Units & Dimensions Test - 4

Kilogram represent unit of physical quantity and not the physical quantity.

The light-year is a unit of length used to express astronomical distances and measures about 9.46 trillion kilometres or 5.88 trillion miles. As defined by the International Astronomical Union, a light-year is the distance that light travels in vacuum in one Julian year. 

The parsec (symbol: pc) is a unit of length used to measure large distances to astronomical objects outside the Solar System. A parsec is defined as the distance at which one astronomical unit subtends an angle of one arcsecond, which corresponds to 648000π astronomical units.

The SI system of units is a modern system and hence involves all the quantities that can't be derived using all the other quantities of the set. While the rest systems are old and local methods and hence are not scientifically accurate and explainable and thus only have three basic quantities while SI have 7.

Joule is the SI unit of energy

The SI unit of pressure is Pascal (represented as Pa) which is equal to one newton per square metre (N/m^-2 or kg m^-1s^-2). Interestingly, this name was given in 1971. Before that pressure in SI was measured in newtons per square metre.

The unit of inductance in the SI system is the henry (H), named after American scientist Joseph Henry, which is the amount of inductance which generates a voltage of one volt when the current is changing at a rate of one ampere per second.

Angular acceleration. Angular acceleration is the rate of change of angular velocity. In three dimensions, it is a pseudovector. In SI units, it is measured in radians per second squared (rad/s²), and is usually denoted by the Greek letter alpha (α).

We know that,       g = G. m₁ m₂ / r₂
Where we know g has dimensions of acceleration
Thus [G] = [ g.r² /m₁ m₂]
= N m²kg^-2

Unit of surface tension is Newton per metre

[F] = MLT^-2
[displacement/time]  = LT^-1
Thus we get [F x displacement/time] = ML²T^-3
Thus the answer is 2

Unit of angular frequency is rad/sec which can be said as angle/time. As angle is dimensionless and time has dimension T, we get the dimension of angular frequency as T^-1

Coefficient of viscosity (η)= Fr/Av   

 F= tangential Force, Area, r= distance between the layers, v= velocity.

Dimensional Formula of Force = M¹L¹T^-2.
Dimensional Formula of Area= M⁰L²T⁰.
Dimensional Formula of distance= M⁰L¹T⁰.
Dimensional Formula of velocity= M⁰L¹T^-1.

Putting these values in above equation we get,

[η]= [M¹L¹T^-2][M⁰L¹T⁰] / [M⁰L²T⁰] [M⁰L¹T^-1] = [M¹L^-1T^-1]

The dimensions of angular momentum are M L²T⁻¹
That of torque is  M L²T⁻²
Also dimension of energy is  M L²T⁻²
Where as same of force is  M LT⁻²
And of power is  M L²T⁻³
Thus we get torque and energy have the same dimensional formulas.

[P] = [F/A] = MLT^-2 / L²  = ML^-1T^-2
[F/V] = MLT^-2 / L3  = ML^-2T^-2
[E/V] = ML²T^-2 / L³  = ML^-1T^-2

[E] = [F][d]
= [P/T][A]^½ 
[E] = P¹A^½T^-1

Correct Answer :- b

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

We know that specific resistance, ⍴ = R A / L
Where R is net resistance, A is area and
L is length
Hence [⍴] = ohm x m.m / m
= ohm.m

Power = energy / time.
energy = power × time.
energy = watt × hour
= 1watt × 1hour
= 1 × (60×60)
= 3600
= 3.6 × 10³ joule

We know that density of Hg = 13600 kg/m.m.m
= 13600 x 1000 g / 100cm x 100cm x 100cm
= 13.6 g/cm.cm.cm

Intensity of magnetisation : It represents the extent to which a specimen is magnetised when placed in a magnetising field. Or in other words the intensity of magnetisation of a magnetic material is defined as the magnetic moment per unit volume of the material. 

It indicates how the sample affected by magnetic field when placed in it.

M = Magnetic moment/volume = μM / V

unit SI : M= Amp.metre^-1

We know that coefficient of viscosity (η)= Fr/Av where F = tangential force, r = distance between the layers , v = velocity and A is the area of the surface.
 Thus we get [η] = MLT^-2.L / L². (L/T)
= M / LT
Thus its unit is kg / m sec

As 'at' and 3 are added in the equation, we get at and 3 have same dimensions i.e.1
Thus a has dimensions same as 1/t.

We know that 10⁵ dyne = 1N
And 10⁴cm²  = 1 m²
Thus we get 10 dyne / cm² = N / m²
Hence 10⁶ dyne / cm² =10⁵ N / m²

The result of an addition or subtraction in the number having different precisions should be reported to the same number of decimal places as present in the number having the least number of decimal places.

The final result should, therefore, be rounded off to one decimal place, i.e. 663.8

We have  1 unit = x meters
So  1 unit² = x² meter²
Hence, we get  1 meter² = 1/x² unit²

multiply the mass / volume value by 1000

v²r/g= (L¹T^-1)2L1/LT^-2=L²
v²/rg= (LT^-1)² L1(L¹ T^-2) ​=M⁰L⁰T⁰
v²g/r​=(LT^-1)2 LT^-2/L​=L²T^-4
v²rg=(LT^-1)²(L1) (LT^-1) =L⁴T^-3
So, option D is correct.

G = 6.67 x 10-11 Nm2/kg2

so, when expressed in VGS units we shall convert N to dynes, m to cm and kg to g, thus

G = 6.67 x 10-11 x [(105dynes x (102)2cm2 ) / (103)2 g2]

=  6.67 x 10-11 x [ (105 x 104 ) / 106 ] 

Thus,

in CGS system

G = 6.67 x 10-8 dyne.cm2/g2