If density of water is 1000 kgm⁻³ and density of copper is 8.9 × 10³ kgm⁻³. What is relative density of copper ?

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Correct — option (1), 8.9. Relative density, also called specific gravity, is defined as the ratio of the density of a substance to the density of water, water being taken at 4 °C where its density is at a maximum of 1000 kg m⁻³, equivalently 1 g cm⁻³. Both numbers needed are printed in the stem, so the working is a single division: 8.9 × 10³ kg m⁻³ divided by 1000 kg m⁻³, which is 8.9 × 10³ divided by 1 × 10³, which is 8.9. That is the whole calculation, and the entire question is what happens to the units while you do it. Because the numerator and the denominator are both densities, the kilograms cancel against kilograms and the cubic metres against cubic metres, and nothing is left. Relative density is therefore a pure number — it is dimensionless, it carries no unit, and it has the same value whatever consistent system of units the two densities were measured in. Work the same problem in grams per cubic centimetre and it comes out identically: copper at 8.9 g cm⁻³ divided by water at 1 g cm⁻³ is again 8.9. That invariance is the practical point of defining the quantity at all, and it is why the relative density of a substance is numerically equal to its density expressed in grams per cubic centimetre, a shortcut worth carrying. So the answer is the bare number 8.9, and any option that attaches a unit to it, or that shifts its power of ten, has misunderstood what a ratio is rather than made an arithmetic slip. It is worth sanity-checking the result against what it means physically: a relative density of 8.9 says that a given volume of copper is 8.9 times as heavy as the same volume of water, which is why copper sinks, and it sits sensibly between aluminium at about 2.7 and lead, mercury and gold higher up the scale.

Why the other options are wrong — (b) This option carries the right number and attaches a unit to it, and that is the error the question was written to expose. Relative density is a ratio of two densities, so the unit of the numerator cancels against the identical unit of the denominator and the result is a pure number with no dimensions at all. Writing 8.9 kg m⁻³ does not merely add a redundant label; it states something physically absurd, because 8.9 kilograms per cubic metre is roughly the density of a gas rather than of a metal — copper's actual density, given in the stem, is a thousand times larger. The habit that prevents this is to carry the units through the arithmetic rather than to attach them at the end: divide kg m⁻³ by kg m⁻³ and watch them disappear. The same discipline settles a whole family of dimensionless quantities that MPSC asks about, including the coefficient of friction, refractive index, strain and the mechanical advantage of a machine, none of which takes a unit.  (c) This is what a candidate gets by multiplying the two densities instead of dividing them: 8.9 × 10³ multiplied by 1 × 10³ is 8.9 × 10⁶. The error is one of operation rather than of arithmetic, and it usually comes from reading 'relative density' as an unfamiliar phrase and reaching for whichever operation the numbers seem to invite. The definition is a quotient, and the order of the quotient matters too — the density of the substance goes on top and the density of water underneath, so that substances denser than water score above 1 and substances lighter than water score below it. A second check catches this option instantly. Relative density is a comparison, and a comparison of a metal with water should produce a number of ordinary size: no everyday solid is millions of times denser than water, and even the densest metals sit around twenty. An answer in the millions is not a slightly wrong value but a value of the wrong order entirely.  (d) This option is the previous one with the sign of the exponent reversed, and no correct handling of the two densities in the stem produces it. It is the answer of a candidate who has decided that a power of ten must appear somewhere and is guessing at its direction — the sort of error that follows from manipulating exponents mechanically rather than dividing 8.9 × 10³ by 1 × 10³ and seeing the powers cancel. The physical check disposes of it even faster than the arithmetic. A relative density of 8.9 × 10⁻⁶ would mean that copper is about a hundred thousand times less dense than water, which would make it lighter than any gas and would put a copper coin somewhere above the atmosphere rather than at the bottom of a bucket. Getting into the habit of asking whether an answer is physically possible, before checking whether it is arithmetically right, disposes of two of the three wrong options here.

Concept: Density is mass per unit volume, measured in kilograms per cubic metre in SI units and often quoted in grams per cubic centimetre in the laboratory, where 1 g cm⁻³ equals 1000 kg m⁻³. Relative density, or specific gravity, is the density of a substance divided by the density of water at 4 °C, the temperature at which water is densest. Because it is a ratio of two like quantities it has no dimensions and no unit, and because water's density is 1 g cm⁻³ the relative density of any substance is numerically the same as its density in grams per cubic centimetre — copper at 8.9, aluminium at about 2.7, mercury at about 13.6, gold at about 19.3, ice at about 0.92. That last figure is why ice floats, and why only about a tenth of an iceberg stands above the water. The quantity matters because it decides floating and sinking: a body placed in a fluid is buoyed up by a force equal to the weight of the fluid it displaces, which is Archimedes' principle, so a body of relative density greater than 1 sinks in water and one of relative density less than 1 floats with the fraction of its volume submerged equal to its relative density. Relative density is measured directly with a hydrometer, a weighted float that sinks to a depth set by the density of the liquid; the lactometer used to check milk and the instruments used to test the electrolyte of a lead-acid battery are hydrometers built for particular ranges.

State PCS 2021 Q52 — Science & Technology PYQ with Answer

State PCS 2021 Q52 — Science & Technology PYQ with Answer