# Two stacked blocks, push using one block, static friction between blocks

1. Sep 30, 2008

### dammitpoo

Newtonian mechanics problem with blocks

Problem:

$$m_{1}$$ and $$m_{2}$$ each interact with $$m_{3}$$ via static friction, with the same $$\mu_{s}$$. The horizontal surface below $$m_{3}$$ is frictionless. An external force $$F_{ext}$$ acts on $$m_{1}$$ from the left and the entire system of three connected masses moves to the right (and obviously accelerates). The idea is to provide a suitable magnitude of $$F_{ext}$$ as to prevent both $$m_{1}$$ and $$m_{2}$$ from moving with respect to $$m_{3}$$ during the acceleration, i.e. we don’t want $$m_{1}$$ to slide down along, nor $$m_{2}$$ to slide “back” along $$m_{3}$$. To make life easy, we let $$m_{1}$$, $$m_{2}$$ and $$m_{3}$$ all have the same mass $$m$$.

a) Find in terms of relevant parameters, the possible range of $$F_{ext}$$ which will allow the desired motion to take place.
b) It might be that, if $$\mu_{s}$$ is less than some critical value $$\mu_{s(critical)}$$, no value of $$F_{ext}$$ will allow the desired motion. Give a simple argument why this might be true, and if so, determine $$\mu_{s(critical)}$$ in terms of relevant parameters.

Relevant equations:

$$\sum F = ma$$

Here is my attempt at the problem:

Part A:
For $$m_{2}$$:
$$N_{2} = m_{2}g$$
$$F_{fr2} = m_{2}a$$
$$\mu_{s}m_{2}g = m_{2}a$$
$$\mu_{s}g = a$$
$$\mu_{s} = \frac{a}{g}$$

For $$m_{1}$$:
$$F_{ext} - N_{1} = m_{1}a$$
$$N_{1} = F_{ext} - m_{1}a$$
$$F_{fr1} = m_{1}g$$
$$\mu_{s} (F_{ext} - m_{1}a) = m_{1}g$$

For $$m_{3}$$:
$$N_{3} - F_{fr1} - N_{2} = m_{3}g$$
$$N_{1} - F_{fr2} = m_{2}a$$

Substitute in for $$N_{1}$$ and $$F_{fr2}$$:
$$N_{1} - \mu_{s}m_{2}g = m_{2}a$$
$$F_{ext} - m_{1}a - \mu_{s}m_{2}g = m_{2}a$$
$$F_{ext} = m_{3}a + m_{1}a + \mu_{s}m_{2}g$$

Since $$\mu_{s}m_{2}g = m_{2}a$$:
$$F_{ext} = m_{3}a + m_{1}a + m_{2}a$$

Since $$m_{1} = m_{2} = m_{3} = m$$:
$$F_{ext} = 3ma$$

Part B:
My guess is that if $$\mu_{s}$$ is infinitely small so that friction is negligible, any magnitude of force applied on the blocks would cause block 1 to slide down and block 2 to slide backwards relative to block 3.

I don't know where to start with the parameters, but here is what I have so far:
$$\mu_{s(critical)} < \mu_{s}$$
$$F_{fr(critical)} < F_{ext} < F_{fr}$$

Any help would be highly appreciated!

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