PhysicsMediumClass 11

Work by Variable Force — Integration, spring force

Work, Energy & Power

JEE Qs

Hard

min

Master the correct setup of the definite integral for work, paying close attention to integration limits and the signs involved with spring forces.

🧮 Key Formulas

W = ∫ F ⋅ dr (general work by variable force)

W = ∫ F_x dx (work by variable force in one dimension)

F_s = -kx (Hooke's Law for spring force, x is displacement from natural length)

W_spring = - (1/2)k(x_f^2 - x_i^2) (Work done by the spring when extension changes from x_i to x_f)

W_ext = (1/2)k(x_f^2 - x_i^2) (Work done by an external agent to change spring extension from x_i to x_f)

1Work done by a variable force F(x) along the x-axis from x_i to x_f is given by ∫ F(x) dx, which corresponds to the area under the F-x graph.
2For a general variable force F acting along a path, the elemental work done is dW = F ⋅ dr (dot product), and total work is found by integrating this expression along the path.
3Hooke's Law defines the restoring force of an ideal spring as F_s = -kx, where 'k' is the spring constant and 'x' is the displacement from its natural (equilibrium) length. The negative sign indicates the force opposes the displacement.
4Distinguish carefully between work done *by* the spring (which is negative when the spring is stretched/compressed) and work done *by an external agent* *on* the spring (which is positive when stretching/compressing the spring from its natural length).
5The Work-Energy Theorem (W_net = ΔK) remains valid for variable forces, where W_net is the total work done by all forces.

✕Incorrectly setting up the limits of integration or missing the differential element (dx, dr) in the integral for work calculation.
✕Sign errors when calculating work done by spring forces, often confusing work done *by* the spring with work done *by an external force* on the spring.
✕Failure to use the dot product (F ⋅ dr) when the force and displacement vectors are not collinear or when the force varies in direction.
✕Using the total length of the spring instead of its extension or compression from the natural length (x) in Hooke's Law (F = -kx) and related work formulas.

Work-Energy Theorem