For springs connected in parallel:
\[ k_{eq} = k_1 + k_2 \]
\[ k_{eq} = 10 + 20 = 30 \text{ N/m} \]
In addition, absolute errors add:
\[ \Delta k_{eq} = \Delta k_1 + \Delta k_2 \]
\[ = 0.2 + 0.3 = 0.5 \]
Percentage error:
\[ \frac{0.5}{30} \times 100 \]
\[ = 1.67\% \]
Correct Answer: 1.67%
Error analysis is one of the most scoring yet conceptually misunderstood topics in JEE Main Physics. It belongs to the experimental physics unit and tests a student’s clarity about measurement uncertainty, propagation of errors, percentage error, and dimensional consistency.
1. Measurement and Uncertainty
No physical quantity can be measured with absolute precision. Every measurement has some uncertainty. If a quantity A is written as:
\[ A = A_0 \pm \Delta A \]
Then A₀ is measured value and ΔA is absolute error.
2. Types of Errors
• Systematic error
• Random error
• Instrumental error
• Personal error
JEE usually focuses on mathematical propagation of error.
3. Absolute, Relative and Percentage Error
Absolute error = ΔA
Relative error:
\[ \frac{\Delta A}{A} \]
Percentage error:
\[ \frac{\Delta A}{A} \times 100 \]
4. Rules of Error Propagation
Addition/Subtraction:
\[ Z = A + B \]
\[ \Delta Z = \Delta A + \Delta B \]
This is directly used in this question.
Multiplication/Division:
\[ \frac{\Delta Z}{Z} = \frac{\Delta A}{A} + \frac{\Delta B}{B} \]
Power Rule:
If Z = Aⁿ
\[ \frac{\Delta Z}{Z} = |n| \frac{\Delta A}{A} \]
5. Spring Constant in Parallel
For parallel springs:
\[ k_{eq} = k_1 + k_2 \]
This behaves like addition of resistances in series analogy reversed.
6. Spring Constant in Series
\[ \frac{1}{k_{eq}} = \frac{1}{k_1} + \frac{1}{k_2} \]
Series combination would require different error propagation rule.
7. Physical Interpretation
Parallel springs share load. Effective stiffness increases.
Series springs increase extension → stiffness decreases.
8. Why Absolute Errors Add in Addition?
Worst-case estimation method assumes maximum possible deviation.
This ensures experimental reliability.
9. JEE Trap Points
• Students mistakenly add percentage errors instead of absolute errors.
• Forget to compute equivalent constant first.
• Apply multiplication rule incorrectly.
10. Conceptual Depth: Why Errors Add Linearly?
If two measured quantities are added, uncertainty in result equals sum of individual uncertainties because deviation may occur in same direction.
This is conservative estimation.
11. Graphical Understanding
If k1 varies between 9.8 to 10.2 and k2 between 19.7 to 20.3, maximum keq variation equals sum of maximum deviations.
12. Experimental Physics Importance
Error analysis ensures scientific reporting accuracy.
Modern experiments depend heavily on uncertainty estimation.
13. Advanced Insight
For complex formulas like:
\[ T = 2\pi \sqrt{\frac{m}{k}} \]
Error propagation becomes:
\[ \frac{\Delta T}{T} = \frac{1}{2} \frac{\Delta m}{m} + \frac{1}{2} \frac{\Delta k}{k} \]
Such expressions are common in JEE Advanced.
14. Dimensional Consistency
Error calculations must maintain units.
Percentage error is dimensionless.
15. Real Life Application
Engineering tolerances use similar percentage error calculations.
Bridge construction, machine design, material testing depend on error bounds.
16. Exam Relevance
Error analysis appears frequently in JEE Main because:
• It is formula based
• Conceptual traps exist
• Requires clarity and speed
17. Common Mistakes
• Adding relative errors in addition case.
• Ignoring unit consistency.
• Not converting final answer into percentage.
18. Master Strategy
Always identify operation first: addition, multiplication, power.
Apply correct rule carefully.
19. Quick Memory Table
Addition → absolute errors add
Multiplication → relative errors add
Power → multiply relative error by exponent
20. Final Conceptual Summary
Spring system + error analysis = direct application of addition rule.
Equivalent stiffness = sum of stiffness.
Absolute errors add.
Percentage error = (total absolute error / total value) × 100.
This clarity ensures full marks in error-based JEE problems.
This content is created strictly for educational and competitive examination preparation purposes only.