# If the force of friction is 20 N for a moving body does acceleration increase with the increase of the mass?

A body is moving and the only force acting on it is the force of friction.

If we put another body on this moving body somehow the force of friction will increase;

does its acceleration change?

## 2 Answers

The force of friction will increase with the addition of the second mass. However, the mass of the system will increase as well, and the net effect is that the acceleration does not change. To see this, note that:

$$a={F \over m}={\mu _kmg \over m}=\mu _kg$$

Where $\mu _k$ is the coefficient of kinetic friction between the moving body and the surface it's sliding on. Notice that it is completely independent of the mass and the frictional force. Thus, changing the mass has no effect on the acceleration.

• when we put the second body will the first body act on it? because it should stay in rest! Feb 26 '15 at 8:34
• My answer relies on the stipulation that no external work is done on the block/floor system when the 2nd is added. If you drop the second block straight down, i.e. with no horizontal velocity, then this assumption is untrue. In that case, when the 2nd block is dropped, the 1st must spend some of its kinetic energy to give kinetic energy to block 2 as well as thermal energy when block 2 momentarily slips while being accelerated. This will temporarily increase the deceleration. If however, you brought block 2 up to speed before dropping it, then the acceleration would not change. Feb 26 '15 at 9:13
• @wgrenard You should add your comment to your answer for clarity and permanence. Oct 28 '15 at 20:55

How the acceleration change may depends on the relation between the two bodies' mass and the coefficient of friction of the surfaces. I will assume the initial friction is exerted by ground. In the following I use U means upper body (your second body), L means lower (your first body), G means ground, 2 stands for 'to'(or any other prepositions).
1.If U is put with initial velocity lower than that of L which means the there's friction between 1st and the 2nd body.
So in this case, L will decelerate because of the friction from both ground and the U.
$a_{L}=\frac{F_{U2L}+F_{L2G}}{m_L}$.
After the upper one get its velocity the same with the lower one, the friction between the two bodies disappears but there's still friction from the ground.Then to the whole system incl. the two bodies, the $a_{final}=\frac{F}{m_{total}}=\frac{\mu m_{total} g}{m_{total}}=a_{original}$
2.If U is put with initial velocity higher than L's,there's still friction between the U and L. But there's difference.The changed accerleration can be calculated in the following expression.
$a_{L}=\frac{F_{L2G}-F_{U2L}}{m_L}$. Note this expression, if the value is positive, it means L decelerate, if it's positive, it accelerates. And also after velocities of U and L get the same, the acceleration of the whole system gets the same as original.
3.If U has exactly same velocity as L, that's easy, the acceleration will keep.