Question

A force vector applied on a mass is represented as $\vec F = 6\hat i - 8\hat j + 10\hat k$ and the magnitude of acceleration of the body is $1\,m{s^{ - 2}}$. What will be the mass of the body in $kg$?
(A) $10\sqrt 2 $
(B) $20$
(C) $2\sqrt {10} $
(D) $10$

Answer 1

Hint: Newton’s second law of motion shows the relationship between the force, acceleration, and mass of the moving object. By applying the force and acceleration value in Newton’s second law of motion equation, the mass of the moving object is determined.

Formula used:
Newton’s second law of motion,
$F = m \times a$
Where,
$F$ is the force of the body
$m$ is the mass of the object
$a$ is the acceleration of the object.

Complete step by step answer:
Given that,
A force vector applied on a mass is represented as $\vec F = 6\hat i - 8\hat j + 10\hat k$,
Here,
$\vec F$ is the force vector
The acceleration of the body, $a = 1\,m{s^{ - 2}}$
Newton’s second law of motion,
$F = m \times a\,...............\left( 1 \right)$
Here we need to find the mass of the body, so keep mass $m$ in one side and other terms in other side,
\[m = \dfrac{F}{a}\]
The force value is given in vector form, so the above equation is written as,
\[m = \dfrac{{\vec F}}{a}\,......................\left( 2 \right)\]
Substitute the force vector value and the acceleration value in the equation (2),
Then,
\[\Rightarrow m = \dfrac{{6\hat i - 8\hat j + 10\hat k\,}}{1}\]
Now the above equation is written as,
\[\Rightarrow m = 6\hat i - 8\hat j + 10\hat k\,\]
Now the equation is in vector form, to remove the vector form, take modulus on both sides,
\[\Rightarrow \,\left| m \right| = \left| {6\hat i - 8\hat j + 10\hat k} \right|\]
On further,
\[\Rightarrow \,\left| m \right| = \sqrt {{{\left( 6 \right)}^2} + {{\left( { - 8} \right)}^2} + {{\left( {10} \right)}^2}} \]
By using the square inside the square root,
\[\Rightarrow \,\left| m \right| = \sqrt {36 + 64 + 100} \]
By adding the terms inside the square root,
\[\Rightarrow \,\left| m \right| = \sqrt {200} \]
For easy simplification of square root, the above equation can be written as,
\[\Rightarrow \,\left| m \right| = \sqrt {100 \times 2} \]
On further simplification,
\[\Rightarrow \,\left| m \right| = 10\sqrt 2 \]

Then, the mass of the body, $m = 10\sqrt 2 \,kg$. Hence, the option (A) is correct.

Note:
In this solution, we use the modulus term for solving the force vector form to the magnitude. In another method, before substituting the force vector to Newton’s second law of motion equation, the magnitude of the force vector is calculated and substituted in Newton’s second law of motion equation to find the mass of the object.