State and prove work energy theorem analytically.

**OR**

Define potential energy. Derive an expression for potential energy of a stretched spring.

Question

State and prove work energy theorem analytically.

**OR**

Define potential energy. Derive an expression for potential energy of a stretched spring.

Accepted Answer

>

> [!TLDR]
> The work-energy theorem states that net work done on a body equals its change in kinetic energy; this follows directly from Newton's second law using $v\,dv = a\,dx$.
>
> **Statement:** The work-energy theorem states that the work done by the net (resultant) force acting on a body is equal to the change produced in its kinetic energy:
> $$W = \Delta KE = KE_f - KE_i$$
>
> **Analytical proof:**
>
> Consider a body of mass $m$ moving along a straight line under the action of a variable force $F$ (acting along the direction of motion). Its velocity changes from an initial value $u$ to a final value $v$ as it moves through a displacement from $x_1$ to $x_2$.
>
> By Newton's second law:
> $$F = ma = m\frac{dv}{dt}$$
>
> Using the chain rule, $\dfrac{dv}{dt} = \dfrac{dv}{dx}\cdot\dfrac{dx}{dt} = v\dfrac{dv}{dx}$, so:
> $$F = mv\frac{dv}{dx}$$
>
> The work done by this force as the body moves from $x_1$ to $x_2$ is:
> $$W = \int_{x_1}^{x_2} F\,dx = \int_{x_1}^{x_2} mv\frac{dv}{dx}\,dx = \int_{u}^{v} mv\,dv$$
>
> Evaluating the integral: …

Q.State and prove work energy theorem analytically. OR Define potential energy. Derive an expression for potential energy of a stretched spring.

Concept understanding — Work Energy Theorem

The Work-Energy Theorem: From Intuition to Precision

The Intuition First

The Precise Statement

Why "Net" Work Matters

A Simple Derivation (for constant force)

What It Tells You (and What It Doesn't) …

Q.State and prove work energy theorem analytically. OR Define potential energy. Derive an expression for potential energy of a stretched spring.

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