Numerical/problem-solving

1. TITLE

How to solve Area under F-x Graph numericals in NEET — Step-by-step

2. PROBLEM TYPE INTRO

This category covers calculating the work done by a variable force. We will need the concept of definite integration, which geometrically means finding the total area under the Force-displacement (F-x) graph. The typical mistake students make is using incorrect geometric formulas or ignoring negative areas below the x-axis.

3. MCQ

Question:
A woman pushes a trunk on a railway platform. She applies a constant force of $100 \text{ N}$ over a distance of $10 \text{ m}$. Thereafter, she gets progressively tired and her applied force reduces linearly with distance to $50 \text{ N}$ over the next $10 \text{ m}$ (the total distance moved is $20 \text{ m}$). If there is a constant opposing frictional force of $50 \text{ N}$, what is the total work done by the applied force?

Options:
(a) $2000 \text{ J}$
(b) $1250 \text{ J}$
(c) $1750 \text{ J}$
(d) $750 \text{ J}$

4. STEP-BY-STEP SOLUTION

SOLUTION

Given:
* Initial applied force ($F_1$) = $100 \text{ N}$ (from $x=0 \text{ m}$ to $x=10 \text{ m}$)
* Final applied force ($F_2$) = $50 \text{ N}$ (decreasing linearly from $x=10 \text{ m}$ to $x=20 \text{ m}$)
* Opposing frictional force ($f$) = $-50 \text{ N}$

To Find:
Total work done by the applied force ($W_F$) over the $20 \text{ m}$ displacement.

Formula:
$W = \text{Area under the } F-x \text{ graph}$
Explain when to use this formula: Use this whenever force varies with position (variable force) and can be mapped on a graph using simple geometric shapes like rectangles, triangles, or trapeziums.

Step 1: Calculate the work done for the first $10 \text{ m}$
From $0$ to $10 \text{ m}$, the force is constant. The shape under the graph is a rectangle.
$\text{Area of Rectangle} = \text{Length} \times \text{Breadth}$
$W_1 = 100 \text{ N} \times 10 \text{ m} = 1000 \text{ J}$

Step 2: Calculate the work done for the next $10 \text{ m}$
From $10 \text{ m}$ to $20 \text{ m}$, the force decreases linearly from $100 \text{ N}$ to $50 \text{ N}$. The shape is a trapezium.
$\text{Area of Trapezium} = \frac{1}{2} \times (\text{Sum of parallel sides}) \times \text{Height}$
$W_2 = \frac{1}{2} \times (100 \text{ N} + 50 \text{ N}) \times (20 \text{ m} - 10 \text{ m})$
$W_2 = \frac{1}{2} \times 150 \text{ N} \times 10 \text{ m} = 750 \text{ J}$

Step 3: Calculate the total work done by the applied force
Add the areas from Step 1 and Step 2.
$W_F = W_1 + W_2 = 1000 \text{ J} + 750 \text{ J}$

Answer:
$1750 \text{ J}$ (Option c is correct)

5. WHERE STUDENTS GO WRONG

  • ❌ Mistake 1: Assuming the force was constant for the entire $20 \text{ m}$ and calculating a simple rectangle ($100 \text{ N} \times 20 \text{ m} = 2000 \text{ J}$). This matches incorrect option (a).
  • ❌ Mistake 2: Calculating the net work instead of the work done by the applied force. If you subtracted the work done by friction ($W_f = -50 \text{ N} \times 20 \text{ m} = -1000 \text{ J}$), you would mistakenly get $1750 \text{ J} - 1000 \text{ J} = 750 \text{ J}$. This matches incorrect option (d).
  • ❌ Mistake 3: Treating the decreasing force section as a triangle rather than a trapezium. If you only took $\frac{1}{2} \times 10 \text{ m} \times 50 \text{ N} = 250 \text{ J}$, you would end up with $1250 \text{ J}$. This matches incorrect option (b).

6. QUICK FORMULA REFERENCE

Formulas Used:

  • $W = \int_{x_i}^{x_f} F(x) dx$ — Use when the variable force is given as a mathematical equation.
  • $W = \text{Area under } F-x \text{ curve}$ — Use when graphing the force over a displacement gives standard geometric shapes (like calculating the area of a rectangle or trapezium).

7. SIMILAR PROBLEM HINT

If NEET changes the values, the method stays the same. Practice with: a variable force that drops below the x-axis into negative values (representing a retarding force). Remember that the area on the negative side of the force axis must be treated as negative work!

8. NCERT REFERENCE

Based on: NCERT Class 11, Chapter 5, Example 5.5

9. CTA

More solved numericals → Work, Energy and Power Complete Guide
Practice problems → [Topic Test: https://knowmyexam.in/mock-tests]