Question

A force of 5N gives a mass ${m_1}$ an acceleration of $8\,{\text{m}}{{\text{s}}^{ - 2}}$ and a mass ${m_2}$ an acceleration of $24\,{\text{m}}{{\text{s}}^{ - 2}}$ . What acceleration would it give if both the masses are tied together?
(A) $6\,{\text{m}}{{\text{s}}^{ - 2}}$
(B) $7\,{\text{m}}{{\text{s}}^{ - 2}}$
(C) $8\,{\text{m}}{{\text{s}}^{ - 2}}$
(D) $5\,{\text{m}}{{\text{s}}^{ - 2}}$

Answer 1

First of all, we will find out the individual mass of the two bodies by applying Newton’s second law of motion. After that we will find the combined mass of the system. Again, Newton’s law of motion is helpful in finding the net acceleration of the combined system.

Complete step by step solution:
we need to understand the situation. We can clearly see in the question that the magnitude of the mass of the two bodies are not given. So, without knowing the respective masses of the two bodies we cannot find the net acceleration of the two bodies when tied together.
Let us proceed to solve the problem. First, we will try to find out the mass of the two bodies.
We will apply the Newton’s second law of motion, whose equation is given by:
$F = ma$ …… (1)
Where,
$F$ indicates the force acting on the body.
$m$ indicates the magnitude of the mass of the body.
$a$ indicates the acceleration of the body.
So, for the first case, we apply the equation (1) and we get:
$5 = {m_1} \times 8 \\\ \Rightarrow {m_1} = \dfrac{5}{8}\,{\text{kg}}$
Therefore, the mass of the first body is $\dfrac{5}{8}\,{\text{kg}}$ .
Again, we apply the equation (1) for the second case too and we get:
$5 = {m_1} \times 24 \\\ \Rightarrow {m_1} = \dfrac{5}{{24}}\,{\text{kg}}$
Therefore, the mass of the second body is $\dfrac{5}{{24}}\,{\text{kg}}$ .
So, we will now calculate the total mass of the bodies when they are tied together:
$M = {m_1} + {m_2}$
Where,
$M$ indicates the combined mass.
$\therefore M = {m_1} + {m_2} \\\ \Rightarrow M = \dfrac{5}{8} + \dfrac{5}{{24}} \\\ \Rightarrow M = \dfrac{{15 + 5}}{{24}} \\\ \Rightarrow M = \dfrac{5}{6}\,{\text{kg}}$
So, the combined mass is found to be $\dfrac{5}{6}\,{\text{kg}}$ .
Now, we apply the equation (1) and we get:
$F = Ma \\\ \Rightarrow 5 = \dfrac{5}{6} \times a \\\ \Rightarrow a = 5 \times \dfrac{6}{5} \\\ \therefore a = 6\,{\text{m}}{{\text{s}}^{ - 2}}$
Hence, the acceleration of the two masses when tied together $6\,{\text{m}}{{\text{s}}^{ - 2}}$ .

The correct option is A.

Note: While solving the problem, it is important to remember that since the force remains constant in the current scenario, so the net acceleration will decrease as compared to the previous magnitude of acceleration. For a constant force, higher the magnitude is, lower is its acceleration. Again, most of the students tend to make mistakes by taking the average of the two values given, which is completely wrong.