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Confirming Pages 1 2-1 Section 2.10: Apparent Weight Jump off a diving board into a pool and you feel “weightless” until you hit the water. In reality, a gravitational force is acting on you during your fall. What is missing is the upward reaction force provided by whatever supports your weight at other times—the seat of a chair, the floor of a room, the diving board, the water. It is useful to distinguish between the actual weight of an object, which is the gravitational force that acts on it, and its apparent weight, which is the force the object exerts on whatever it rests on. You can think of your apparent weight as the reading of a bathroom scale you are standing on. If you are falling freely, the scale is a=0 dropping just as fast as you are, and your apparent weight is then zero. Suppose you are standing on a scale in an elevator, as in Figure 1. When the elevator’s upward acceleration is a, the upward force FY on you is the sum of your actual weight mg and the force that accelerates you upward, which is ma. Hence Fy 5 mg 1 ma The reaction force to Fy has the same magnitude and is the force you exert on the scale, which is your apparent weight wapp. Hence wapp 5 mg 1 ma Apparent Apparent Actual weight weight weight a 700 N At rest or constant velocity a 800 N Upward acceleration If the acceleration a is positive, corresponding to an upward acceleration, then wapp > mg, and the scale reading will be greater than your actual weight. If a is negative, corresponding to a downward acceleration, wapp < mg, and the scale reading will be less than your actual weight. If the cable that supports the elevator breaks and the elevator falls freely, a 5 2g and wapp 5 mg 2 mg 5 0. When the elevator is at rest or moving at constant speed up or down, a 5 0 and wapp 5 mg. g 600 N Downward acceleration 0 Free fall FIGURE 1 The actual weight of this person is 700 N. When the elevator is accelerated upward, her apparent weight is greater, and when the elevator is accelerated downward, her apparent weight is smaller. In free fall, her apparent weight is zero. kra1392X_ch02_online.indd A-1 17/12/12 6:29 PM Confirming Pages 2 2-2 Section 2.10 Example You are standing on a scale in an elevator. When the elevator is at rest on the fourteenth floor, the scale reads 700 N. When the elevator starts to move, the scale reads 600 N. Describe the motion of the elevator. Solution Here your actual weight is 700 N and your apparent weight is 600 N. Because wapp and mg are different, the elevator must be accelerating, and because wapp < mg, the acceleration must be negative, that is, downward. Your mass is mg ________ 700 N m 5 ___ g 5 9.80 m/s2 5 71.4 kg Solving the apparent-weight equation for the elevator’s acceleration gives wapp 600 N 2 2 2 ______ a 5 ____ m 2 g 5 71.4 kg 2 9.80 m/s 5 (8.40 2 9.80) m/s 5 2 1.40 m/s Section 2.12: Example A road has a hump 12 m in radius. What is the minimum speed at which a car will leave the road at the top of the hump? Solution The car will leave the hump when the required centripetal force mv2/r is more than the car’s weight of mg, since it is this weight that is providing the centripetal force. Hence 2 mv , mg 5 ____ r 2 v, g 5 __ r __ v 5 √ rg We note that the mass of the car does not matter here. (We will find the same formula later as the orbital speed a satellite of the earth must have.) Substituting for r and g gives ______________ v 5 √ (12 m)(9.8 m/s2) 5 10.8 m/s This is about 24 mi/h. Driving faster over the hump is not recommended. kra1392X_ch02_online.indd A-2 17/12/12 6:29 PM