Question

A body, under the action of a force $\vec F = 6\hat i - 8\hat j + 10\hat k$ , acquires an acceleration of $1\,m{s^{ - 2}}$. The mass of this body must be:
(A) $10\sqrt 2 \,kg$
(B) $2\sqrt {10} \,kg$
(C) $10\,kg$
(D) $20\,kg$

Answer 1

Hint: The mass is determined by using the relation of the force and acceleration because in the question the force and the acceleration are given. The force equation gives the relation of the mass and acceleration, by using the force equation, the mass of the body is determined.

Useful formula
The force equation is given by,
$F = ma$
Where, $F$ is the force of the object, $m$ is the mass of the object and $a$ is the acceleration of the object.

Complete step by step solution
Given that,
The body is in the action of force, then the force is, $\vec F = 6\hat i - 8\hat j + 10\hat k$,
The acceleration of the force is, $a = 1\,m{s^{ - 2}}$
Now,
The force equation is given by,
$F = ma\,.................\left( 1 \right)$
But the force is given in the vector form, so we have to change the vector form of the force to the magnitude, so it can be done by taking the square root of the sum of the individual square of the i, j and k components. Then
$\left| {\vec F} \right| = \sqrt {{6^2} + {{\left( { - 8} \right)}^2} + {{10}^2}} $
By using the square in the above equation, then the above equation is written as,
$\left| {\vec F} \right| = \sqrt {36 + 64 + 100} $
By adding the terms in the above equation, then the above equation is written as,
$\left| {\vec F} \right| = \sqrt {200} $
To make the square root easy, then the above equation is written as,
$\left| {\vec F} \right| = \sqrt {100 \times 2} $
Now by taking the square root, then the above equation is written as,
$\left| {\vec F} \right| = 10\sqrt 2 \,.....................\left( 2 \right)$
By substituting the equation (2) in the equation (1), then the equation (1) is written as,
$10\sqrt 2 = ma$
By keeping the mass of the object in one side and the other terms in other side, then
$m = \dfrac{{10\sqrt 2 }}{a}$
By substituting the acceleration value in the above equation, then
$m = \dfrac{{10\sqrt 2 }}{1}$
By dividing the above equation, then
$m = 10\sqrt 2 \,kg$

Hence, the option (A) is the correct answer.

Note: When the force is given in the vector form, the conversion of the vector equation to the magnitude value is not much difficult. It can be easily done by taking the square root of the sum of the individual squares of the $\hat i$ , $\hat j$ and $\hat k$ components.