Using a line integral to find the work done by a vector field example | Khan Academy

Khan Academy
26 Feb 201011:32
EducationalLearning
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TLDRThis video script delves into the concept of work done by a vector field on a moving particle along a path, specifically a counterclockwise circle. The vector field is defined in the x-y plane with a function associating a force vector with every point. The particle's path is parameterized by cosine and sine functions, creating a circular motion. Through mathematical analysis, it's demonstrated that the work done by the field is negative, indicating the field opposes the particle's motion. The script effectively illustrates the application of vector calculus in physics.

Takeaways
  • ๐ŸŒ€ The concept being discussed is the work done by a vector field on an object moving along a path within the field.
  • ๐Ÿ“ฑ The vector field is defined over the x-y plane, associating a vector with every point, specifically y times i minus x times j.
  • ๐Ÿ–ผ๏ธ The vector field is visualized as force vectors at various points, such as (1,0) resulting in a vector of -1j and (2,0) resulting in a vector of -2j.
  • ๐Ÿ”„ The example path of the object is a counterclockwise circle described by the parameterization x(t) = cos(t) and y(t) = sin(t) from t = 0 to t = 2ฯ€.
  • ๐Ÿ›ฃ๏ธ The work done by the vector field is calculated using a line integral, which involves the dot product of the vector field and the differential of the path's movement.
  • ๐ŸŽฏ The derivative of the position vector function with respect to time (dr/dt) is determined to facilitate the calculation of the line integral.
  • ๐Ÿ”ข The dot product of the vector field and dr results in an expression that can be integrated over the path of the object.
  • ๐ŸŒŸ The field is observed to oppose the motion of the object at every point, suggesting that the work done by the field will be negative.
  • ๐Ÿ“ˆ The integral simplifies to the integral of 1 with respect to t from 0 to 2ฯ€, which evaluates to -2ฯ€, confirming the field's opposition to the object's motion.
  • ๐Ÿ’ก The result of the work done calculation reinforces the concept that negative work corresponds to a force opposing the motion of an object.
Q & A

  • What is the vector field described in the script defined over?

    -The vector field is defined over R^2 for the x-y plane.

  • What is the specific vector field given in the example?

    -The vector field is given by y times the unit vector i minus x times the unit vector j.

  • How is the force vector represented at the point (1, 0) in the x-y plane?

    -At the point (1, 0), the force vector is represented as 0i - 1j, which looks like a vector pointing straight down along the negative y-axis.

  • What is the parameterization of the curve c that describes the path of the particle?

    -The parameterization of the curve c is x(t) = cos(t) and y(t) = sin(t), where t ranges from 0 to 2ฯ€.

  • What does the parameterization of the curve c represent?

    -The parameterization represents a counterclockwise circle with a radius of 1 unit.

  • How is the work done by the vector field on the curve defined?

    -The work done by the vector field on the curve is defined as the line integral over the contour of the field, which is the dot product of the vector field and the differential of the movement dr.

  • What is the expression for the differential of the position vector dr in terms of t?

    -The differential of the position vector dr in terms of t is given by dr = -sin(t)dt * i + cos(t)dt * j.

  • How is the vector field f(t) expressed in terms of the parameterization of the curve c?

    -The vector field f(t) is expressed as f(t) = sin(t)i - cos(t)j, using the values of y(t) and x(t) from the curve's parameterization.

  • What is the result of the line integral for the work done by the field on the particle?

    -The result of the line integral for the work done by the field on the particle is -2ฯ€.

  • What does the negative result of the work done indicate?

    -The negative result indicates that the field is doing work against the direction of the particle's motion, hindering its movement around the circle in a counterclockwise direction.

  • How does the script relate the concept of work done by a field to everyday experiences?

    -The script relates the concept by comparing it to lifting an object against gravity, where gravity does negative work while the person applies a force to do positive work.

Outlines
00:00
๐Ÿ“ Introduction to Vector Field Work

This paragraph introduces the concept of work done by a vector field on a moving object. It describes a vector field defined over the x-y plane, associating a vector with every point. The vector field is given by y times the unit vector i minus x times the unit vector j. The paragraph then sets up a scenario where a particle moves along a path described by a curve c, with its parameterization given by x(t) = cos(t) and y(t) = sin(t), from t = 0 to t = 2ฯ€, representing a counterclockwise circle. The goal is to calculate the work done by the vector field on the moving curve.

05:01
๐Ÿ”„ Negative Work Intuition

In this paragraph, the speaker discusses the intuition behind negative work in the context of the vector field. It is observed that the vector field seems to oppose the motion of the particle at every point along its path. The speaker uses the analogy of lifting an object against gravity, where positive work is done by the person and negative work is done by gravity. The paragraph sets the stage for a mathematical calculation to confirm this intuition and further explore the concept of negative work in the given scenario.

10:02
๐Ÿงฎ Calculating the Work Done

The final paragraph focuses on the calculation of the work done by the vector field on the moving particle. It begins by finding the derivative of the position vector function with respect to time, dr/dt, and then proceeds to calculate the differential dr. The vector field is then rewritten in terms of t to find the force from the field along the path. The line integral is set up as an integral from 0 to 2ฯ€ of the dot product of the field and the differential movement. After performing the dot product and integrating, the result simplifies to the integral of 1 with respect to t from 0 to 2ฯ€, which evaluates to -2ฯ€. This confirms the negative work done by the field, as it opposes the particle's motion.

Mindmap
Keywords
๐Ÿ’กVector Field
A vector field is a mathematical representation that assigns a vector to each point in a space, defining both magnitude and direction. In the context of the video, the vector field is described over the x-y plane and is given by the function y times the unit vector i minus x times the unit vector j. This vector field represents a force field, where each point in the plane has an associated force vector.
๐Ÿ’กWork Done
In physics, work done refers to the amount of energy transferred by a force acting upon an object as it moves along a path. The work done is calculated as the integral of the force vector dot product with the differential displacement vector along the path. In the video, the work done by the vector field on a particle moving along a path is determined by integrating the dot product of the force vector from the field and the differential displacement of the particle's path.
๐Ÿ’กLine Integral
A line integral is a mathematical concept used to compute the integral of a function over a curve or path. It is used when the function's input is a vector and the output is a scalar. In the video, the line integral is used to calculate the work done by the vector field on the moving particle, where the function is the force vector field and the path is the curve described by the particle's motion.
๐Ÿ’กParameterization
Parameterization is a mathematical technique used to represent curves or surfaces in a space by assigning a set of parametric equations. In the video, the curve traced by the particle's motion is parameterized by x(t) = cos(t) and y(t) = sin(t), which describe the position of the particle as a function of time t.
๐Ÿ’กCounterclockwise Circle
A counterclockwise circle refers to a circular path where the direction of travel is opposite to the direction of the hands of a clock. In the video, the particle's path is described as a counterclockwise circle with the parameterization x(t) = cos(t) and y(t) = sin(t), which means the particle moves around the unit circle in a counterclockwise direction.
๐Ÿ’กDifferential dr
The differential dr represents an infinitesimally small change in position along a curve or path. In the context of the video, dr is used to denote the differential displacement vector of the particle as it moves along its path. The differential dr is calculated by taking the derivative of the position vector function with respect to the parameter t and then multiplying by the differential dt.
๐Ÿ’กDot Product
The dot product is an operation on two vectors that results in a scalar value. It is calculated by multiplying the corresponding components of the two vectors and summing them up. In the video, the dot product is used to calculate the work done by the vector field on the particle by taking the dot product of the force vector from the field and the differential displacement vector dr of the particle's path.
๐Ÿ’กUnit Vectors i and j
Unit vectors i and j are vectors of length one used to represent directions in the x and y coordinates of the Cartesian coordinate system, respectively. In the video, the vector field is described using unit vectors i and j, with the force vectors being y times i and minus x times j, indicating the direction and magnitude of the force at each point in the x-y plane.
๐Ÿ’กNegative Work
Negative work refers to a situation where the work done by a force is in the opposite direction to the displacement of the object. In the video, the work done by the vector field on the particle is negative because the force vectors are always opposite to the direction of the particle's motion, indicating that the field is opposing the particle's movement.
๐Ÿ’กTrigonometric Functions
Trigonometric functions, such as sine and cosine, are mathematical functions that relate the angles and sides of a triangle. In the video, the parameterization of the particle's path uses sine and cosine functions to describe the x and y coordinates as a function of time t, which traces a counterclockwise circle on the x-y plane.
๐Ÿ’กAntiderivative
An antiderivative, also known as an indefinite integral, is a function whose derivative is the given function. In the video, the antiderivative of 1 is t, which is used to evaluate the integral of 1 with respect to t, resulting in the value of t at the upper limit of integration minus the value at the lower limit.
Highlights

The application of vector field work on a particle moving along a path is discussed.

The vector field is defined over the x-y plane with a function of x and y.

The vector field associates a force vector with every point on the plane.

A specific vector field is given, where y times the unit vector i is subtracted by x times the unit vector j.

The force vector at the point (1, 0) is described as -1 j.

The vector field is visualized across the x-y plane with varying points.

A particle's path is described by a curve c, parameterized by x(t) = cos(t) and y(t) = sin(t).

The path of the particle is a counterclockwise circle from t=0 to t=2ฯ€.

The work done by the vector field on the curve is calculated using a line integral.

The differential dr is derived from the parameterization of the path.

The vector field is rewritten in terms of t to match the path's parameterization.

The dot product of the vector field and the differential dr is calculated.

The work done by the field is found to be negative, indicating opposition to the particle's movement.

The negative work is analogous to gravity doing negative work when lifting an object.

The line integral is simplified to an integral of 1 with respect to t from 0 to 2ฯ€.

The final result of the work done by the field is -2ฯ€, confirming the field's oppositional force.

The process provides a concrete example of the theoretical concepts discussed in the previous video.

The mathematical derivation and visualization help to understand the interaction between the vector field and the particle's motion.

Transcripts

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Related Tags
Vector FieldsWork CalculationForce AnalysisMathematical Physics2D MotionParticle DynamicsLine IntegralsTrigonometric FunctionsCounterclockwise MotionNegative Work