Question

A force produces an acceleration of ${a_1}$ in a body and the same force produces an acceleration of ${a_2}$ in another body. If the two bodies are combined and the same force is applied on the combination, then the acceleration produced in it is
A. ${a_1} + {a_2}$
B. $\dfrac{{{a_1} + {a_2}}}{{{a_1}{a_2}}}$
C. $\dfrac{{{a_1}{a_2}}}{{{a_1} + {a_2}}}$
D.$\sqrt {{a_1}{a_2}}$

Answer 1

In this question, we will assume a force F and let it be applied on each of the two masses ${m_1}$ and ${m_2}$separately. We will calculate the corresponding acceleration and then form a body having mass m = ${m_1}$+${m_2}$. And then calculate the acceleration when the same force applied by using the formula F=ma.

Complete answer:
Let us assume two bodies having masses ${m_1}$ and ${m_2}$ and F be the force which is applied on the bodies separately.
We know that according to Newton’s second law of motion:
F=ma. (1)
Now on applying this on body of mass ${m_1}$, we have:
F = ${m_1}$ ${a_1}$
On dividing both sides by ${m_1}$, we get:
${m_1} = \dfrac{F}{{{a_1}}}$ (2)
Now on applying this on body of mass ${m_2}$, we have:
F = ${m_2}$ ${a_2}$
On dividing both sides by ${m_2}$, we get:
${m_2} = \dfrac{F}{{{a_2}}}$ (3)
The equivalent mass m = ${m_1} + {m_2}$.
Now on applying equation 1 on this equivalent mass, we have:
F = (${m_1} + {m_2}$)a
On dividing both sides by${m_1} + {m_2}$, we get:
${a_{}} = \dfrac{F}{{{m_1} + {m_2}}}$
Putting the value of ${m_1}$ and ${m_2}$in above equation, we get:
${a_{}} = \dfrac{F}{{\dfrac{F}{{{a_1}}} + \dfrac{F}{{{a_2}}}}} = \dfrac{{{a_1}{a_2}}}{{{a_1} + {a_2}}}$

So, the correct answer is “Option C”.

Note:
In this question, you should know about the Newton law of motion, especially Newton’s second law. Newton’s second law states that acceleration produced in the body of mass m is directly proportional to force applied and inversely to the mass of the body. i.e. F = ma. If F is constant, the acceleration produced is inversely to mass of the body.