Test: Units & Measurement - 2 - JEE MCQ

• A physical quantity is a physical property of a phenomenon, body, or substance, that can be quantified by measurement.
• A physical quantity can be expressed as the combination of a magnitude expressed by a number – usually a real number – and a unit.
• All these given above can be expressed as explained so these are physical quantity.

If radian correction is not considered in specific heat measurement. The measured value of specific heat will be more than its actual value.

► Time can be converted as t = (8 x 60) + 20s = 500 seconds
► Now speed given= 1 unit
We know speed = Distance / Time
which implies distance = Speed x Time
Hence, the distance becomes 1 x 500 = 500 units.

Length , l = 4,234 m
Breadth, b = 1.005 m
Thickness , t = 2.01 x 10^-2 m
Volume = l x b x t
⇒ V = 4.234 x 1.005 x 0.0201 = 0.0855289 = 0.0855 m³ (significant figure = 3)

There are three rules for determining how many significant figures are in a number: 

• Non-zero digits are always significant.
• Any zeros between two significant digits are significant.
• A final zero or trailing zeros in the decimal portion ONLY are significant.

Keeping these rules in mind, we can say that there are 4 significant digits.

The Measurement of a given quantity is essentially an act or result of comparison between a quantity whose magnitude (amount) is unknown, with a similar quantity whose magnitude (amount) is known, the latter quantity being called a Standard.

The relative error in a physical quantity raised to the power k is the k times the relative error in the individual quantity.

Suppose Z = A²,
Then,
ΔZ/Z = (ΔA/A) + (ΔA/A) = 2 (ΔA/A).
Hence, the relative error in A² is two times the error in A.

There are three rules on determining how many significant figures are in a number:

• Non-zero digits are always significant.
• Any zeros between two significant digits are significant.
• A final zero or trailing zeros in the decimal portion ONLY are significant.

So, keeping these rules in mind, there are 1 significant digit.

There are three rules on determining how many significant figures are in a number: 

• Non-zero digits are always significant.
• Any zeros between two significant digits are significant.
• A final zero or trailing zeros in the decimal portion ONLY are significant.

Keeping these rules in mind, we can say that there are 5 significant digits.

Unit is the reference used as the standard measurement of a physical quantity.

• The unit in which the fundamental quantities are measured are called fundamental unit.
• Units used to measure derived quantities are called derived units.

Zeros occurring between non zero digits also contribute to the number accuracy and thus all the digits that contribute towards the accuracy of the number are considered significant.

Difference in masses = 20.17 – 20.15 = 0.02 g
In subtraction, the final result should retain as many decimal places as there are in the number with the least decimal places.

• The horsepower (hp) is a unit in the foot-pound-second ( fps ) or English system, sometimes used to express the rate at which mechanical energy is expended.
• It was originally defined as 550 foot-pounds per second (ft-lb/s).
• A power level of 1 hp is approximately equivalent to 746 watt.

Explanation

To determine the number of significant digits in the number 8.1000, we can follow a systematic approach based on the established rules of significant figures

1Identify non-zero digits. In the number 8.1000, the digits '8' and '1' are non-zero. According to the rules of significant figures, all non-zero digits are considered significant. Therefore, we have two significant digits from these numbers

2Examine the zeros. The zeros in the number 8.1000 need to be evaluated based on their position. The first zero appears after the '1', and since it is sandwiched between non-zero digits (1 and 0), it is significant

3Analyze trailing zeros. The last three zeros in 8.1000 come after the decimal point and follow the last non-zero digit (which is '1'). According to the rules, trailing zeros in a decimal number are significant. Therefore, these three zeros also count as significant digits

4Count the total significant digits. Adding the significant digits identified: 2 (from '8' and '1') + 1 (from the zero between '1' and '0') + 3 (trailing zeros) gives us a total of 5 significant digits

The SI base units and their physical quantities are the metre for measurement of length, the kilogram for mass, the second for time, the ampere for electric current, the kelvin for temperature, the candela for luminous intensity, and the mole for amount of substance.

The dimension of the units of a derived physical quantity may be defined as the number of times the fundamental units of mass, length and time appear in the physical quantity.

The expression for velocity obtained above is said to be its dimensional formula. Thus, the dimensional formula for velocity is [M⁰L¹T​​​​​​^-1].

Significant Digits

Significant digits are the digits in a number that carry meaning contributing to its precision. They are used to indicate the reliability of a measurement.

Rules for determining significant digits:

1. Non-zero digits are always significant.
2. Zeros between non-zero digits are always significant.
3. Leading zeros (zeros to the left of the first non-zero digit) are never significant.
4. Trailing zeros in a number with a decimal point are significant.

Applying the rules to 0.000532:

• The first non-zero digit is 5.
• The digits following 5, which are 3 and 2, are non-zero and therefore significant.

So, the significant digits in 0.000532 are 5, 3, and 2.

The zeros before the 5 are leading zeros and do not contribute to the precision of the number, hence they are not significant.

for an approximate result, divide the length value by 1.609

► One kilometer is equals to 0.62 miles.

157900  implies that the population is believed to be within the range of about 157850 to about 157950.

In other words, the population is 157900 ± 50. The “plus-or-minus 50” appended to this number means that we consider the absolute uncertainty of the population measurement to be 50 – (–50) = 100.

We can also say that the relative uncertainty is 100/157900, which we can also express as 1 part in 1579, or 1/1579 = 0.000633, or about 0.06 percent.

Derived units are units which may be expressed in terms of base units by means of mathematical symbols of multiplication and division.
Example: the SI derived unit of area is the square metre (m^{​​​​​​2}), and the SI derived unit of density is the kilogram per cubic metre (kg/m³ or kg m^{​​​​​​-3}).

The names of SI units are written in lowercase.

It would be more exact to say that "Quantities with the same units can be added and subtracted with no problem."

Quantities with the same dimensions can often be added and subtracted as long as the units are correctly converted.

There are 20 divisions in 4 cm of main scale
⇒ 1 cm of main scale will have 20/4 = 5 divisions

Thus,  1 MSD  = 1/5 = 0.2 cm Vernier scale has 10 divisions 
Thus, L.C. = 0.2/10 = 0.02 cm 

The joule (symbol J), is a derived unit of energy in the International System of Units.

It is equal to the energy transferred to (or work done on) an object when a force of one newton acts on that object in the direction of its motion through a distance of one metre (1 newton metre or N⋅m).

The principle of homogeneity of dimensions states that “For an equation to be dimensionally correct, the dimensions of each term on LHS must be equal to the dimensions of each term on RHS.”

As no indication of its precision is given. In this case, all we have to go on is the number of digits contained in the data.

Thus the quantity “0.42 cm” is specified to 0.01 unit in 0.42, or one part in 42.

The implied relative uncertainty in this figure is 1/42, or about 2%.

The smallest value that can be measured by the measuring instrument is called its least count.

Measured values are good only up to this value. The least count error is the error associated with the resolution of the instrument.

Dimensional Analysis (also called Factor-Label Method or the Unit Factor Method) is a problem-solving method that uses the fact that any number or expression can be multiplied by one without changing its value. It is a useful technique.

Another use of dimensional analysis is in checking the correctness of an equation which you have derived after some algebraic manipulation. Even a minor error in algebra can be detected because it will often result in an equation which is dimensionally incorrect.

The rules for rounding off are following:

• If the first non-significant digit is less than 5, then the least significant digit remains unchanged.
• If the first non-significant digit is greater than 5, the least significant digit is incremented by 1.
• If the first non-significant digit is 5, the least significant digit can either be incremented or left unchanged.
• All non-significant digits are removed.

So rounding off 34.216 upto 4 digits is 34.22 and upto 3 digits is 34.2

Resolution is the smallest amount of input signal change that the instrument can detect reliably. This term is determined by the instrument noise (either circuit or quantization noise).

If an equation fails this consistency test, it is proved wrong, but if it passes, it is not proved right.

Thus, a dimensionally correct equation need not be actually an exact (correct) equation, but a dimensionally wrong (incorrect) or inconsistent equation is definitely wrong.