6 - Circular Motion and Other Applications of Newton's Laws

On the pilot act downward and so the net downward

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Unformatted text preview: r as it moves in a vertical circle at constant speed. (b) Free-body diagram for the pilot at the bottom of the loop. In this position the pilot experiences an apparent weight greater than his true weight. (c) Free-body diagram for the pilot at the top of the loop. n bot Top A ntop mg mg (b) (c) Bottom (a) a magnitude n top Fr ntop ntop m ntop mg mg v2 r mg (2.70 mg. Applying Newton’s second law yields m v2 r mg v2 rg In this case, the magnitude of the force exerted by the seat on the pilot is less than his true weight by a factor of 0.913, and the pilot feels lighter. Exercise Determine the magnitude of the radially directed force exerted on the pilot by the seat when the aircraft is at point A in Figure 6.8a, midway up the loop. 1 (225 m/s)2 103 m)(9.80 m/s2) 1 0.913mg Answer nA 1.913mg directed to the right. Quick Quiz 6.3 A bead slides freely along a curved wire at constant speed, as shown in the overhead view of Figure 6.9. At each of the points , , and , draw the vector representing the force that the wire exerts on the bead in order to cause it to follow the path of the wire at that point. QuickLab Hold a shoe by the end of its lace and spin it in a vertical circle. Can you feel the difference in the tension in the lace when the shoe is at top of the circle compared with when the shoe is at the bottom? Figure 6.9 6.2 NONUNIFORM CIRCULAR MOTION In Chapter 4 we found that if a particle moves with varying speed in a circular path, there is, in addition to the centripetal (radial) component of acceleration, a tangential component having magnitude dv/dt. Therefore, the force acting on the 6.2 Nonuniform Circular Motion 159 Some examples of forces acting during circular motion. (Left) As these speed skaters round a curve, the force exerted by the ice on their skates provides the centripetal acceleration. (Right) Passengers on a “corkscrew” roller coaster. What are the origins of the forces in this example? Figure 6.10 When the force acting on a particle moving in a circular path has a tangential component Ft , the particle’s speed changes. The total force exerted on the particle in this case is the vector sum of the radial force and the tangential force. That is, F Fr Ft . F Fr Ft particle must also have a tangential and a radial component. Because the total acceleration is a ar at , the total force exerted on the particle is F Fr Ft , as shown in Figure 6.10. The vector Fr is directed toward the center of the circle and is responsible for the centripetal acceleration. The vector Ft tangent to the circle is responsible for the tangential acceleration, which represents a change in the speed of the particle with time. The following example demonstrates this type of motion. EXAMPLE 6.8 Keep Your Eye on the Ball A small sphere of mass m is attached to the end of a cord of length R and whirls in a vertical circle about a fixed point O, as illustrated in Figure 6.11a. Determine the tension in the cord at any instant when the speed of the sphere...
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