UNITS AND DIMENTSIONS

SYNOPSIS

1. Physics is a quantitative science, based on measurement of physical quantities. Certain physical quantities have been chosen as fundamental or base quantities (such as length, mass, time, electric current, thermodynamic temperature, amount of substance, and luminous intensity).

2. Each base quantity is defined in terms of a certain basic, arbitrarily chosen but properly standardised reference standard called unit (such as metre, kilogram, second, ampere, kelvin, mole and candela). The units for the fundamental or base quantities are called fundamental or base units.3. Other physical quantities, derived from the base quantities, can be expressed as acombination of the base units and are called derived units. A complete set of units,both fundamental and derived, is called a system of units.4. The International System of Units (SI) based on seven base units is at presentinternationally accepted unit system and is widely used throughout the world.5. The SI units are used in all physical measurements, for both the base quantities andthe derived quantities obtained from them. Certain derived units are expressed bymeans of SI units with special names (such as joule, newton, watt, etc).6. The SI units have well defined and internationally accepted unit symbols (such as m formetre, kg for kilogram, s for second, A for ampere, N for newton etc.).7. Physical measurements are usually expressed for small and large quantities in scientificnotation, with powers of 10. Scientific notation and the prefixes are used to simplifymeasurement notation and numerical computation, giving indication to the precisionof the numbers.8. Certain general rules and guidelines must be followed for using notations for physicalquantities and standard symbols for SI units, some other units and SI prefixes forexpressing properly the physical quantities and measurements.9. In computing any physical quantity, the units for derived quantities involved in therelationship(s) are treated as though they were algebraic quantities till the desiredunits are obtained.10. Direct and indirect methods can be used for the measurement of physical quantities.In measured quantities, while expressing the result, the accuracy and precision ofmeasuring instruments along with errors in measurements should be taken into account.11. In measured and computed quantities proper significant figures only should be retained.Rules for determining the number of significant figures, carrying out arithmeticoperations with them, and rounding off the uncertain digits must be followed.12. The dimensions of base quantities and combination of these dimensions describe thenature of physical quantities. Dimensional analysis can be used to check the dimensionalconsistency of equations, deducing relations among the physical quantities, etc. Adimensionally consistent equation need not be actually an exact (correct) equation,but a dimensionally wrong or inconsistent equation must be wrong.

Exercise1. The moon is observed from two diametrically opposite points A and B on Earth. The angle (subtended at the moon by the two directions of observation is 1054. Given the diameter of the Earth to be about 1, compute the distance of the moon from the Earth.2. The Suns angular diameter is measured to be 1920 . The distance D of the Sun from the Earth is . What is the diameter of the Sun?3. If the size of a nucleus (in the range of ) is scaled up to the tip of a sharp pin, what roughly is the size of an atom? Assume tip of the pin to be in the range.4. Two clocks are being tested against a standard clock located in a national laboratory. At 12:00:00 noon by the standard clock, the readings of the two clocks are :Clock 1

Clock 2Monday

12:00:05

10:15:06Tuesday 12:01:15

10:14:59Wednesday 11:59:08

10:15:18Thursday 12:01:50

10:15:07Friday

11:59:15

10:14:53Saturday 12:01:30

10:15:24Sunday

12:01:19

10:15:11

If you are doing an experiment that requires precision time interval measurements, which of the two clocks will you prefer?5. We measure the period of oscillation of a simple pendulum. In successive measurements, the readings turn out to be 2.63 s, 2.56 s, 2.42 s, 2.71s and 2.80s. Calculate the absolute errors, relative error or percentage error.6. The temperatures of two bodies measured by a thermometer are t1 = 20o C 0.5o C and t2 = 50o C 0.5o C. Calculate the temperature difference and the error theirin.7. The resistance R = V/I where V = (100 5)V and I = (10 0.2)A. Find the percentage error in R.8. Two resistors of resistances R1 = 100 3 ohm and R2 = 200 4 ohm are connected (a) in series, (b) in parallel. Find the equivalent resistance of the (a) series combination, (b) parallel combination. Use for (a) the relation = + , and for (b) and9. The period of oscillation of T = 2 L/g. Measured value of L is 20.0 cm known to 1 mm accuracy and time for 100 oscillations of the pendulum is found to be 90 s using a wrist watch of 1 s resolution. What is the accuracy in the determination of g?10. 5.74 g of a substance occupies 1.2 cm3. Express its density by keeping the significant figures in view.11. Each side of a cube is measured to be 7.203 m. What are the total surface area and the volume of the cube to appropriate significant figures?12. Let us consider an equation where m is the mass of the body, its velocity, is the acceleration due to gravity and is the height. Check whether this equation is dimensionally correct.13. A student measures the thickness of a human hair by looking at it through a microscope of magnification 100. He makes 20 observations and finds that the average width of the hair in the field of view of the microscope is 3.5 mm. What is the estimate on the thickness of hair?