5circular motion, Lecture notes of Physics

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Chapter 5

Dynamics of Uniform

Circular Motion

5.1 Uniform Circular Motion DEFINITION OF UNIFORM CIRCULAR MOTION Uniform circular motion is the motion of an object traveling at a constant speed on a circular path.

5.1 Uniform Circular Motion Example 1: A Tire-Balancing Machine The wheel of a car has a radius of 0.29m and it being rotated at 830 revolutions per minute on a tire-balancing machine. Determine the speed at which the outer edge of the wheel is moving.

1. 2 10 min revolution 830 revolutions min (^1)  3  
2. 2 10 min 0. 072 s 3     T

25 m s

1. 072 s 2 2 0.29 m      T r v

5.2 Centripetal Acceleration In uniform circular motion, the speed is constant, but the direction of the velocity vector is not constant. 

     90    5

5.2 Centripetal Acceleration The direction of the centripetal acceleration is towards the center of the circle; in the same direction as the change in velocity. r v a c 2 

5.2 Centripetal Acceleration Conceptual Example 2: Which Way Will the Object Go? An object is in uniform circular motion. At point O it is released from its circular path. Does the object move along the straight path between O and A or along the circular arc between points O and P?

5.2 Centripetal Acceleration

a v r

c 2

a g

c

35 m s 3. 6

33 m

34 m s 2

2

a g

c

48 m s 4. 9

24 m

34 m s 2

2

5.3 Centripetal Force

F  a

m

m

F

a

Recall Newton’s Second Law When a net external force acts on an object of mass m , the acceleration that results is directly proportional to the net force and has a magnitude that is inversely proportional to the mass. The direction of the acceleration is the same as the direction of the net force.

5.3 Centripetal Force Example 5: The Effect of Speed on Centripetal Force The model airplane has a mass of 0.90 kg and moves at constant speed on a circle that is parallel to the ground. The path of the airplane and the guideline lie in the same horizontal plane because the weight of the plane is balanced by the lift generated by its wings. Find the tension in the 17 m guideline for a speed of 19 m/s. r v F T m c 2       19 N 17 m 19 m s

1. 90 kg 2 T  

5.3 Centripetal Force Conceptual Example 6: A Trapeze Act In a circus, a man hangs upside down from a trapeze, legs bent over and arms downward, holding his partner. Is it harder for the man to hold his partner when the partner hangs straight down and is stationary of when the partner is swinging through the straight-down position?

5.4 Banked Curves On a frictionless banked curve, the centripetal force is the horizontal component of the normal force. The vertical component of the normal force balances the car’s weight.

5.4 Banked Curves r v F F m c N 2  sin   FN cos^   mg

5.4 Banked Curves Example 8: The Daytona 500 The turns at the Daytona International Speedway have a maximum radius of 316 m and are steely banked at 31 degrees. Suppose these turns were frictionless. As what speed would the cars have to travel around them? rg v 2

tan   v  rg tan 

 316 m  9. 8 m s  tan 31 43 m s  96 mph 2    v

5.5 Satellites in Circular Orbits There is only one speed that a satellite can have if the satellite is to remain in an orbit with a fixed radius.