Work Done By A Nonconstant Force

From Physics Book
Jump to navigation Jump to search

Claimed by Amanda Barber Spring 2018

This page explains work done by non-constant forces. In addition, it provides three levels of difficulty worked examples and analytical models will help readers develop a more thorough understanding.

Let's get started by first understanding what work is! Below is a fun cartoon explaining work! Work Cartoon

The Main Idea

Before you can understand work done by a nonconstant force, you have to understand work done by a constant force.

For a better understanding of what a force is reference this video: [What is Force?]

Work done by a Constant Force

Work done by a constant force is dependent on the amount of newtons executed on the object and the distance traveled by the object. Above is an image depicting the formula W = F*d, where F is the force and d (or X) is the distance travelled. The formula W=F*d only holds true when a constant force is applied to the system.

While this formula is useful, it is not realistic to assume force will be constant in every system.

Work done by a Nonconstant Force

Work done by a nonconstant force is more commonly seen in every day life than work done by a constant force. You can tell if a force is nonconstant if the object moves a distance with a changing force at points along the path. Two examples of nonconstant forces are spring forces and gravitational forces. You can tell that gravitational force is a nonconstant force by observing the Moon's orbit and comparing different points on its path. If you choose points at different distances from the Earth to calculate the gravitational force on the Moon, you would observe that the gravitational force is greater at a closer distance from the Earth. Another example of a nonconstant force is a spring. If a spring had a constant force, the spring would forever stretch or compress rather than oscillate.

To calculate a nonconstant force you must use a different formula than W=F*d. An integral is needed to calculate the work done along a path of nonconstant force, where work is the is change in energy of a system by a force.

Mathematical Model

Particle with nonconstant force

The total amount of work done on a system is equal to the sum of the work done by all individual forces, therefore, the total amount of work done can be calculated by the summation of each force multiplied by the distance. Iterative calculations are used in order to calculate non-constant forces and predict an object's motion. Given initial and final states of a system under non-constant force, small displacement intervals should be used to calculate the object's trajectory.

Below is the formula used to calculate each iteration then add them together to calculate total work.

[math]\displaystyle{ {{W}_{total} = {W}_{1} + {W}_{2} + {W}_{3} + ... + {W}_{n} = \overrightarrow{F}_{1}\bullet\overrightarrow{dr}_{1} + \overrightarrow{F}_{2}\bullet\overrightarrow{dr}_{2} + \overrightarrow{F}_{3}\bullet\overrightarrow{dr}_{3} + ... + \overrightarrow{F}_{n}\bullet\overrightarrow{dr}_{n}} }[/math]

[math]\displaystyle{ {{W}_{total} = \sum\overrightarrow{F}\bullet\Delta\overrightarrow{r}} }[/math]

This method, while possible, can get tedious and repetitive. If you make the intervals you calculate indefinitely small, it is the same as integrating. The most common formula used for work with a nonconstant force is the integral from the first point of a path to the last point.

[math]\displaystyle{ {{W}_{total} = \int\limits_{i}^{f}\overrightarrow{F}\bullet\overrightarrow{dr}} }[/math]

Here is a graphical example of the integral.

Computational Model

Spring-Mass Motion along an axis

[<> Model of an Oscilating Spring]

This model shows both the total work and the work done by a spring on a ball attached to a vertical spring. The work done by the spring oscillates because the work is negative when the ball is moving away from the resting state and is positive when the ball moves towards it.

Because gravity causes the ball’s minimum position to be further from the spring’s resting length than its maximum position could be, the work is more negative when the ball approaches its minimum height.

The code works by using small time steps of 0.01 seconds and finding the work done in each time step. Work is the summation of all of the work done in each time step, so another step makes sure the value for work is cumulative.

Modeling Non-Constant Forces in VPython

Spring-Mass Motion in a 2-D plane

As shown in this trinket model, Planer Motion of a Spring-Mass System, computational models can also be used in predicting non-constant forces in multiple directions.

#intialize conditions
#calculation loop
  #calculate/update force at every time step
    L = ball.pos - spring.pos
    Lhat = norm(L)
    s = mag(L) - L0
    Fspring = -(ks * s) * Lhat
  #apply momentum principle
    ball.p = ball.p + (Fspring + Fgravity) * deltat
 #update positions
 #update time




A box is pushed to the East 10 meters by a force of 40 N, then it is pushed to the north 8 meters by a force of 60 N.
Calculate the total work done on the box.


[math]\displaystyle{ W = \sum\overrightarrow{F}\bullet\Delta\overrightarrow{r} }[/math]

[math]\displaystyle{ W = 40N \bullet\ 10m + 60N \bullet\ 8m }[/math]

[math]\displaystyle{ W = 40N \bullet\ 10m + 60N \bullet\ 8m }[/math]

[math]\displaystyle{ W = 880 J }[/math]


We know that the formula for force is [math]\displaystyle{ F=ks }[/math], where [math]\displaystyle{ s }[/math] is the distance the spring is stretched. If we integrate this with respect to [math]\displaystyle{ s }[/math], we find that [math]\displaystyle{ W=.5ks^2 }[/math] is the formula for work.

[math]\displaystyle{ W=\int\limits_{i}^{f}\overrightarrow{k}\bullet\overrightarrow{ds} = .5ks^2 }[/math]


Say that we want to find the work done by a horizontal spring with spring constant k=70 N/m as it moves an object 10 cm.


Using the formula W=.5ks2 that we derived from F=ks, we can calculate that the work done by the spring is 0.35 J.

[math]\displaystyle{ W=\int\limits_{0}^{10}70\bullet\overrightarrow{ds}=.5ks^2=.5(70)(0.10^2)=0.35 J }[/math]



The earth does work on an asteroid approaching from an initial distance d. How much work is done on the asteroid by gravity before it hits the earth’s surface?


First, we must recall the formula for gravitational force.

Because [math]\displaystyle{ G }[/math], [math]\displaystyle{ M }[/math], and [math]\displaystyle{ m }[/math] are constants, we can remove them from the integral. We also know that the integral of [math]\displaystyle{ -1\over r^2 }[/math] is [math]\displaystyle{ 1\over r }[/math]. We then must calculate the integral of [math]\displaystyle{ –GMm\over d^2 }[/math] from the initial radius of the asteroid, [math]\displaystyle{ R }[/math], to the radius of the earth, [math]\displaystyle{ r }[/math].

[math]\displaystyle{ W=-GMm\bullet\int\limits_{R}^{r}{-1\over d^2}\bullet dr }[/math]

[math]\displaystyle{ W=-GMm\bullet({1\over r}-{1\over R}) }[/math]

Our answer will be positive because the forces done by the earth on the asteroid and the direction of the asteroid's displacement are the same.


The sector of physics that interests me the most is when I can see what I am computing. This unit covers spring forces. I can understand how the force is not constant because I can see the spring oscillating and changing positions and the amount of force.

Work done by a nonconstant force is related to my major, Industrial Engineering, because my major focuses heavily on optimization. Understanding how nonconstant forces work will help with machines in the workplace to help optimize efficiency.

On an industrial level, the work needed to fill empty tanks depends on the weight of the liquid, which varies as the tanks fill and empty. Energy conversion in hydroelectric dams depends on the work done by water against turbines, which depends on the flow of water. Windmills work in the same way.


Gaspard-Gustave de Coriolis, famous for discoveries such as the Coriolis effect, is credited with naming the term “work” to define force applied over a distance. Later physicists combined this concept with Newtonian calculus to find work for non-constant forces.

See also

Further Reading


Matter and Interactions - 4e Chabay and Sherwood


[Non Constant Force]

Iterative Prediction of Spring-Mass System

External Links

[Work done by Non Constant Force]

[Work done by Constant Force]


[1] [2] [3]

Created by Justin Vuong

Edited by Chris Mickas

Edited by Yunqing Jia

Edited by Noemi Nath