Questions: Units and Measurements
Units and Measurements24 min readLesson 4 of 5
Q1. Why does a measurement always carry a unit, and why is a number alone not a measurement?
Answer: A measurement compares an unknown quantity with an agreed reference standard, and the unit names that standard. The number says how many copies of the standard are present, so a value without the unit does not identify what was measured. Writing 5 says nothing until the unit is supplied, because 5 m and 5 cm differ by two orders of magnitude.
Hinglish: Measurement ka matlab hai kisi quantity ko ek agreed standard se compare karna. Number batata hai kitne copies mile, aur unit batata hai woh standard kya hai. Sirf 5 likhne se kuch nahi pata, 5 m aur 5 cm mein do order ka farak hai. Base units agreement se fix hote hain, isliye unit zaroori hai.
Q2. State the rules for counting significant figures and apply them to 0.06900, 2.308 cm and 123 m.
Answer: Non-zero digits count, zeros between them count, leading zeros before the first non-zero digit do not count, trailing zeros after a decimal point count and trailing zeros in a whole number without a decimal point do not count. For 0.06900 the leading zero is a place holder and the two zeros after 69 are significant, so the count is four. For 2.308 cm all four digits count. For 123 m the count is three; written as 123.0 m it would become four.
Hinglish: Non-zero digits count hote hain, beech ke zeros bhi, par first non-zero se pehle wale zeros place holder hote hain. Decimal ke baad trailing zeros significant hote hain, bina decimal whole number ke nahi. Unit change se count nahi badalta. 0.06900 aur 2.308 cm mein chaar, 123 m mein teen significant figures hain.
Q3. The length and breadth of a sheet are 16.2 cm and 10.1 cm, each measured with a metre scale. Find the area with its uncertainty.
Answer: Each measurement is 16.2 cm plus or minus 0.1 cm and 10.1 cm plus or minus 0.1 cm, so the relative uncertainties are about 0.6 percent and about 1 percent. The product is 163.62 cm^2, and the relative uncertainty of a product is the sum, about 1.6 percent, which is about 2.6 cm^2. The result is quoted as 164 plus or minus 3 cm^2. The value keeps three significant figures and the uncertainty keeps one.
Hinglish: Dono readings 0.1 cm tak accurate hain, isliye relative errors 0.6 percent aur 1 percent hain. Product 163.62 cm^2 aata hai, aur product mein relative errors add hote hain, total 1.6 percent, matlab 2.6 cm^2. Final answer 164 plus or minus 3 cm^2, uncertainty ek significant figure tak.
Q4. Distinguish between accuracy and precision, and give one example where a measurement is precise but not accurate.
Answer: Accuracy measures how close a reading is to the true value, while precision measures how closely repeated readings agree with each other. A vernier caliper with a positive zero error of 0.07 cm can be read very consistently, so its precision is high, but every reading is larger than the true value by the same amount, so accuracy is poor. Random errors reduce precision, and systematic errors reduce accuracy. Averaging repeated readings reduces random error but cannot remove a systematic error.
Hinglish: Accuracy true value se doori batati hai, precision repeated readings ki consistency. Positive zero error wale caliper mein har reading consistent aayegi par sach se 0.07 cm zyada hogi, isliye precision high aur accuracy kam. Random error se precision girta hai aur systematic error se accuracy; repeat aur average se random error kam hota hai, systematic nahi hatata.
Q5. What is dimensional analysis, and what can it do and not do?
Answer: Dimensional analysis writes every quantity as powers of the base dimensions and uses the fact that only like dimensions can be added or equated. It can check consistency, convert a relation between unit systems and deduce the form of a relation up to a dimensionless constant. It cannot find that constant, cannot distinguish quantities with the same dimensions and cannot replace the physical derivation. A dimensionally consistent equation may still be wrong, but a dimensionally inconsistent one is certainly wrong.
Hinglish: Dimensional analysis har quantity ko base dimensions ki powers mein likhta hai. Isse equation ki consistency check hoti hai, units convert hote hain, aur relation ka form dimensionless constant tak nikalta hai. Constant yeh method nahi deta aur derivation ki jagah nahi le sakta. Dimensions match hona zaroori hai par kaafi nahi, ulta mismatch pakka galat hai.
Q6. Check dimensionally the equation v^2 = u^2 + 2as.
Answer: The left side is a velocity squared, with dimensions L^2 T^-2. On the right side u^2 has the same dimensions. The term 2as is an acceleration multiplied by a displacement, L T^-2 times L, which is also L^2 T^-2. Every term carries the same dimensions, so the equation is dimensionally consistent. The 2 is a pure number with no dimensions, and consistency does not prove it; only the derivation does.
Hinglish: v^2 ki dimensions L^2 T^-2 hain, aur u^2 ki bhi wahi. 2as mein acceleration guna displacement hai, L T^-2 guna L, matlab L^2 T^-2. Sab terms same dimensions rakhti hain, isliye equation dimensionally consistent hai. 2 ek pure number hai, uski koi dimension nahi. Consistency se yeh proof nahi hota ki 2 hi sahi coefficient hai, woh derivation se aata hai.
Q7. The period of a simple pendulum is claimed to depend on its length, the mass of the bob and the acceleration due to gravity. Use dimensions to find the form of the relation.
Answer: Assume T = k l^x g^y m^z with k dimensionless. On the left the dimension is T; on the right l contributes L^x, g contributes L^y T^-2y and m contributes M^z. Comparing powers, the exponent of M gives z = 0, the exponent of T gives -2y = 1 so y = -1/2, and the exponent of L gives x + y = 0 so x = 1/2. Therefore T is proportional to the square root of l/g, and the mass of the bob does not appear. The constant k stays undetermined; it equals 2 pi in the exact result.
Hinglish: T = k l^x g^y m^z maan lo. Left side T hai, right side mein l se L^x, g se L^y T^-2y aur m se M^z. M ka power zero deta hai z = 0, T ka power -2y = 1 se y = -1/2, aur L ka power x + y = 0 se x = 1/2. Isliye T, l/g ke square root ke proportional hai, aur bob ka mass cancel ho jaata hai. Constant k dimensions se nahi milta, exact result mein woh 2 pi hai.
Q8. The mass and the radius of a sphere are measured with percentage errors of 1 percent and 2 percent. Find the percentage error in the density.
Answer: The density of a sphere is 3m / (4 pi r^3), so its relative error is the relative error of the mass plus three times the relative error of the radius. That is 1 percent plus 3 times 2 percent, which gives 7 percent. The power 3 supplies the large multiplier. A percentage error quoted on the diameter equals that on the radius, so the 3 is unchanged.
Hinglish: Sphere ki density 3m / (4 pi r^3) hoti hai, isliye relative error mass ke error plus radius ke error ka teen guna hota hai. Yeh 1 percent plus 3 guna 2 percent, matlab 7 percent. Power 3 hi bada multiplier banata hai. Diameter diya ho tab bhi percentage error barabar hota hai, isliye 3 hi lagta hai.
Q9. One exercise describes a screw gauge whose spindle advances 1.0 mm in a full turn and whose circular scale is divided into 200 parts. Find the least count, and say whether a longer scale alone can keep improving the accuracy.
Answer: The least count is the pitch divided by the number of circular scale divisions, so 1.0 mm / 200 = 0.005 mm. More divisions would in principle reduce that number, but the scale has a limit: the marks become too fine to read and the play in the screw swamps the finer reading, so better precision needs a better instrument.
Hinglish: Least count = pitch / circular scale divisions = 1.0 mm / 200 = 0.005 mm. Divisions badhane se least count ghat sakta hai, lekin practical limit hoti hai: divisions bahut mahin ho jaati hain, reference line ki motai aur screw ka play finer reading ko kha jaate hain. Isliye divisions badha kar accuracy infinitely nahi badha sakte.
Q10. Why do we use scientific notation, and what is the order of magnitude of 1.06 x 10^-10 m?
Answer: Scientific notation writes a value as a number from 1 to 10 multiplied by a power of 10. It removes the ambiguity about trailing zeros because every digit in the leading number counts. For 1.06 x 10^-10 m the leading number is close to 1, so the order of magnitude is -10, the size of an atom.
Hinglish: Scientific notation value ko 1 se 10 ke beech ke number aur 10 ki power ke roop mein likhta hai. Isse trailing zeros ki ambiguity khatam ho jaati hai aur chhote ya bade numbers compare karna aasan ho jaata hai. 1.06 x 10^-10 m mein leading number 1 ke paas hai, isliye order of magnitude -10, jo atom ke size ke barabar hai.
Quiz below: five questions on SI units, significant figures, error combination and dimensional formulae. Attempt them without the formula sheet first; each wrong answer will send you to one specific page.