Question

A calorie is a unit of heat or energy and heat equals about 4.2 J where 1 J = 1 Kg m² s⁻². Suppose we employ a system of units in which the unit of mass equals to a kg, the unit of length equals to b m and the unit of time is g s.Show that a calorie has a magnitude 4.2 a⁻¹ b⁻² G² in terms of the new units.

Answer 1

Hii dear,

# Proof-
Presently,
1 cal = 4.2 J = 4.2 kg.m^2/s^2 ...(1)

Suppose in new system fundamental units be Kg', m' and s' for mass, length and time respectively.
Kg' = a Kg Hence, Kg = Kg'/a
m' = b m Hence, m = m'/b
s' = g s Hence, s = s'/g

Putting these values in eqn (1)
1 cal = 4.2 (Kg'/a).(m'/b)^2.(s'/g)^-2
1 cal = 4.2 a^-1.b^-2.g^2 Kg'.m'^2.s'^-2

Hope this is helpful.

Answer 2

Answer:

Given that,

1 Calorie=4.2 J = 4.2 Kg m2 s-2 ...... (i)

As new unit of mass = α Kg

∴ 1 Kg = 1/α new unit of mass

Similarly, 1 m = β-1 new unit of length

1 s = γ-1 new unit of time

Putting these values in (i), we get

1 calorie = 4.2 (α-1 new unit of mass) (β-1 new unit of length)2 (γ-1 new unit of time)-2

= 4.2 α-1 β-2 γ2 new unit of energy

(Proved)