Question

If the units of force, energy and velocity are \[10{\rm{ N}}\], \[100{\rm{ J}}\] and \[5{\rm{ }}{{\rm{m}} {\left/
 {\vphantom {{\rm{m}} {\rm{s}}}} \right.
 } {\rm{s}}}\] then the unit of mass is:
A. \[1{\rm{ kg}}\]
B. \[{\rm{2 kg}}\]
C. \[3{\rm{ kg}}\]
D. \[4{\rm{ kg}}\]

Answer 1

Hint: From the Einstein equation concept, we can say that a body's energy is directly proportional to the product of its mass and square of its velocity. In other words, we can say that this equation gave us the relationship between energy, speed, and mass of a body.

Complete step by step answer:
Given:
The value of force is \[F = 10{\rm{ N}}\].
The value of energy is \[E = 100{\rm{ J}}\].
The value of velocity is \[v = 5{\rm{ }}{{\rm{m}} {\left/
 {\vphantom {{\rm{m}} {\rm{s}}}} \right.
 } {\rm{s}}}\].
We know that the value of energy and mass of the body is given, and we can evaluate the mass of this body using the Einstein equation and hence unit of mass, which will be the final answer.
From the concept of Einstein's equation of energy, we can write:
\[E = m{v^2}\]
Substitute \[100{\rm{ J}}\] for E and \[5{\rm{ }}{{\rm{m}} {\left/
 {\vphantom {{\rm{m}} {\rm{s}}}} \right.
 } {\rm{s}}}\] for v in the above expression.
\[100{\rm{ J}} = m{\left( {5{\rm{ }}{{\rm{m}} {\left/
 {\vphantom {{\rm{m}} {\rm{s}}}} \right.
 } {\rm{s}}}} \right)^2}\]……(1)
We know that Joule can be expressed into its base units as below.
\[{\rm{J}} = {\rm{kg}} \cdot {{{{\rm{m}}^2}} {\left/
 {\vphantom {{{{\rm{m}}^2}} {{{\rm{s}}^2}}}} \right.
 } {{{\rm{s}}^2}}}\]
Substitute \[{\rm{kg}} \cdot {{{{\rm{m}}^2}} {\left/
{\vphantom {{{{\rm{m}}^2}} {{{\rm{s}}^2}}}} \right.
} {{{\rm{s}}^2}}}\] for Joule in equation (1).
\[
100{\rm{ J}} = m{\left( {5{\rm{ }}{{\rm{m}} {\left/
{\vphantom {{\rm{m}} {\rm{s}}}} \right.
} {\rm{s}}}} \right)^2}\\
\therefore m = 4{\rm{ kg}}
\]

Therefore, the value of mass is \[4{\rm{ kg}}\] , and option (D) is correct.

Note:Einstein gives the equation of energy in the special theory of relativity. According to this equation, Einstein said the body's mass and energy could be transformed into each other as they are directly proportional. Based on this equation's significance, the birth of the bomb and various energy systems took place. This equation is mainly used to establish a relationship between energy and mass of a body in the universe.