Work as area under curve | Work and energy | Physics | Khan Academy

Khan Academy Physics
29 Jul 201612:31
EducationalLearning
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TLDRThe video script explores the concept of work done in physics, particularly when pushing a hamburger as an example. It explains that work, defined as the force applied in the direction of motion times the displacement, can be calculated using the work formula or by understanding the geometrical significance of force over distance. The script introduces an alternate method for calculating work, especially useful for varying forces, by considering the area under the force-graph. This is demonstrated through a step-by-step explanation of how the area under a rectangle (for constant force) or a triangle (for varying force) can represent the work done. The concept is further clarified with examples of negative work and varying force, emphasizing the practical application of this approach in solving complex physics problems without calculus.

Takeaways
  • πŸ‘ The work done on an object, like pushing a hamburger, can be calculated by multiplying the force applied in the direction of motion by the distance moved.
  • πŸ“ˆ A constant force applied over a distance results in work that can be visualized as the area under a force vs. position graph.
  • πŸ’­ An alternative to the standard work formula involves understanding the geometric significance of work through the area under the force graph.
  • πŸ”¨ When the force is constant, the area under the force vs. position graph (a rectangle) directly represents the work done.
  • πŸ€” For varying forces, calculating work becomes more complex as the force applied changes over the distance.
  • πŸ‘€ Demonstrates that for non-constant forces, the area under the force vs. position graph still equals the work done, using a conceptual approach involving infinitesimally small rectangles.
  • πŸ“Š The area under a triangle in a force vs. position graph can represent work done by a force that diminishes to zero over a distance.
  • βœ… Highlights the power of this geometric approach to work, enabling the calculation of work done without relying on calculus for varying forces.
  • πŸ“ƒ Explains that the concept of calculating work as the area under the force graph applies universally, not just for constant forces or simple shapes like rectangles and triangles.
  • ⚑ Warning that when calculating the total work from a force vs. position graph, one must consider the area between the force line and the x-axis, especially when forces can have negative values.
Q & A
  • What is the scenario described in the transcript?

    -The scenario involves pushing a hamburger to the right with a force of four newtons over a distance of five meters and discussing the concept of work done in this process.

  • How much work was done in the hamburger example with a constant force?

    -The work done was 20 joules, calculated as the force (4 newtons) times the displacement (5 meters).

  • What is the significance of the horizontal line in the force graph?

    -The horizontal line signifies that a constant force of four newtons was applied throughout the entire displacement of five meters.

  • Why is the cosine theta term not needed in this example?

    -The cosine theta term is not needed because the force was already in the direction of motion, making the angle zero, and cosine of zero is one, which does not change the value.

  • What is the alternative method to calculate work when the force is not constant?

    -The alternative method is to find the area under the force versus position graph, which represents the work done by the force, even when it is varying.

  • How does the area under the force graph represent work done?

    -The area under the force graph represents the work done because it accounts for the total effect of the force applied over the displacement, considering the varying magnitude of the force if applicable.

  • What is the process of breaking down the force into infinitesimal rectangles to find the area under a varying force graph?

    -The process involves dividing the force into many small constant forces over tiny displacements, calculating the area of each small rectangle (force times displacement), and summing these areas to approximate the total area under the graph, which equals the work done.

  • How is negative work represented in the context of this problem?

    -Negative work is represented when the force is applied in the opposite direction of motion, resulting in a negative area under the force graph, indicating that the force is doing work against the direction of displacement.

  • What is the formula for work when the force is constant?

    -The formula for work when the force is constant is W = Fd cos(theta), where W is work, F is the force, d is the displacement, and theta is the angle between the force and displacement vectors.

  • How does the shape of the force graph affect the calculation of work?

    -The shape of the force graph determines how the area under it is calculated. Different shapes (e.g., rectangles, triangles) require different formulas to find the area, but the principle that the area under the graph equals the work done remains the same.

  • What is the total work done if the force starts negative and becomes positive, as in the example of pushing to the left then to the right?

    -The total work done is the sum of the work done in each segment. For the portion where the force was negative (pushing to the left), the work was -1 joule, and for the portion where the force was positive (pushing to the right), the work was +4 joules, resulting in a total work of +3 joules.

Outlines
00:00
πŸ” Understanding Work Done with Constant Force

This paragraph introduces a scenario where a hamburger is pushed to the right with a constant force of four newtons over a distance of five meters. The instructor explains that the work done can be calculated using the work formula, but instead, they present an alternative method. This method involves visualizing the force as a horizontal line on a graph and recognizing that the area under this line (which forms a rectangle) represents the work done. The instructor calculates the work as positive 20 joules and emphasizes that when the force is constant, the area under the force graph equals the work done. This concept is further extended to handle more complex situations where the force is not constant.

05:03
πŸ“ˆ Calculating Work with Varying Force Using Graphs

In this paragraph, the instructor discusses how to calculate work done when the force is not constant. They propose breaking down the force into infinitesimally small rectangles, each representing a small displacement with a constant force. By summing the areas of these rectangles, the total work done is found, which is equivalent to the area under the force graph. The concept is illustrated with an example of pushing the hamburger with decreasing force, resulting in a triangular force graph. The instructor explains that the area under this triangle can be found using the formula for the area of a triangle, yielding the work done as ten joules. The key takeaway is that the area under any force versus position graph, regardless of the force's constancy, equals the work done, providing a powerful tool for calculating work in various situations without the need for calculus.

10:04
πŸ”„ Dealing with Negative Work and Variable Forces

The final paragraph addresses a situation where the force starts negative and becomes positive, representing a change in direction of the force. The instructor explains how to calculate the total work done by considering the area under the force curve from the force line to the x-axis. They clarify that the work done is represented by the finite area under the curve, not an infinite area. The example given involves pushing the hamburger first to the left (negative work) and then to the right (positive work). The areas under the respective force triangles are calculated, with negative work being represented by a negative area. The total work done is the sum of the positive and negative work, resulting in a net work of three joules. The paragraph emphasizes the importance of correctly identifying the area under the graph that corresponds to the work done and highlights the versatility of using force graphs to calculate work regardless of the force's variability.

Mindmap
Keywords
πŸ’‘work
In the context of physics, work is defined as the measure of energy transfer that occurs when an object is moved by applying a force over a distance. In the video, work is calculated as the product of the force applied to the hamburger and the displacement it undergoes. The concept is crucial as it helps in understanding energy transfer and mechanical processes. For instance, the instructor explains that pushing the hamburger with a force of four newtons over five meters results in 20 joules of work done, calculated as positive 20 joules.
πŸ’‘force
Force is a physical quantity that describes the interaction between two objects, causing them to accelerate or decelerate. In the video, the force is applied to a hamburger to move it from one position to another. The constant force of four newtons is used to illustrate the concept of work and its calculation. The force is a fundamental concept in the explanation of work, as it is the driving factor behind the movement and the energy transfer that occurs.
πŸ’‘displacement
Displacement refers to the change in position of an object and is a vector quantity that has both magnitude and direction. In the video, the hamburger is displaced to the right by five meters as a result of the applied force. Displacement is key to calculating work because it represents the distance over which the force is applied, and it must be in the direction of the force for work to be done.
πŸ’‘newtons
Newtons is the unit of force in the International System of Units (SI), named after Sir Isaac Newton. In the video, the force applied to the hamburger is described as four newtons, which gives a clear measure of the magnitude of the force used to move the hamburger. Understanding newtons helps in quantifying the force and subsequently calculating the work done.
πŸ’‘joules
Joules are the unit of work or energy in the International System of Units (SI). In the context of the video, the work done on the hamburger is measured in joules, with the calculation resulting in 20 joules of work done. Joules are essential for quantifying the amount of energy transferred when work is performed.
πŸ’‘cosine theta
Cosine theta (cos ΞΈ) is a trigonometric function used in the formula for work to account for the angle between the direction of the force and the direction of motion. In the video, since the force is applied in the same direction as the motion (horizontal), the angle is zero, and cosine of zero is one, making the cosine term unnecessary for this particular example. However, it's important for understanding the concept of work in scenarios where the force and displacement are not aligned.
πŸ’‘area under the graph
The concept of the area under the graph is used to visualize and calculate the work done by a force, especially when the force is not constant. In the video, the instructor explains that the work done can be found by calculating the area under the force versus position graph. This method provides an alternative way to understand and calculate work, especially useful when dealing with varying forces, by breaking the force into infinitesimal segments and summing their areas.
πŸ’‘calculus
Calculus is a branch of mathematics that deals with rates of change and accumulation. In the context of the video, calculus could be used to calculate the work done by a varying force. However, the instructor presents the area under the graph as an alternative method for calculating work without the need for calculus, which is particularly useful for those not familiar with advanced mathematical concepts.
πŸ’‘infinitesimal
Infinitesimal refers to an extremely small quantity or an infinitesimally small change. In the video, the concept is used to describe the division of the force into infinitesimally small segments to approximate the area under a varying force graph. This method allows for the calculation of work done by a force that changes over time or distance, by summing the areas of these infinitesimally small rectangles, which converge to the actual area under the curve.
πŸ’‘negative work
Negative work occurs when the force applied is in the opposite direction to the displacement of the object. In the video, the instructor explains that if the force initially pushes the hamburger to the left (negative direction) and then to the right (positive direction), the work done in the initial part is negative because the force and displacement are in opposite directions. This concept is important for understanding the sign convention in work calculations and the directionality of forces and displacements.
πŸ’‘mechanical processes
Mechanical processes refer to the physical actions or movements that involve the application of force and result in work. In the video, the mechanical process is the pushing of the hamburger, which is a simple example used to illustrate the principles of work, force, and energy transfer. Understanding mechanical processes is essential for analyzing and solving problems in physics and engineering.
Highlights

Exploring the concept of work done in physics, specifically in the context of pushing a hamburger.

Introducing an alternate way to think about work beyond the traditional formula.

Using a geometrical approach to understand work as the area under a force graph.

Demonstrating that work done is equivalent to the area of a rectangle when force is constant.

Explaining that the work formula can be simplified by removing the cosine theta term when the force is in the direction of motion.

Calculating the work done on the hamburger as positive 20 joules using the area under the force graph.

Discussing the application of this concept for varying forces, where calculus might otherwise be necessary.

Describing a method for calculating work done by a varying force by breaking it down into infinitesimal constant forces.

Using the area under a triangle to find the work done when force diminishes over a distance.

Providing a clear explanation of how to handle negative work and the significance of the direction of force.

Clarifying that the total work done is the sum of the work done over discrete segments of the force graph.

Emphasizing that the area under the force graph represents work done, even when the force is not constant.

Explaining that the area under the force graph must be considered from the force line to the x-axis to accurately represent work done.

Illustrating how to find the work done with a changing force by calculating the area of triangles under the graph.

Summarizing the key takeaway that the area under any force versus position graph equals the work done, providing a powerful tool for various scenarios.

Reiterating that for constant forces, the traditional work formula can be used, but for varying forces, the area under the graph is the preferred method.

Concluding with the practical application of determining work done by simply finding the area under a force versus position graph, regardless of the force's constancy.

Transcripts
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