Ch5 Uniform Circular Motion Uniform circular motion is the motion of an object traveling at a constant uniform speed on a circular path r Period T is the time required to travel once around the circle that is to make one complete revolution 2 r v T 1 Example 1 A Tire Balancing Machine The wheel of a car has a radius of r 0 29m and is being rotated at 830 revolutions per minute rpm on a tire balancing machine Determine the speed in m s at which the outer edge of the wheel is moving 2 r The speed v can be obtained directly from v T but first the period T is needed It must be expressed in seconds 2 830 revolutions in one minute 1 1 2 10 3 min revolution 830 revolutions min T 1 2 10 3 min which corresponds to 0 072s 2 r 2 0 29m v 25m s T 0 072s 3 Uniform circular motion emphasizes that 1 The speed or the magnitude of the velocity vector is constant 2 Direction of the vector is not constant 3 Change in direction means acceleration 4 Centripetal acceleration it points toward the center of the circle 4 Centripetal Acceleration Magnitude ac of the centripetal acceleration depends on the speed v of the object and the radius r of the circular path ac v2 r 5 v in velocity divided by the elapsed time t or a v t Sector of the circle COP t is very small the arc length OP is approximately a straight line whose length is the distance v t traveled by the object 6 COP is an isosceles triangle Both triangles have equal apex angles v v t v r v t ac v2 r The direction is toward the center of the circle 7 Conceptual Example 2 Which way will the object go An object on a guideline is in uniform circular motion The object is symbolized by a dot and at point O it is release suddenly from its circular path If the guideline is cut suddenly will the object move along OA or OP 8 Newton s first law of motion guides our reasoning An object continues in a state of rest or in a state of motion at a constant speed along a straight line unless compelled to changes that state by a net force When the object is suddenly released from its circular path there is no longer a net force being applied to the object In the case of a model airplane the guideline cannot apply a force since it is cut Gravity certainly acts on the plane but the wings provide a lift force that balances the weight of the plane 9 In the absence of a net force then the plane or any object would continue to move at a constant speed along a straight line in the direction it had at the time of release This speed and direction are given in Figure 5 4 by the velocity vector v As a result the object would move along the straight line between points O and A not on the circular arc between points O and P 10 Example 3 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 11 From ac v2 r it follows that Radius 33m 2 34m s ac 35m s 2 3 6 g 33m Radius 24m 2 34m s 2 ac 48m s 4 9 g 24m 12 Conceptual Example 4 Uniform Circular Motion and Equilibrium A car moves at a constant speed and there are three parts to the motion It moves along a straight line toward a circular turn goes around the turn and then moves away along a straight line In each of three parts is the car in equilibrium 13 An object in equilibrium has no acceleration according to the definition given in Section 4 11 As the car approaches the turn both the speed and direction of the motion are constant Thus the velocity vector does not change and there is no acceleration The same is true as the car moves away from the turn For these parts of the motion then the car is in equilibrium As the car goes around the turn however the direction of travel changes so the car has a centripetal acceleration that is characteristic of uniform circular motion Because of this acceleration the car is not in equilibrium during the turn In general an object that is in uniform circular motion can never be in equilibrium 14 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 15 a The velocity is due south and the acceleration is due west b The velocity is due west and the acceleration is due north 16 Centripetal Force 17 Concepts at a glance Newton s second law indicates that whenever an object accelerates there must be a net force to create the acceleration Thus in uniform circular motion there must be a net force to produce the centripetal acceleration As the Concept at aglance chart the second law gives this net force as the product of the object s mass m and its acceleration v2 r This chart is an expanded version of the chart shown previously in Figure 4 9 The net force causing the centripetal acceleration is called the centripetal force FC and points in the same direction as the accelerationthat is toward the center of the circle 18 mv FC r 2 centripetal force does not denote a new and separate force created by nature 19 Example 5 The Effect of Speed on Centripetal Force The model airplane has a mass of 0 90 kg and moves at a constant speed on a circle that is parallel to the ground The path of the airplane and its 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 T in the guideline length 17m for speeds of 19 and 38m s 20 Equation 5 3 gives the tension directly FC T mv2 r Speed 19m s 0 90kg 19m s 2 T 19 N 17m Speed 38m s 0 90kg 38m s 2 T 76 N 17m 21 Conceptual Example 6 A Trapeze Act In a circus a man hangs upside down from a trapeze legs bent over the bar 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 or when the partner is swinging through the straight down position 22 Reasoning and Solution When the man and his …