Work-Energy Theorem — Revision Notes

⚡ 30-Second Revision

Work-Energy Theorem: — $W_{\text{net}} = Delta K = K_f - K_i$
Kinetic Energy: — $K = \frac{1}{2}mv^2$
Work by Constant Force: — $W = Fd cos\theta$
Work by Variable Force: — $W = int vec{F} cdot dvec{r}$
Work by Spring Force (from $x_1$ to $x_2$): — $W_s = \frac{1}{2}k(x_1^2 - x_2^2)$ (or $rac{1}{2}kx^2$ from $x$ to $0$)
Net Work: — Sum of work done by all forces (applied, gravity, friction, normal, etc.)
Positive Work: — Increases kinetic energy.
Negative Work: — Decreases kinetic energy.
Zero Work: — No change in kinetic energy (force perpendicular to displacement).

2-Minute Revision

The Work-Energy Theorem is a powerful principle stating that the total work done by all forces acting on an object ($W_{\text{net}}$) equals the change in its kinetic energy ($Delta K$). Mathematically, $W_{\text{net}} = K_f - K_i$.

Kinetic energy is the energy of motion, $K = \frac{1}{2}mv^2$. Work is done when a force causes displacement; for a constant force, $W = Fd cos\theta$, and for a variable force, $W = int F dx$. Remember to include work done by all forces: applied, gravitational, frictional, and spring forces.

Forces perpendicular to displacement (like normal force) do no work. Friction always does negative work. This theorem is universally applicable, even when non-conservative forces are present, making it more general than the conservation of mechanical energy.

It's particularly useful for problems involving variable forces or when time is not a factor, allowing direct calculation of speeds or distances without complex kinematic equations. Always identify initial and final kinetic energies and sum up all work contributions carefully.

5-Minute Revision

The Work-Energy Theorem is a fundamental concept in mechanics, providing a direct link between the work done on an object and its change in kinetic energy. The core statement is $W_{\text{net}} = Delta K = K_f - K_i$. Here, $W_{\text{net}}$ is the algebraic sum of work done by *all* forces acting on the object, and $Delta K$ is the difference between the final ($K_f = \frac{1}{2}mv_f^2$) and initial ($K_i = \frac{1}{2}mv_i^2$) kinetic energies.

Key Steps for Application:

1
Identify the System and Initial/Final States: — Define the object and its initial and final velocities (or speeds) to determine $K_i$ and $K_f$.
2
List All Forces: — Identify every force acting on the object during its displacement (e.g., applied force, gravity, normal force, friction, spring force).
3
Calculate Work Done by Each Force:

* Constant Force: $W = Fd cos\theta$. Remember $heta$ is the angle between the force and displacement. If $heta=0^circ$, $W=Fd$. If $heta=180^circ$, $W=-Fd$. If $heta=90^circ$, $W=0$. * Variable Force: For a spring, work done by the spring force as it moves from $x_1$ to $x_2$ is $W_s = \frac{1}{2}k(x_1^2 - x_2^2)$.

If expanding from compression $x$ to equilibrium ($0$), $W_s = \frac{1}{2}kx^2$. * Gravity: $W_g = mgh$ (positive if moving down, negative if moving up). * Friction: $W_f = -f_k d = -mu_k N d$.

1
Sum for Net Work: — $W_{\text{net}} = W_1 + W_2 + W_3 + dots$
2
Apply the Theorem: — Set $W_{\text{net}} = K_f - K_i$ and solve for the unknown quantity.

Example: A $1,\text{kg}$ block is pushed $2,\text{m}$ on a rough horizontal surface ($mu_k = 0.2$) by a $10,\text{N}$ force. Initial speed is $0$. Find final speed ($g=10,\text{m/s}^2$).

$K_i = 0$.
$N = mg = 1 \times 10 = 10,\text{N}$. $f_k = mu_k N = 0.2 \times 10 = 2,\text{N}$.
$W_{\text{app}} = 10,\text{N} \times 2,\text{m} = 20,\text{J}$.
$W_f = -2,\text{N} \times 2,\text{m} = -4,\text{J}$.
$W_{\text{net}} = 20 - 4 = 16,\text{J}$.
$W_{\text{net}} = K_f - K_i Rightarrow 16 = \frac{1}{2}(1)v_f^2 - 0 Rightarrow v_f^2 = 32 Rightarrow v_f = sqrt{32} approx 5.66,\text{m/s}$.

This theorem is a scalar approach, often simpler than vector-based force analysis, especially with variable forces or complex paths. It's a must-know for NEET.

Prelims Revision Notes

The Work-Energy Theorem is a scalar relationship that states the net work done on an object equals its change in kinetic energy: $W_{\text{net}} = Delta K = K_f - K_i$. This is a fundamental principle for NEET UG Physics.

Key Formulas to Remember:

Kinetic Energy: — $K = \frac{1}{2}mv^2$. It's always positive and depends on speed, not velocity.
Work by a Constant Force: — $W = Fd cos\theta$, where $heta$ is the angle between the force vector ($vec{F}$) and displacement vector ($vec{d}$).

* If $heta = 0^circ$ (force in direction of motion), $W = Fd$ (positive work). * If $heta = 90^circ$ (force perpendicular to motion, e.g., normal force, centripetal force), $W = 0$ (no work). * If $heta = 180^circ$ (force opposite to motion, e.g., friction, braking force), $W = -Fd$ (negative work).

Work by Gravity: — $W_g = pm mgh$. Positive when object moves down, negative when object moves up.
Work by Kinetic Friction: — $W_f = -f_k d = -mu_k N d$. Always negative as it opposes motion.
Work by a Spring Force: — When a spring is compressed or stretched from equilibrium ($x=0$) to a position $x$, the work done by the spring force is $W_s = -\frac{1}{2}kx^2$. The work done *by an external agent* to compress/stretch it is $+\frac{1}{2}kx^2$. When the spring expands from compression $x$ to equilibrium, the work done *by the spring* on the attached mass is $+\frac{1}{2}kx^2$.

Application Strategy:

1
Identify all forces — doing work. Forces like normal force and centripetal force do no work as they are perpendicular to displacement.
2
Calculate the work done by each force, paying close attention to the sign.
3
Sum all individual works — to find the net work, $W_{\text{net}}$.
4
Equate $W_{ ext{net}}$ to $Delta K$ — ($K_f - K_i$) and solve for the unknown (e.g., final speed, displacement, or force).

Common Traps:

Forgetting to include all forces in $W_{\text{net}}$.
Incorrectly assigning signs to work (especially for friction or gravity).
Confusing work done *by* a force with potential energy changes.
Using velocity instead of speed for kinetic energy ($v^2$ term).

This theorem is highly efficient for problems where time is not a factor, forces are variable, or paths are complex, offering a scalar approach to mechanics problems.

Vyyuha Quick Recall

Work Equals Change in Kinetic Energy: WECKE (pronounced 'weck-ee')

Work ($W_{\text{net}}$) is the Effect of Causing a Kinetic Energy change ($Delta K$).

Remember: Work is Net, Everything Together!