Area Under Graphs: Finding Work Done

In summary, the conversation discusses the concept of finding work done from a force-distance graph. It explains that for a graph with a constant force, the area under the line is equal to force x distance, which matches the equation for work. However, for a graph with a proportional force, the area under the line is a triangle, causing the equation to be 1/2 force x distance. The conversation also explains that this is because work is the integral of force over distance, which is equal to the area under the curve. Finally, the conversation clarifies that regardless of the amount of force or distance, the equation work = force x distance always applies.
  • #1
IQScience
6
0
Okay, so I've got a question on graphs and areas under curves.

For example, a force-distance graph, with force on the y-axis and distance on the x-axis, where you find work done from that graph.
I understand that a constant force will produce a horizontal line on the graph, like so:

Force
|
|-------------
|
|
|___________distance

So the area under the line is just force x distance.
This matches the equation, work = force x distance.

But I'm having trouble understanding a proportional graph, or one that looks like a triangle, like this:

Force
|------/-----
|-----/-----
|----/------
|---/-------
|--/--------
|-/---------
|/_________distance

i.e. force increases at a constant rate.

To find work done from this, you find the area under the line, which is a triangle.
But this doesn't match the equation (work = force x distance) which is confusing me.
Why is it not force x distance like the equation?
Why is is 1/2 force x distance now?
I understand that this is because the area under the line is a triangle, but why is work done the area under the graph?
 
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  • #2
You know that work done is ##\int{Fdx}## in one dimension and you know that integral is equal to the area under the curve so work done would be equal to the area under the curve.
 
  • #3
What is the average force under the second diagram?

Remember, work = force * distance applies regardless of the amount of force or the distance over which it acts. If you understand the concept of integration, you will see that to determine the work under a force-distance diagram, you are calculating incrementally all of the little bits of work done by a constant force acting over a small distance. In the limit, as these small increments of distance go to zero, then the total work is found.
 
  • #4
Ahhhhh, I see, that makes sense.

Thank you very much :smile:
 
  • #5


As a scientist, it is important to understand the relationship between graphs and the physical quantities they represent. In this case, we are looking at the relationship between force and work done.

First, let's clarify the equation for work done: work = force x distance only applies when the force is constant. When the force is changing, we need to use a different equation: work = average force x distance.

Now, let's look at the graph with a constant force. As you correctly stated, the area under the line is equal to the work done. This is because the force is constant, so the average force is the same as the actual force, and the equation becomes work = force x distance.

However, when the force is changing, the average force is not the same as the actual force. In the case of a proportional graph, the force is increasing at a constant rate, meaning the average force is half of the maximum force. This is why the equation becomes work = 1/2 force x distance.

To answer your question, the reason why work is equal to the area under the graph is because work is defined as the force applied through a distance. When we represent this on a graph, the area under the graph is a visual representation of the force applied over a certain distance. In other words, the area under the graph is a way to quantify the work done.

I hope this explanation helps you to better understand the relationship between graphs and work done. Keep asking questions and exploring the connections between different physical quantities – that's what being a scientist is all about!
 

What is the concept of area under a graph?

The area under a graph is a measure of the total amount of space between the graph and the x-axis. It represents the accumulated value of the data over a given interval and can be used to find the work done, or the total amount of energy expended, in a system.

Why is finding work done important in scientific research?

Finding work done is important in scientific research because it allows us to quantify and understand the energy changes that occur in a system. This information is crucial in fields such as physics, engineering, and chemistry, where the transfer of energy is a fundamental concept.

How is the area under a graph calculated?

The area under a graph can be calculated in a few different ways, depending on the type of graph. One method is to use basic geometry formulas, such as the area of a rectangle or triangle, to find the individual areas and then sum them together. Another method is to use numerical integration techniques, such as the trapezoidal rule or Simpson's rule, to approximate the area under the curve.

What are some real-world applications of finding work done?

Finding work done has many real-world applications, such as calculating the amount of energy needed to lift an object, determining the power output of a machine, and understanding the efficiency of a chemical reaction. It is also used in fields like economics, where it can be used to measure the productivity of a company or industry.

How can the area under a graph be used to analyze data?

The area under a graph can be used to analyze data by providing a visual representation of the data and allowing for comparisons between different intervals. It can also be used to identify patterns and trends in the data, as well as to make predictions about future values. Additionally, the area under a graph can be used in statistical analysis to calculate important parameters, such as the mean and standard deviation, which can provide valuable insights into the data.

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