Machinery's Handbook, 31st Edition
Energy 189 where M = mass in kilograms, and V = velocity in meters per second. Kinetic energy due to rotation is expressed by the formula E KR = 1 ∕ 2 J MO ω 2 , where J MO = moment of inertia in kg·m 2 , and ω = the angular velocity in radians per second. Total kinetic energy ET = 1 ∕ 2 MV 2 + 1 ∕ 2 J MO ω 2 joules = 1 ∕ 2 M ( V 2 + k 2 ω 2 ) joules, where k = radius of gyration in meters. Potential Energy: A common example of a body having potential energy because of its position in a field of force is that of a body elevated to some height above the earth. The field of force is the gravitational field of the earth. The potential energy of a body weighing W pounds elevated to height S feet above the earth’s surface is E PF = WS lb. If the body is permitted to drop from this height its potential energy will be converted to kinetic energy. Thus, after falling through height S the kinetic energy of the body will be WS ft-lbs. In metric SI units, the potential energy E PF of a body of mass M kilograms elevated to a height of S meters, is MgS joules. After it has fallen a distance S , the kinetic en ergy gained will thus be MgS joules. Another type of potential energy is elastic potential energy , such as possessed by a spring that has been compressed or extended. The amount of work in ft-lbs done in com- pressing the spring S feet is equal to KS 2 ⁄2, where K is the spring constant in pounds per foot. Thus, when the spring is released to act against some resistance, it can perform KS 2 ⁄2 ft-lbs of work, which is the amount of elastic potential energy E PE stored in the spring. Using metric SI units, the amount of work done in compressing the spring a dis tance S meters is KS 2 ⁄2 joules, where K is the spring constant in newtons per meter. Work Performed by Forces and Couples.— The work U done by a force F in moving an object along some path is the product of the distance S the body is moved and the compo nent F cos α of the force F in the direction of S . cos U FS = α where U = work in ft-lbs; S = distance moved in feet; F = force in lbs; and α = angle between line of action of force and the path of S . If the force is in the same direction as the motion, then cos α = cos 0 = 1, and this formula reduces to: U FS = Similarly, the work done by a couple T turning an object through an angle θ is: U T i = where T = torque of couple in pounds-feet and θ = the angular rotation in radians. The above formulas can be used with metric SI units: U in joules; S in meters; F in newtons, and T in newton-meters. Relation Between Work and Energy.— Theoretically, when work is performed on a body and there are no energy losses (such as those due to friction, air resistance, etc.), the energy acquired by the body is equal to the work performed on the body; this energy may be either potential, kinetic, or a combination of both. In actual situations, however, there may be energy losses that must be taken into account. Thus, the relation between work done on a body, energy losses, and the energy acquired by the body can be stated as: Work Performed Energy Losses – = Energy Acquired U –Losses E T =
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