Question

A force produces an acceleration of \[4m/{{s}^{2}}\] in a body of mass \[{{m}_{1}}\] and the same force produces an acceleration of \[6m/{{s}^{2}}\] in another body of mass \[{{m}_{2}}\] . If the same force is applied to \[({{m}_{1}}+{{m}_{2}})\] then the acceleration will be
A. 1.6 \[m{{s}^{-2}}\]
B. 2 \[m{{s}^{-2}}\]
C. 2.4 \[m{{s}^{-2}}\]
D. 3.2 \[m{{s}^{-2}}\]

Answer 1

Hint: In this question we have been asked to calculate the acceleration of given two mass systems under the given conditions. It is given that A force acts on two bodies of mass \[{{m}_{1}}\]and \[{{m}_{2}}\]. This force produces acceleration of \[4m/{{s}^{2}}\] in body of mass \[{{m}_{1}}\] and \[6m/{{s}^{2}}\] in body of mass\[{{m}_{2}}\]. Therefore, to solve this question, we shall first write the equation to calculate force. We shall use Newton's second law to calculate the acceleration of the joint system.

Formula used: \[F=ma\]

Complete step by step solution:
It is given that a force say F acts on two bodies \[{{m}_{1}}\]and \[{{m}_{2}}\]. This produces acceleration of \[4m/{{s}^{2}}\] and \[6m/{{s}^{2}}\]respectively on both bodies.
Using Newton’s second law, we can write,
\[F={{m}_{1}}\times 4\]
Therefore,
\[{{m}_{1}}=\dfrac{F}{4}\] …………….. (1)
Similarly,
\[F={{m}_{2}}\times 6\]
Therefore,
\[{{m}_{2}}=\dfrac{F}{6}\] ……………….. (2)
Now, let the acceleration due to combine mass system \[({{m}_{1}}+{{m}_{2}})\] be ‘a’
From Newton’s second law, we know
\[F=ma\]
Substitute \[({{m}_{1}}+{{m}_{2}})\] in above equation
\[F=({{m}_{1}}+{{m}_{2}})a\]
Solving for a,
We get,
\[a=\dfrac{F}{({{m}_{1}}+{{m}_{2}})}\]…………………. (3)
From (1), (2) and (3)
We get,
\[a=\dfrac{F}{(\dfrac{F}{4}+\dfrac{F}{6})}\]
On solving,
\[a=2.4m/{{s}^{2}}\]

Therefore, the correct answer is option C.

Note:
Newton's second law states that when a constant force is applied on a body, the body is set in motion. The force causes the body to accelerate and change its velocity at constant rate. The direction of acceleration is the same as the direction of force causing the body to accelerate. The mathematical statement for second law is: For a body of mass m, the force applied on a body is directly proportional to the rate of change of velocity.