University:
Santa Fe College
Course:
PHY 2004 | Applied Physics 1
Academic year:

2024
Views:

487

Pages:

5
Author:

Jabin F.

# Conservative force: A force is said to be conservative if work done by the force is independent of path. It only depends on the initial & final points. W = ∫F.dl (Independent of path) Theorem: Work done in a closed path is zero in case of conservative forces. Proof: ∫F.dl = ∫F.dl + ∫F.dl + ∫F.dl + ∫F.dl ∫F.dl = ∫F.dl + ∫F.dl + ∫F.dl + ∫F.dl ∫F.dl = 0 Alternate using Stokes' theorem: ∫F.dl = ∫(∇×F).dS = 0 General formula of curl of a vector: A = A₁î + A₂ĵ + A₃k̂ ∇ × A = (∂A₃/∂y - ∂A₂/∂z)î + (∂A₁/∂z - ∂A₃/∂x)ĵ + (∂A₂/∂x - ∂A₁/∂y)k̂ Question: Check whether force F = k(x + z)î is conservative or not. Solution: ∇ × F = (∂/∂x, ∂/∂y, ∂/∂z) × (kx + kz, 0, 0) = (0, 0, 0) ⇒ F is conservative. # Some examples of conservative forces: Gravitational force between two masses: F = -Gm₁m₂/r^2 Coulombic force between two point charges: F = kq₁q₂/r^2 (not applicable for fast-moving charges) Spring Force: F = -kx (elongation or compression) In the figure shown, if the ball is moved x distance horizontally, then at the displaced position, both the springs will apply a force to restore back the springs.