiment 1: The sum of the forces exerted on an object moving at constant speed along a circular path points toward the center of that circle in the same direction as the object's acceleration = © 2014 Pearson Education, Inc. Driving over a Rise Roller-Coaster A car of mass 1500 kg goes over a hill at a speed of 20 m/s. The shape of the hill is approximately circular, with a radius of 60 m, as in the figure at right. When the car is at the highest point of the hill, • A roller-coaster vehicle has a mass of 500 kg when fully loaded with passengers. If the vehicle has a speed of 20.0 m/s at point A, what is the force of the track on the vehicle at this point? a. What is the force of gravity on the car? b. What is the normal force of the road on the car at this point? Slide 6-26 Example Example 4.5: Toy airplane • A toy airplane flies around in a horizontal circle at constant speed. The airplane is attached to the end of a 46-cm string, which makes a 25° angle relative to the horizontal while the airplane is flying. A scale at the top of the string measures the force that the string exerts on the airplane. • Predict the period of the airplane's motion (the time interval for it to complete one circle). In the hammer throw, an athlete spins a heavy mass in a circle at the end of a chain, then lets go of the chain. For male athletes, the “hammer” is a mass of 7.3 kg at the end of a 1.2 m chain. A world-class thrower can get the hammer up to a speed of 29 m/s. If an athlete swings the mass in a horizontal circle centered on the handle he uses to hold the chain, what is the tension in the chain? © 2014 Pearson Education, Inc. Slide 6-25 2 10/28/2014 Example 4.7: Texas Motor Speedway Newton's law of universal gravitation • Texas Motor Speedway is a 2.4-km (1.5-mile)-long oval track. One of its turns is about 200 m in radius and is banked at 24° above the horizontal. • How fast would a car have to move so that no friction is needed to prevent it from sliding sideways off the raceway (into the infield or off the track)? © 2014 Pearson Education, Inc. Example 4.8: Geostationary satellite • You are in charge of launching a geostationary satellite into orbit. • At which altitude above the equator must the satellite orbit be to provide continuous communication to a stationary dish antenna on Earth? • The mass of Earth is 5.98 x 1024 kg. © 2014 Pearson Education, Inc. Answers • Drive over a rise: n=4700 N • Roller Coaster: n=24900 N. • Hammer throw: T=5116 N © 2014 Pearson Education, Inc. 3