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Sketch the situation Show and label the distance that the spring has stretched and its equilibrium position Known:
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x F 18 cm 56 N
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The minus sign can be dropped because it just means that the force is restoring Substitute F 56 N, x 018 m
56 N
310 N/m b PEsp
1 2 kx 2 1 (310 N/m)(018 m)2 2
Math Handbook
Substitute k 310 N/m, x 018 m Operations with Significant Digits pages 835 836
50 J
Evaluate the Answer
Are the units correct N/m are the correct units for the spring constant (N/m)(m2) N m J, which is the correct unit for energy Is the magnitude realistic The spring constant is consistent with a scale used, for example, to weigh groceries The energy of 50 J is equal to the value obtained from W Fx mgh, when the average force of 28 N is applied
Section 141 Periodic Motion
1 How much force is necessary to stretch a spring 025 m when the spring constant is 95 N/m 2 A spring has a spring constant of 56 N/m How far will it stretch when a block weighing 18 N is hung from its end 3 What is the spring constant of a spring that stretches 12 cm when an object weighing 24 N is hung from it 4 A spring with a spring constant of 144 N/m is compressed by a distance of 165 cm How much elastic potential energy is stored in the spring 5 A spring has a spring constant of 256 N/m How far must it be stretched to give it an elastic potential energy of 48 J
When the external force holding the object is released, as in Figure 14-3c, the net force and the acceleration are at their maximum, and the velocity is zero As the object passes through the equilibrium point, Figure 14-3d, the net force is zero, and so is the acceleration Does the object stop No, it would take a net downward force to slow the object, and that will not exist until the object rises above the equilibrium position When the object comes to the highest position in its oscillation, the net force and the acceleration are again at their maximum, and the velocity is zero The object moves down through the equilibrium position to its starting point and continues to move in this vibratory manner The period of oscillation, T, depends upon the mass of the object and the strength of the spring Automobiles Elastic potential energy is an important part of the design and building of today s automobiles Every year, new models of cars are tested to see how well they withstand damage when they crash into barricades at low speeds A car s ability to retain its integrity depends upon how much of the kinetic energy it had before the crash can be converted into the elastic potential energy of the frame after the crash Many bumpers are modified springs that store energy as a car hits a barrier in a slow-speed collision After the car stops and the spring is compressed, the spring returns to its equilibrium position, and the car recoils from the barrier
Figure 14-4 Fnet , the vector sum of Ft and Fg, is the restoring force for the pendulum
Pendulums
Simple harmonic motion also can be demonstrated by the swing of a pendulum A simple pendulum consists of a massive object, called the bob, suspended by a string or light rod of length l After the bob is pulled to one side and released, it swings back and forth, as shown in Figure 14-4 The string or rod exerts a tension force, FT, and gravity exerts a force, Fg, on the bob The vector sum of the two forces produces the net force, shown at three positions in Figure 14-4 At the left and right positions shown in Figure 14-4, the net force and acceleration are maximum, and the velocity is zero At the middle position in Figure 14-4, the net force and acceleration are zero, and the velocity is maximum You can see that the net force is a restoring force; that is, it is opposite the direction of the displacement of the bob and is trying to restore the bob to its equilibrium position
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