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Work and Kinetic EnergyFinding an Expression for Work Let us find an expression for work by considering a bead that can slide along a frictionless wire, which is stretched along a horizontal where
Solving this equation for The first term on the left side of the equation is the kinetic energy
If we know values for To calculate the work done on an object by a force during a displacement, we use only the force component along the object's displacement. The force component perpendicular to the displacement does zero work. From Fig. 7-2, we see that we can write
This equation is useful for calculating the work if we know values for
(You may wish to review the discussion of scalar products in Section 3-7.) Equation 7-8 is especially useful for calculating the work when Cautions: There are two restrictions to using Eqs. 7-6 through 7-8 to calculate work done on an object by a force. First, the force must be a constant force; that is, it must not change in magnitude or direction as the object moves. (Later, we shall discuss what to do with a variable force that changes in magnitude.) Second, the object must be particle-like. This means that the object must be rigid; all parts of the object must move together, in the same direction. In this chapter we consider only particle-like objects, such as the bed and its rider being pushed in Fig. 7-3.
Fig. 7-5 A contestant in a bed race. We can approximate the bed and its rider as being a particle for the purpose of calculating the work done on them by the force applied by the student. Signs for work. The work done on an object by a force can be either positive work or negative work. For example, if the angle A force does positive work when it has a vector component in the same direction as the displacement, and it does negative work when it has a vector component in the opposite direction. It does zero work when it has no such vector component. Units for work. Work has the SI unit of the joule, the same as kinetic energy. However, from Eqs. 7-6 and 7-7 we can see that an equivalent unit is the newton-meter (N • m). The corresponding unit in the British system is the foot-pound (ft • lb). Extending Eq. 7-2, we have 1 J = kg • m2/s2 = 1 N • m = 0.738 ft • lb.(7-9) Net work done by several forces. When two or more forces act on an object, the net work done on the object is the sum of the works done by the individual forces. We can calculate the net work in two ways. (1) We can find the work done by each force and then sum those works. (2) Alternatively, we can first find the net force Work-Kinetic Energy Theorem Equation 7-5 relates the change in kinetic energy of the bead (from an initial
which says that
which says that These statements are known traditionally as the work-kinetic energy theorem for particles. They hold for both positive and negative work: If the net work done on a particle is positive, then the particle's kinetic energy increases by the amount of the work. If the net work done is negative, then the particle's kinetic energy decreases by the amount of the work. For example, if the kinetic energy is initially 5 J and there is a net transfer of 2 J to the particle (positive net work), then the final kinetic energy is 7 J. If, instead, there is a net transfer of 2 J from the particle (negative net work), then the final kinetic energy is 3 J.
Fig. 7-6 A particle-like tomato of mass m thrown upward slows from velocity 70 to velocity ~v during displacement d because the gravitational force Fg acts on it. A kinetic energy gauge indicates the resulting change in the kinetic energy of the object, from ËÃ, (= {mv§) to Kf{= \mv2).
Date: 2015-01-12; view: 1669
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