Work by Variable Force — Revision Notes

⚡ 30-Second Revision

Definition — Work done by variable force $W = \int \vec{F} \cdot d\vec{r}$.
1D Case — $W = \int_{x_1}^{x_2} F(x) dx$.
Graphical — Area under $F-x$ curve (signed area).
Spring Force — $F_s = -kx$. Work done *by* external agent to stretch/compress from $x_1$ to $x_2$: $W = \frac{1}{2}k(x_2^2 - x_1^2)$.
Work-Energy Theorem — $W_{net} = \Delta K = K_f - K_i$.
Conservative Force — Work done is path-independent, $W = -Delta U$.
Units — Joules (J).

2-Minute Revision

Work done by a variable force is a crucial concept in physics, extending the basic definition of work to situations where the force's magnitude or direction changes with position. Unlike a constant force, where $W = Fdcos\theta$, a variable force requires integral calculus.

The fundamental formula is $W = \int \vec{F} \cdot d\vec{r}$, which means summing up infinitesimal work contributions $dW = vec{F} cdot dvec{r}$ along the path. For one-dimensional motion, this simplifies to $W = \int_{x_1}^{x_2} F(x) dx$.

Graphically, the work done is the signed area under the Force-displacement ($F-x$) curve. Areas above the x-axis are positive work, and below are negative. A key application is the spring force, $F_s = -kx$.

The work done by an external agent to stretch or compress a spring from $x_1$ to $x_2$ is $W = \frac{1}{2}k(x_2^2 - x_1^2)$. Remember to convert units (e.g., cm to m). This concept is often combined with the Work-Energy Theorem, $W_{net} = \Delta K$, to find changes in kinetic energy or velocity.

Distinguish between conservative forces (work is path-independent, like gravity and spring force) and non-conservative forces (work is path-dependent, like friction). Mastery of basic integration and graphical interpretation is essential for NEET.

5-Minute Revision

Work done by a variable force is a cornerstone of mechanics, essential for understanding energy transfer in dynamic systems. When a force changes its magnitude or direction as an object moves, we cannot use the simple $W = Fdcos\theta$ formula.

Instead, we conceptualize the path as a series of infinitesimally small displacements, $dvec{r}$. Over each $dvec{r}$, the force $vec{F}$ is considered constant, and the infinitesimal work done is $dW = vec{F} cdot dvec{r}$.

The total work is the sum of these infinitesimal contributions, obtained through integration: $W = \int_{\vec{r_1}}^{\vec{r_2}} \vec{F} \cdot d\vec{r}$.

For one-dimensional motion along the x-axis, where $vec{F} = F(x)hat{i}$ and $dvec{r} = dxhat{i}$, the formula simplifies to $W = \int_{x_1}^{x_2} F(x) dx$. This integral represents the area under the $F-x$ curve. It's crucial to remember that this is a *signed* area: areas above the x-axis contribute positive work, and areas below contribute negative work.

Key Applications & Formulas:

1
Spring Force — The force exerted by an ideal spring is $F_s = -kx$ (Hooke's Law). The work done *by an external agent* to change the spring's extension from $x_1$ to $x_2$ is $W = \frac{1}{2}k(x_2^2 - x_1^2)$. The work done *by the spring* is the negative of this value. Always ensure $x_1$ and $x_2$ are in meters.

*Mini-Example*: A spring ($k=100,\text{N/m}$) is stretched from $x=0.1,\text{m}$ to $x=0.3,\text{m}$. Work done = $\frac{1}{2}(100)((0.3)^2 - (0.1)^2) = 50(0.09 - 0.01) = 50(0.08) = 4,\text{J}$.

1
General 1D Force — If $F(x) = (2x+3),\text{N}$ and a particle moves from $x=1,\text{m}$ to $x=4,\text{m}$.

*Mini-Example*: $W = \int_{1}^{4} (2x+3) dx = [x^2 + 3x]_{1}^{4} = ((4^2 + 3(4)) - (1^2 + 3(1))) = (16+12) - (1+3) = 28 - 4 = 24,\text{J}$.

Important Considerations for NEET:

Units — Always convert all quantities to SI units (meters, Newtons) before calculation.
Conservative vs. Non-Conservative Forces — For conservative forces (e.g., gravity, spring, electrostatic), work done is path-independent and can be related to potential energy ($W = -Delta U$). For non-conservative forces (e.g., friction, air resistance), work is path-dependent.
Work-Energy Theorem — This theorem states that the net work done on an object equals its change in kinetic energy ($W_{net} = \Delta K$). This is frequently used to link work calculations to motion parameters.

Mastering these aspects will ensure you can confidently solve problems involving work by variable forces in NEET.

Prelims Revision Notes

Work by Variable Force: NEET Revision Notes

1. Definition and Formula:

When force $\vec{F}$ varies with position, work done $W$ is calculated using integration.
General formula: $W = \int_{\vec{r_1}}^{\vec{r_2}} \vec{F} \cdot d\vec{r}$. This is a line integral.
For 1D motion along x-axis: $W = \int_{x_1}^{x_2} F(x) dx$.
For 2D motion: $W = \int (F_x dx + F_y dy)$. If force is conservative, integrate each component separately over its limits.

2. Graphical Interpretation (1D):

Work done is the signed area under the Force-displacement ($F-x$) graph.
Area above x-axis = positive work.
Area below x-axis = negative work.
Be careful with geometric shapes (triangles, rectangles, trapezoids) and their areas.

3. Spring Force (Hooke's Law):

Restoring force of an ideal spring: $F_s = -kx$, where $k$ is spring constant, $x$ is extension/compression from equilibrium.
Work done by external agent to stretch/compress spring from $x_1$ to $x_2$: $W_{ext} = \frac{1}{2}k(x_2^2 - x_1^2)$.
Work done by the spring during this process: $W_{spring} = -\frac{1}{2}k(x_2^2 - x_1^2)$.
Elastic Potential Energy stored in spring: $U = \frac{1}{2}kx^2$.
Crucial — Always convert $x$ from cm to m before calculations.

4. Conservative vs. Non-Conservative Forces:

Conservative Forces — Work done depends only on initial and final positions, not on the path taken. Examples: Gravitational force, Spring force, Electrostatic force. For these, $W = -\Delta U$.
Non-Conservative Forces — Work done depends on the path taken. Examples: Friction, Air resistance. These forces dissipate mechanical energy.

5. Work-Energy Theorem:

The net work done by all forces (constant and variable, conservative and non-conservative) on an object equals the change in its kinetic energy.
$W_{net} = \Delta K = K_{final} - K_{initial} = \frac{1}{2}mv_f^2 - \frac{1}{2}mv_i^2$.
This theorem is frequently combined with variable force calculations.

6. Common Force Functions for Integration:

$F(x) = ax+b \implies \int (ax+b) dx = \frac{ax^2}{2} + bx + C$
$F(x) = ax^n \implies \int ax^n dx = \frac{ax^{n+1}}{n+1} + C$ (for $n \neq -1$)
$F(x) = A/x^2 \implies \int A/x^2 dx = -A/x + C$

7. Key Traps:

Incorrect unit conversions (cm to m).
Sign errors (positive vs. negative work).
Confusing $x_2^2 - x_1^2$ with $(x_2 - x_1)^2$.
Misinterpreting area under $F-x$ graph (e.g., summing absolute areas).
Errors in basic integration or evaluation of definite integrals.

Vyyuha Quick Recall

To calculate Work by a Variable Force, remember 'W.V.F. = Integrate F.dr'.

Work is Variable, so Force is Integrated, From Displacement Range.

Work = Integral
Force = $F(x)$ (or $\vec{F}$)
Displacement = $dx$ (or $d\vec{r}$)
Range = Limits of integration ($x_1$ to $x_2$)