Work Done by a Force Field on a Helix Path | Query Point Official

 

Work Done by a Force Field on a Helix Path

Problem Statement

Find the work done by the force field

`F(x,y,z) = −½x î − ½y ĵ + ¼ k̂`

on a particle that moves along the helix

`r(t) = cos t î + sin t ĵ + t k̂`

from the starting point (1,0,0) to the end point (−1,0,3π).

Concept: Work as a Line Integral

In vector calculus and physics, the work done by a force field on a particle moving along a curve C is computed using a line integral:

`W = ∫_C F · dr`

where `F` is the force vector and `dr` is the differential displacement along the path. Only the component of the force in the direction of displacement contributes to work.

Step 1: Parameterize the Path

The helix is given as:

`r(t) = cos t î + sin t ĵ + t k̂`

So, the coordinates are:

`x = cos t`, `y = sin t`, `z = t`

Step 2: Compute Differential Displacement

Differentiate `r(t)` with respect to `t`:

`dr = (−sin t î + cos t ĵ + k̂) dt`

Step 3: Substitute into Force Field

Substitute `x, y, z` into `F(x,y,z)`:

`F = −½(cos t) î − ½(sin t) ĵ + ¼ k̂`

Step 4: Dot Product F · dr

Compute the dot product of force and displacement:

`F · dr = (−½cos t)(−sin t) + (−½sin t)(cos t) + (¼)(1) dt`

Simplify the trigonometric terms:

`F · dr = ¼ dt`

Step 5: Evaluate the Work Integral

Integrate from `t = 0` to `t = 3π`:

`W = ∫_0^{3π} ¼ dt = ¼ [t]_0^{3π} = ¾π`

Final Answer

The total work done by the force field along the helix is:

`W = ¾π`

Explanation

The work integral simplifies because the trigonometric part cancels out, leaving a constant integrand. This makes the calculation straightforward once the line integral expression is formed. Note that the path is not straight but a helix, and integration accounts for the actual displacement along the curve.

FAQ

Q1: What does `F · dr` mean?

`F · dr` is the dot product of force and differential displacement. It measures how much of the force acts in the direction of motion.

Q2: Is the work path dependent?

In general, yes — work done by a non-conservative force depends on the path taken. This helix example is a demonstration of that concept.

Q3: Where else is this used?

This concept appears in calculus 3, physics (mechanics), and engineering problems involving force fields and particle motion.

Related Posts

See VECTORS and CLASSICAL MECHANICS Notes & Problems for more conserved and non-conserved work problems.