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Check Your Understanding Can Earth’s gravity ever be a constant force for all paths?

No, it’s only approximately constant near Earth’s surface.

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Work done by forces that vary

In general, forces may vary in magnitude and direction at points in space, and paths between two points may be curved. The infinitesimal work done by a variable force can be expressed in terms of the components of the force and the displacement along the path,

d W = F x d x + F y d y + F z d z .

Here, the components of the force are functions of position along the path, and the displacements depend on the equations of the path. (Although we chose to illustrate dW in Cartesian coordinates, other coordinates are better suited to some situations.) [link] defines the total work as a line integral, or the limit of a sum of infinitesimal amounts of work. The physical concept of work is straightforward: you calculate the work for tiny displacements and add them up. Sometimes the mathematics can seem complicated, but the following example demonstrates how cleanly they can operate.

Work done by a variable force over a curved path

An object moves along a parabolic path y = ( 0.5 m −1 ) x 2 from the origin A = ( 0 , 0 ) to the point B = ( 2 m, 2 m ) under the action of a force F = ( 5 N/m ) y i ^ + ( 10 N/m ) x j ^ ( [link] ). Calculate the work done.

A graph of y in meters versus x in meters is shown. A parabolic path labeled as y of x starts at 0, 0 and curves up and to the right. The point (2, 2) is on the parabola. Vector F of x, y is shown at a point between the origin and coordinate 2, 2. Vector F points to the right and up, at some angle to the curve y of x.
The parabolic path of a particle acted on by a given force.

Strategy

The components of the force are given functions of x and y . We can use the equation of the path to express y and dy in terms of x and dx ; namely,

y = ( 0.5 m −1 ) x 2 and d y = 2 ( 0.5 m −1 ) x d x .

Then, the integral for the work is just a definite integral of a function of x .

Solution

The infinitesimal element of work is

d W = F x d x + F y d y = ( 5 N/m ) y d x + ( 10 N/m ) x d y = ( 5 N/m ) ( 0.5 m 1 ) x 2 d x + ( 10 N/m ) 2 ( 0.5 m 1 ) x 2 d x = ( 12.5 N/m 2 ) x 2 d x .

The integral of x 2 is x 3 / 3 , so

W = 0 2 m ( 12.5 N/m 2 ) x 2 d x = ( 12.5 N/m 2 ) x 3 3 | 0 2 m = ( 12.5 N/m 2 ) ( 8 3 ) = 33.3 J .

Significance

This integral was not hard to do. You can follow the same steps, as in this example, to calculate line integrals representing work for more complicated forces and paths. In this example, everything was given in terms of x - and y -components, which are easiest to use in evaluating the work in this case. In other situations, magnitudes and angles might be easier.

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Check Your Understanding Find the work done by the same force in [link] over a cubic path, y = ( 0.25 m −2 ) x 3 , between the same points A = ( 0 , 0 ) and B = ( 2 m, 2 m ) .

W = 35 J

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You saw in [link] that to evaluate a line integral, you could reduce it to an integral over a single variable or parameter. Usually, there are several ways to do this, which may be more or less convenient, depending on the particular case. In [link] , we reduced the line integral to an integral over x , but we could equally well have chosen to reduce everything to a function of y . We didn’t do that because the functions in y involve the square root and fractional exponents, which may be less familiar, but for illustrative purposes, we do this now. Solving for x and dx , in terms of y , along the parabolic path, we get

x = y / ( 0.5 m −1 ) = ( 2 m ) y and d x = ( 2 m ) × 1 2 d y / y = d y / ( 2 m −1 ) y .

The components of the force, in terms of y , are

F x = ( 5 N/m ) y and F y = ( 10 N/m ) x = ( 10 N/m ) ( 2 m ) y ,

Questions & Answers

a particle projected from origin moving on x-y plane passes through P & Q having consituents (9,7) , (18,4) respectively.find eq. of trajectry.
rahul Reply
definition of inertia
philip Reply
the reluctance of a body to start moving when it is at rest and to stop moving when it is in motion
charles
An inherent property by virtue of which the body remains in its pure state or initial state
Kushal
why current is not a vector quantity , whereas it have magnitude as well as direction.
Aniket Reply
why
daniel
the flow of current is not current
fitzgerald
bcoz it doesn't satisfy the algabric laws of vectors
Shiekh
The Electric current can be defined as the dot product of the current density and the differential cross-sectional area vector : ... So the electric current is a scalar quantity . Scalars are related to tensors by the fact that a scalar is a tensor of order or rank zero .
Kushal
what is binomial theorem
Tollum Reply
hello are you ready to ask aquestion?
Saadaq Reply
what is binary operations
Tollum
What is the formula to calculat parallel forces that acts in opposite direction?
Martan Reply
position, velocity and acceleration of vector
Manuel Reply
hi
peter
hi
daniel
hi
Vedisha
*a plane flies with a velocity of 1000km/hr in a direction North60degree east.find it effective velocity in the easterly and northerly direction.*
imam
hello
Lydia
hello Lydia.
Sackson
What is momentum
isijola
hello
Saadaq
A rail way truck of mass 2400kg is hung onto a stationary trunk on a level track and collides with it at 4.7m|s. After collision the two trunk move together with a common speed of 1.2m|s. Calculate the mass of the stationary trunk
Ekuri Reply
I need the solving for this question
philip
is the eye the same like the camera
EDWIN Reply
I can't understand
Suraia
same here please
Josh
I think the question is that ,,, the working principal of eye and camera same or not?
Sardar
yes i think is same as the camera
muhammad
what are the dimensions of surface tension
samsfavor
why is the "_" sign used for a wave to the right instead of to the left?
MUNGWA Reply
why classical mechanics is necessary for graduate students?
khyam Reply
classical mechanics?
Victor
principle of superposition?
Naveen Reply
principle of superposition allows us to find the electric field on a charge by finding the x and y components
Kidus
Two Masses,m and 2m,approach each along a path at right angles to each other .After collision,they stick together and move off at 2m/s at angle 37° to the original direction of the mass m. What where the initial speeds of the two particles
MB
2m & m initial velocity 1.8m/s & 4.8m/s respectively,apply conservation of linear momentum in two perpendicular directions.
Shubhrant
A body on circular orbit makes an angular displacement given by teta(t)=2(t)+5(t)+5.if time t is in seconds calculate the angular velocity at t=2s
MB
2+5+0=7sec differentiate above equation w.r.t time, as angular velocity is rate of change of angular displacement.
Shubhrant
Ok i got a question I'm not asking how gravity works. I would like to know why gravity works. like why is gravity the way it is. What is the true nature of gravity?
Daniel Reply
gravity pulls towards a mass...like every object is pulled towards earth
Ashok
An automobile traveling with an initial velocity of 25m/s is accelerated to 35m/s in 6s,the wheel of the automobile is 80cm in diameter. find * The angular acceleration
Goodness Reply
(10/6) ÷0.4=4.167 per sec
Shubhrant
what is the formula for pressure?
Goodness Reply
force/area
Kidus
force is newtom
Kidus
and area is meter squared
Kidus
so in SI units pressure is N/m^2
Kidus
In customary United States units pressure is lb/in^2. pound per square inch
Kidus
Practice Key Terms 2

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Source:  OpenStax, University physics volume 1. OpenStax CNX. Sep 19, 2016 Download for free at http://cnx.org/content/col12031/1.5
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