# Boxes A and B are in contact on a horizontal, frictionless surface. Box A has mass 20.0 kg and box B has mass 5.0 kg. A horizontal force of 250 N is exerted on box A. What is the magnitude of the force that box A exerts on box B?

The aim of this question is to understand and apply Newton’s laws of motion to moving objects.

According to Newton’s motion laws, a body can’t just move by itself. Instead, an agent called the force acts on a body to move it from rest or to stop it. This force causes the change in speed, thereby creating acceleration that is proportional to the mass of the body. In reaction to this force, the body exerts a reaction force on the object causing the first force. Both of these action and reaction forces have equal magnitudes with opposite directions such that they try to cancel out each other in a broader sense.

Mathematically, Newton’s second law of motion dictates that the relationship between force $F$ acting on a body of mass $m$ and the acceleration $a$ is given by the following formula:

$F \ = \ m a$

Given:

$\text{ Total Mass } \ = \ m \ = \ m_{ A } \ + \ m_{ B } \ = \ 20 \ + \ 5 \ = \ 25 \ kg$

$\text{ Total Force } \ =\ F \ = \ 250 \ N$

According to the second law of motion:

$F \ = \ m a$

$\Rightarrow a \ = \ \dfrac{ F }{ m }$

Substituting values in the above equation:

$\Rightarrow a \ = \ \dfrac{ 250 }{ 25 }$

$\Rightarrow a \ = \ 10 \ m/s^{ 2 }$

Since both boxes A and B are in contact with each other, both of them must move with the same acceleration. So for the case of box B:

$\text{ Mass of Box B} \ = \ m_{ B } \ = \ 5 \ kg$

$\text{ Acceleration of Box B} \ = \ a_{ B } \ = \ a \ = \ 10 \ m/s^{ 2 }$

According to the second law of motion:

$F_{ B } \ = \ m_{ B } a_{ B }$

Substituting values:

$F_{ B } \ = \ ( 5 ) ( 10 )$

$\Rightarrow F_{ B } \ = \ 100 \ N$

## Numerical Result

$F_{ B } \ = \ 50 \ N$

## Example

If the mass of box A was 24 kg and that of box B was 1 kg, how much force will be exerted on B in this case provided that the force acting on box A remains the same?

Given:

$\text{ Total Mass } \ = \ m \ = \ m_{ A } \ + \ m_{ B } \ = \ 24 \ + \ 1 \ = \ 25 \ kg$

$\text{ Total Force } \ =\ F \ = \ 250 \ N$

According to the second law of motion:

$F \ = \ m a$

$\Rightarrow a \ = \ \dfrac{ F }{ m }$

Substituting values in the above equation:

$\Rightarrow a \ = \ \dfrac{ 250 }{ 25 }$

$\Rightarrow a \ = \ 10 \ m/s^{ 2 }$

Since both boxes A and B are in contact with each other, both of them must move with the same acceleration. So for the case of box B:

$\text{ Mass of Box B} \ = \ m_{ B } \ = \ 1 \ kg$

$\text{ Acceleration of Box B} \ = \ a_{ B } \ = \ a \ = \ 10 \ m/s^{ 2 }$

According to the second law of motion:

$F_{ B } \ = \ m_{ B } a_{ B }$

Substituting values:

$F_{ B } \ = \ ( 1 ) ( 10 )$

$\Rightarrow F_{ B } \ = \ 10 \ N$