JEE Advanced · Physics · 1. Math in Physics

In an experiment to determine the acceleration due to gravity g, the formula used for the time period of a periodic motion is $T = 2 π \sqrt{\frac{7 (R - r)}{5 g}} .$ The values of R and r are measured to be $(60 \pm 1) m m a n d (10 \pm 1) m m,$ respectively. In five successive measurements, the time period is found to be 0.52s, 0.56s, 0.57s, 0.54s and 0.59s. The least count of the watch used for the measurement of time period is 0.01s. Which of the following statements (s) is (are) true?

A The error in the measurement of r is 10%.
B The error in the measurement of T is 3.57 %.
C The error in the measurement of T is 2%.
D The error in the determined value of g is 11%.

Verified Solution

Answer & Solution

Correct Answer

(A) The error in the measurement of r is 10%.

Step-by-step Solution

Detailed explanation

	T	Absolute error
1	0.52	$-0.04$
2	0.56	$00$
3	0.57	$+0.01$
4	0.54	$-0.02$
5	0.59	$+0.03$
	T_{avg = 0.556 = 0.56}

Avg. absolute error

\frac{0.04 + 0 + 0.01 + 0.02 + 0.03}{5} = 0.02

Percentage error in time period

= \frac{∆ T}{T} \times 100 % = \frac{0.02}{0.56} \times 100 % \approx 3.57 %

Percentage error in radius

= \frac{∆ r}{r} \times 100 % = \frac{1}{10} \times 100 = 10 %

Also,

g = \frac{28 π (R - r)}{5 T^{2}}

Percentage error in

g

\frac{∆ g}{g} = \frac{∆ R + ∆ r}{R - r} + \frac{2 ∆ T}{T}

\frac{∆ g}{g} \times 100 % = \frac{1 + 1}{50} \times 100 % + 2 (3.57) %

\approx 11 %

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

Same subject

One mole of a monatomic ideal gas undergoes four thermodynamic processes as shown schematically in the PV - diagram below. Among these four processes, one is isobaric, one is isochoric, one is isothermal and one is adiabatic. Match the processes mentioned in List-1 with the corresponding statements in List-II.

	LIST -I		LIST -II
A)	In process $I$	P)	Work done by the gas is zero
B)	In process $I I$	Q)	Temperature of the gas remains unchanged
C)	In process $I I I$	R)	No heat is exchanged between the gas and its surroundings
D)	In process $I V$	S)	Work done by the gas is $6 P_{0} V_{0}$

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	T	Absolute error
1	0.52	\(-0.04\)
2	0.56	\(00\)
3	0.57	\(+0.01\)
4	0.54	\(-0.02\)
5	0.59	\(+0.03\)
	T_{avg = 0.556 = 0.56}