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Physics problems are full of phrases like "for h much less than R" or "for small oscillations". Each one is an instruction to use an approximation, and each approximation has a limit. Students lose marks in two ways: they do not spot when to approximate, or they approximate where it is not allowed. This post covers the two approximations that do most of the work, with four worked examples that also check how large the error is.
What are the two approximations?
Binomial: for |x| much less than 1, (1 + x)ⁿ ≈ 1 + nx. This holds for any power n, including negative and fractional powers.
Small angle: for small θ in radians, sin θ ≈ θ, tan θ ≈ θ and cos θ ≈ 1 − θ²/2.
The Syllabus for JEE (Main) 2026 lists the binomial theorem for a positive integral index in the mathematics units. Physics uses the approximate form for any power, which follows from the series you meet later. If you want the exact theorem first, see our guide to binomial theorem problems.
Worked example 1: g above Earth's surface
The NCERT chapter on gravitation writes g(h) = g/(1 + h/R)² and, for h much less than R, uses the binomial expression to get g(h) ≈ g(1 − 2h/R). NCERT takes R = 6400 km.
At h = 64 km, h/R = 0.01. Approximation: g(h) ≈ g(1 − 0.02) = 0.980g. Exact: 1/1.01² = 0.9803g. The error is 0.03 percent.
At h = 640 km, h/R = 0.1. Approximation: 0.80g. Exact: 1/1.1² = 0.826g. Now the error is about 3 percent.
So the shortcut is excellent for hills and aircraft. For a satellite hundreds of kilometres up, use the exact form.
Worked example 2: the simple pendulum
The NCERT chapter on oscillations writes the restoring torque with sin θ, then replaces sin θ by θ to get simple harmonic motion and T = 2π√(L/g). It notes that for θ as large as 20 degrees, sin θ is nearly the same as θ in radians.
Check it yourself:
10° = 0.1745 rad, sin 10° = 0.1736. θ is 0.5 percent larger.
20° = 0.3491 rad, sin 20° = 0.3420. θ is 2.1 percent larger.
Always convert to radians first. sin 10 in a calculator set to radians is the sine of 10 radians, a completely different number.
Worked example 3: the electric dipole far away
The NCERT chapter on electric charges and fields gives the exact field on the axis of a dipole (charges ±q, separation 2a) as E = q × 4ar / [4πε₀(r² − a²)²], and for r much greater than a reduces it to E = 4qa/(4πε₀r³), that is 2p/(4πε₀r³) with p = 2qa.
The step is a binomial expansion once you take out the large term:
(r² − a²)⁻² = r⁻⁴ (1 − a²/r²)⁻² ≈ r⁻⁴ (1 + 2a²/r²)
Keeping only the leading 1 gives the textbook result. How good is it? At r = 10a, the exact field is 1/(1 − 0.01)² = 1.020 times the approximate one, a 2 percent difference. At r = 5a it is 1.085 times, about 8.5 percent.
The habit to build: factor out the biggest quantity so that the bracket looks like (1 + small number).
Worked example 4: how errors grow through powers
From T = 2π√(L/g), g = 4π²L/T². Suppose your timing makes T 1 percent too long. Then g is multiplied by (1.01)⁻² ≈ 1 − 2(0.01) = 0.98, so it comes out about 2 percent low. The exact factor is 0.9803.
This is why a quantity raised to the power n carries n times the fractional error into the result. It is the same binomial step, used for estimating uncertainty instead of simplifying a formula.
Common mistakes and how to avoid them
Approximating before factoring. (R + h)⁻² is not ≈ R⁻² − 2h. Write it as R⁻²(1 + h/R)⁻² first.
Using degrees. θ ≈ sin θ only works with θ in radians.
Keeping x² in one term and dropping it in another. Decide the order you are working to and apply it everywhere.
Approximating when the question wants the exact answer. If the options differ by less than your approximation error, you cannot tell them apart.
Sign of the power. (1 + x)⁻² gives 1 − 2x, not 1 + 2x.
A practice routine
Rewrite three exact formulas from your notes (g at height, dipole field, a pendulum period) in the form (1 + small)ⁿ.
For each, compute the exact and approximate values at x = 0.01, 0.1 and 0.3, and write the percentage error.
In mixed practice, underline every "much less than" or "small" in a question before solving.
Once a week, take one solved problem and ask: which approximation was used, and where would it fail?
For a fuller treatment of oscillations where the small-angle step decides the answer, see our post on the physical pendulum and angular SHM. And if you are unsure why an expansion works, the Fermi Concept Library covers every concept in physics, chemistry, biology, mathematics and computing.
Start with example 1: work out g at 320 km both ways and see how far apart the answers are.
