Reaching Equilibrium?????

Question

Given that the mass of block A is 8.30 kg and the coefficient of static friction between block A and the surface on which it rests is 0.340, what is the maximum mass block B can have and the system still remain in equilibrium? (Both blocks are at rest).

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Thanks sooo much for your help!! God Bless!

Answer 1

If the system is in equilibrium, acceleration is 0 so the net force on every object must be 0 too.

First let's deal with gravity.

Force = Mass x Acceleration

for gravity, at the surface of the Earth the acceleration is "g", which has a value of about 9.8 m/s^2. This allows us to go back and forth between masses and weights.

Now look at block A.

The only two vertical forces are its weight Wa and the normal force N. They are pointing in opposite directions and the net force is 0 so:

Wa = N

There are also two horizontal forces, the tension on the rope, Ta and the force of friction Da. They are pointing in opposite directions and cancel so:

Ta = Da

We also know that the frictional force is related to the normal force by the coefficient of friction Cs:

Da = N x Cs

Next look at block B. It has only two forces acting on it, its weight, Wb, and the tension of the rope Tb. They are in opposite directions and cancel so:

Tb = Wb

Finally, there is the knot where the three ropes come together. It has three forces, the three tensions acting on it.

Ta acts in the horizontal, Tb acts in the vertical, but Tw acts at an angle so it has both horizontal and vertical components.

If @ is the angle of the rope to the horizontal, then the horizontal component of Tw is:
Th = Tw x cos @

and the vertical component is
Tv = Tw x sin @

Again, the horizontal and vertical components all cancel so:

Ta = Th
Tb = Tv

Now you have all the equations. Just solve them in sequence:

from the mass of block A you get Wa
from Wa so you also know N
you know Cs so you also know Da
with Da you also know Ta
you know @ and so can compute cos @
with that and Ta you can compute Tw
with Tw and sin @ you get Wb
from Wb you get the mass of B.