![]()
![]()
Coaster Dynamics
Physics Primer
Chapter 10
Forces and Work
Newton's Second Law of Motion is the underlying principle in almost every aspect of the study of motion dynamics. In certain cases with a constant force, such as free fall, F = ma can be applied directly to yield useful conclusions. In other cases, the differential form of Newton's Second Law, F = dp/dt, can be employed (usually with the help of a computer) to analyze cases with variable forces. However, in many other situations, the concepts of work and energy -- stemming from Newton's Second Law -- are extremely powerful tools which allow easy analysis of a large number of problems. In this chapter, we first introduce the concept of work.
Work
In the specialized world of physics, the term work has a precise meaning. Work is equal to the energy required to move an object. Specifically, work is defined to be the product of an object's displacement and the force producing it:
(10-1) W = Fs
where W is the work, s is the magnitude of the displacement, and F is the magnitude of the force in the direction of the displacement.
Note that work is a scalar quantity... equal to the product of the magnitude of two vectors.
Also notice that, by definition, no work is done if the displacement is zero. Thus, unlike other references to work within the common vernacular, if you try to push a large stone and it doesn't move, no matter how hard you may push, or how tired your muscles may get, you have done no work.
Work only occurs in the direction of the displacement. So, to be more precise:
(10-2) W = ( Fcos(ß) )(s)
where ß is the angle between the applied force and the direction of the displacement, as illustrated in Figure 10-A.
Figure 10-A. Work for a Constant Force.
In the situation illustrated in Figure 10-B,
Figure 10-B. Work for an Inclined Constant Force.
by application of vector resolution, the total work is equal to the sum of the individual contributions in the component directions:
(10-3) W = (Fx)(sx) + (Fy)(sy) = ( Fcos(ß) + Fsin(ß) )(s).
Work for a Non-Constant Force
For situations when the applied force is not constant, the calculation of work is a little more complex.
Consider the case of a force that is always in the x-direction, but varies as shown in Figure 10-C.
Figure 10-C. Work for a Non-Constant Force.
To calculate the total work, imagine dividing the graph of F(x) versus x into small increments of
x, as illustrated in the figure. For each increment, if we approximate the force as a constant, Fx, the incremental work is equal to:
(10-4)
W = (Fx)(
x).
The net work in the x-direction would now equal
(10-5) Wx = Si=1,n{ (Fx,i)(
xi) },
where the notation, Si=1,n{ -- }, means the summation of the enclosed quantity for all increments from 1 to n (n = total number of increments).
Notice that the incremental work,
W, is equal to the product of Fx and
x -- which is also equal to the area of the increment in the F(x) versus x graph. Thus, the total work in the x-direction, Wx, is equal to the total area under the curve of the F(x) versus x graph.
As the increments,
x, get smaller and smaller, the calculated value for the net work, Wx, will approach its true value.
Combining equation (10-5) with the extension of equation (10-3) into the three component directions, x, y, and z, we find that the total work is equal to:
(10-6) W = Si=1,n{ (Fx,i)(
xi) + (Fy,i)(
yi) + (Fz,i)(
zi ) }.
And, as the increments,
x,
y, and
z, get smaller and smaller, the calculated value for the total work will approach its true value.
ADVANCED STUDY
Differential Form of the Work Equation
If we let the increment,
x, in equation (10-5) approach zero, the exact amount of work, Wx, would equal
(10-7) Wx = lim
x --> 0, Si=1,n{ (Fx,i)(
xi) }.
Those familiar with calculus will recognize the right hand side of equation (10-7) as the definition of the integral. Therefore, the exact amount of work in the x-direction, Wx, is equal to the integral,
(10-8) Wx = §( (F(x) )dx.
Similarly, the exact amount of total work, W, is equal to
(10-9) W = §( (F(x) + (F(y) + (F(z) )dxdydz.
Copyright © 2001, Cyclone Software, Pleasanton, CA, USA. All rights reserved.