Study Notes on Work Done Calculations and Line Integrals

Total Work Done in Various Scenarios

Work Done in Moving a Particle in a Given Force Field

  • Definition of Work: Work done by a force on an object is defined as the integral of the force along the path taken by the object.
  • Given Force: The force is represented as
    extbfF=Feextbfi+zextbfj+xextbfkextbf{F} = Fe extbf{i} + z extbf{j} + x extbf{k}
  • Displacement and Path: The path is given by
    N=extCon′t,exty=extsin(t),extz=tN = ext{Con't}, ext{ } y = ext{sin}(t), ext{ } z = t
      with the limits of integration from
    t=aexttot=bt = a ext{ to } t = b.
  • Total Work Calculation: The total work done can be calculated using the formula:
    W=extintegralextofextbfFextalongthecurveW = ext{integral} ext{ of } extbf{F} ext{ along the curve}

Work Done in Moving a Particle Around a Circular Path

  • Circle Definition: To find the work done in moving a particle once around a circular path in the xy-plane:
      - Center: The center of the circle is at the origin
      - Radius: The radius of the circle is 3.
  • Force Field: The force field is described as
    extbfF=(2n−y+z)extbfi+(n+y−2)extbfjextbf{F} = (2n - y + z) extbf{i} + (n + y - 2) extbf{j}
  • Integration Path: The particle moves along the path of the circle
  • Work Done Calculation:
    W=extintegralextbfF∙dextbfrextalongthecircleW = ext{integral} extbf{F} \bullet d extbf{r} ext{ along the circle}

Line Integrals Over a Specific Curve

  • Given Curve: The curve is defined by the parametric equations where
      - n=t2,exty=2+t,extz=2−t3n = t^2, ext{ } y = 2 + t, ext{ } z = 2 - t^3
      - The limits for t are from t=0t = 0 to t=1t = 1.
  • Evaluating the Line Integral: Evaluate the line integral
    extintegralextbfF∙dextbfrext{integral} extbf{F} \bullet d extbf{r} where
    extbfF=(−3t2+5xy)extbf{F} = (-3t^2 + 5xy)
  • Path Details: The curve specified in the xy-plane is
    y=2n2y = 2n^2
      from the point (0,0)(0, 0) to the point (1,4)(1, 4)

Summary of Steps for Calculating Work Done

  1. Define the force vector field accurately.
  2. Understand the path along which the particle moves, verifying parametric forms and limits for integration.
  3. Apply the work formula, evaluating the integral across the specified limits by substituting the parametrization into the force equation.
  4. Calculate the line integral carefully, ensuring correct application of vector operation and limits.