Question

(a) Define 1 kgf. How is it related to the S.I unit of force?
(b) Calculate refractive index of air with respect to diamond if the refractive index of diamond with respect to air is 2.4.

Answer 1

Kg-force is the gravitational unit of force. It is the force at which the earth attracts the body. The earth’s attraction is the force of gravity, the acceleration produced by it is called the acceleration due to gravity, g and it is always directed towards the centre of the earth.
Hence, from Newton's second law, $F = ma$, where weight of a body of mass m is formulated as $W = mg$, where g= acceleration due to gravity at the place. The value of g varies at different places and hence also the value of gravitational force/weight also changes.
When a light passes from a medium a into another medium b, the ratio of the sine of the angle of incidence (i) to the sine of the angle of refraction (r) is called the refractive index of medium b with respect to medium a, is denoted by $_a{\mu _b}$, so,
$_a{\mu _b} = \dfrac{{\sin i}}{{\sin r}}$, this is called a relative refractive index.
Since light path is reversible, so,
$_b{\mu _a} = \dfrac{{\sin r}}{{\sin i}}$, where, $_b{\mu _a}$= refractive index of medium a with respect to medium b.
So, $_a{\mu _b}{ \times _b}{\mu _a} = \dfrac{{\sin i}}{{\sin r}} \times \dfrac{{\sin r}}{{\sin i}} = 1$
Now, $_a{\mu _b} = \dfrac{1}{{_b{\mu _a}}}$

Complete step by step answer:
(a) 1kgf is the force with which the earth attracts a body of mass 1kg towards its centre.
1kgf is the weight of a mass of 1 kg at a place where acceleration due to gravity,$g = 9.80665m.{s^{ - 2}}$.
Now, $W = mg \\\ \Rightarrow 1kgf = 1kg \times 9.80665m \cdot {s^{ - 2}} = 9.80665kg \cdot m \cdot {s^{ - 2}} = 9.80665N \\\$
So, $1kgf = 9.80665N$, this is the required relation.

(b) The refractive index of diamond with respect to air, $_{air}{\mu _{diamond}} = 2.4$
Let, refractive index of diamond with respect to air=$_{diamond}{\mu _{air}}$
From the relation in the hint, $_a{\mu _b} = \dfrac{1}{{_b{\mu _a}}}$, where a and b are two mediums.
So, $_{diamond}{\mu _{air}} = \dfrac{1}{{_{air}{\mu _{diamond}}}}$
${ \Rightarrow _{diamond}}{\mu _{air}} = \dfrac{1}{{2.4}} = \dfrac{5}{{12}} = 0.417$
Then, refractive index of diamond with respect to air= 0.417

Note:
(a) 1kg-wt and 1kgf units practically have the same value. Only when very precise measurements are needed, kgf and kg-wt can be differentiated. 1kg-wt is variable as the value of g is different at different places on earth.
(b) Refractive index of medium a with respect to medium b${ = _b}{\mu _a} = \dfrac{{{v_a}}}{{{v_b}}}$ where, $v_a$= velocity of light in medium a and $v_b$= velocity of light in medium b.
$_b{\mu _a} = \dfrac{{{v_a}}}{{{v_b}}} = \dfrac{{\dfrac{c}{{{v_b}}}}}{{\dfrac{c}{{{v_a}}}}} = \dfrac{{{\mu _b}}}{{{\mu _a}}}$