Solutions to Exercises on Introduction to Physics HC Verma's Concepts of Physics Part 1 (5-9)

 Q#5

Find the dimensions of Planck's constant h from the equation E = h𝞶 where E is the energy and 𝞶 is the frequency.      

Answer:
Since, E = h𝞶
h = E/𝞶
Energy = (Force)(distance) {same as work done}

[E] = [MLT⁻²][L] = [ML²T⁻²]

The dimensions of frequency, [𝞶] = [T⁻¹]
Hence the dimensions of Planck's constant,
[h] = [ML²T⁻²]/[T⁻¹] = [ML²T⁻¹]            

Q#6
Find the dimensions of
(a) specific heat capacity c,
(b) the coefficient of linear expansion α and
(c) the gas constant R.
Some of the equations involving these quantities are

Q = mc(T₂ – T₁), Lₜ = L₀[1 + α(T₂ – T₁)] and PV = nRT.      

Answer:
(a) Since Q = mc(T₂ – T₁)

c = Q/m(T₂ – T₁)

The dimension of (T₂-T₁) = The dimension of temperature = [K]

The dimension of mass, [m] = [M]

The dimensions of heat, Q = the dimension of energy = [ML²T⁻²]

Hence the dimensions of specific heat capacity,

[c] = [ML²T⁻²]/[M][K] = [L²T⁻²K⁻¹]  

(b) Since Lₜ = L₀ {1 + α(T₂ – T₁)}

Lₜ – L₀ = L₀α(T₂ – T₁)
α = (Lₜ – L₀)/{L₀(T₂ – T₁)}

The dimensions of (Lₜ – L₀) and L₀ = the dimensions of length = [L]
The dimensions of (T₂ – T₁) = dimension of temperature = [K]

Hence the dimensions of the coefficient of linear expansion, [α] = [L]/[L][K] = [K⁻¹]

(c) Since PV = nRT

R = PV/nT
The dimensions of pressure, [P] = Force/area = [MLT⁻²]/[L²] = [ML⁻¹T⁻²]

The dimensions of volume, [V] = [L³]
The dimension of temperature, T = [K]
The dimensions of amount of gas, n = [N]
Hence the dimensions of the gas constant,

[R] = [ML⁻¹T⁻²][L³]/[N][K]
      = [ML²T⁻²K⁻¹N⁻¹]             

Q#7
Taking force, length and time to be the fundamental quantities find the dimensions of
(a) density,
(b) pressure
(c) momentum, and
(d) energy.      

Answer:
Note that instead of mass force is taken as a fundamental quantity. Hence we shall express the mass in terms of force where ever it comes. For example, Force = mass x acceleration
mass = Force/acceleration

(a) Density = mass per unit volume = mass/volume

= Force/(acceleration x volume)
Dimension of force = [F]
Dimensions of acceleration = [LT⁻²]
Dimensions of volume = [L³]
Hence the dimensions of density = [F]/{[LT⁻²][L³]} = [FL⁻⁴T²]
             
(b) Pressure = Force/Area = [F]/[L²] = [FL⁻²]

(c) Momentum = Mass x velocity
     = {Force/(Acceleration)}{Length/Time}
     = {[F]/[LT⁻²]}{[L]/[T]}
     = [FL⁻¹T²][LT⁻¹]
     = [FT]

(d) Energy = Force x Distance = [F] x [L] = [FL]

Q#8
Suppose the acceleration due to gravity at a place is 10 m/s². Find its value in cm/(minute)².      

Answer:
1 m = 100 cm, and 1 s = 1/60 minute

Putting these in the units,
10 m/s² = 10 x (100 cm)/(1/60 minute)²
             = 1000 x 60² cm/(minute)²
             = 1000 x 3600 cm/(minute)²
 10 m/s = 3.6 x 10⁶ cm/(minute)²                

Q#9
The average speed of a snail is 0.020 miles/hour and that of a leopard is 70 miles/hour. Convert these speeds in SI units.      

Answer:
The SI unit of length is m (meter), and of time is s (seconds).

1 mile = 1.6 km = 1600 m, 1 hour = 3600 s

Hence 1 mile/hour = (1600 m)/(3600 s) = 4/9 m/s

So, average speed of a snail = 0.020 miles/hour

= 0.020(4/9) m/s = 0.08/9 m/s = 0.00889 m/s
= 8.89 x 10⁻³ m/s

And the average speed of a leopard = 70 miles/hour

= 70(4/9) m/s = 280/9 m/s = 31.11 m/s   

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