A system is composed of three objects with masses m, 5m and 6m. The coefficient of kinetic friction between the objects m, 5m and the surface is 0.5. Find the acceleration of the objects.

###### Solution.

First, let’s take a look at the system. The maximum acceleration of the object with mass 6m may be g. Then, the maximum acceleration of the object with mass m is 2g. We can write Newton’s second law for this object with mass m as

T – μmg = m2g

or

T = m2g + μmg = 2.5mg.

We can see that the maximum tension is not enough to move the object with mass 5m.

Thereby, we know that only the objects with masses 6m and m are moving.

Now, we can write Newton’s second law for these objects.

The acceleration of the object with mass 6m is a and the acceleration of the object with mass m is 2a.

For the object with mass 6m, the equation is

-2T + 6mg = 6ma (1)

and for the object with mass m, the equation is

T – μmg = m2a

or multiplying by 2 we can get

2T – 2*0.5mg = m4a (2)

We can add equations (1) and (2)

5mg = 10ma

or

a = 0.5g = 0.5*9.8 = 4.9 m/s^{2}.

Finally, the acceleration of the mass 6m is 4.9 m/s^{2} and acceleration of the mass m is g or 9.8 m/s^{2}.