units and measurement

physical quantites

matter that can be measured

it is expressed with a number with a unit

it is of two types; fundamental and derived

fundamental quantities

also called base quantities. they are independent of each other

length - metre

mass - kilogram

time - second

electric current - ampere

thermodynamic temperature - kelvin

amount of substance - mole

luminous intensity - candela

derived quantities

made by multipl. and divis of fundamental quantities

eg: speed = length/time

radian and steradian

they are two other fundamental units

radian = plane angle = length of arc/ radius = ds/r

steradian = solid angle

prefixes of units

very large quantities can be expressed w a prefix before the unit signifying a power of 10

kilo = 10³

hecto = 10²

deka = 10^1

deci = 10^-1

centi = 10^-2

milli = 10^-3

micro = 10^-6

nano = 10^-9

pico = 10^-12

dimensions

an expression of a derived quantity by showing the powers of each fundamental quantity

eg: speed = L/T = L^1T^-1 [this is called dimensional formula]

homogeneity of dimensions in an equation

dimensions of every term in an equation is the same

different dimensions cannot be added & subtracted

no.s are dimensionless

eg: x = ut + 1/2at² => L = L + L

conversion of units

Q (quantity) = n1u1 = n2u2

=> n2 = [M1/M2]^a [L1/L2]^b [T1/T2]^c

finding equations using dimensions

Q = kQ1^a Q2^b Q3^c [k = dimensionless constant]

eg: time period of pendulum depends on length, mass, g

t = k l^a m^b g^c

T = k[L]^a[M]^b[LT^-2]^c

T = k[L]^a+c [M]^b [T]^-2c

a = 0 ; c = -1/2 ; b = ½

t = k square root l/g

limitations of finding equations using dimensions

the expression can only be product type (no + or -)

k is unknown

significant figures

they are all the certain digits & one uncertain digit (least count)

eg: 10.46cm has 4 SFs and 6 is uncertain

least count

it is the smallest subdivision of a scale & is uncertain when measuring

eg: mm is smallest subdivision of a cm scale (0.1 cm)

least count of screw gauge = pitch/ no. of divisions

order of magnitude

it is the expression of big numbers in the form N x 10^n where N is less than 10

eg: 95200000 = 9.52 × 10^5

rules to determine no. of SFs

1- all non-zero digits are SF [1234 = 4 SFs]

2- all zeros btw two non-zero digits are SF [10.07, 1007 = 4SFs]

3- in whole no.s, trailing zeros are NOT SF [3210 = 3 SFs]

4- if no. is less than 1, all zeros b4 the 1st non-zero digit are NOT SF [0.005704 = 4 SFs]

5- in decimals, trailing zeros are SF [3.500 = 4 SFs]

6- factors are exact & have infinite SF [r = d/2 => 2 = infinite SF]

7- no. of SF does not depend on the unit [164cm, 0.164m, 0.00164km = 3 SFs]

precision and accuracy

precision = closeness of 2 measurements to each other

accuracy = closeness of a measurement to the actual value

rounding off SFs

7.82 = 7.8

6.87 = 6.9

16.351 = 16.4

3.250 = 3.2

3.750 = 3.8

addition and subtraction of SFs

the result should have the same SF as the term with the least decimal places

multiplication and division of SFs

the result should have the same SFs as the term with the least SFs

relative error

eg: R.E of 9.24kg

m = 9.24 +- 0.01

RE = (0.01/9.24) x 100 = +- 0.1%

RE = (least count/meaurement) x 100

arial and linear magnification

arial magnification = area of image/ area of object

linear magnification = sq. root of arial magnification