Normal reactions at the bases of two light supports

[brotherbobby]

Homework Statement

Shown in the figure below are two light plastic supports pinned at their top. One of the supports is 4 m long while the other is 3 m long and they are arranged to form a right angled triangle with their base 5 m. A mass of 300 N is hung from the pin at the top. If the ground is frictionless, calculate the reactions $n_1$ and $n_2$ at the bases of the two supports.

Relevant Equations

For equilibrium, $\Sigma \vec F = 0$ for the system as a whole and $\Sigma \vec \tau = 0$ for any point on the system where torque $\vec \tau = \vec r \times \vec F$.

For equilibrium, using $\Sigma \vec F = 0$, we get $n_1 + n_2 = 300\; \text{N}$.

Taking the system as a whole and applying $\Sigma \vec \tau = 0$ about the hinge (pin) at the top from where the load is hung, we get $n_1 \times (0.8) \times 4 = n_2 \times (0.6) \times 3$, by taking those components of the normal forces perpendicular to the supports and using trigonometry.

Hence, $3.2 n_1 = 1.8 n_2 \Rightarrow n_2 = \frac{16}{9} n_1$.

Thus, going to the earlier equation, $n_1 + \frac{16}{9} n_1 = 300 \Rightarrow \frac{25}{9} n_1 = 300 \Rightarrow \boxed{n_1 = 108 \; \text{N}}$.
Also, $\boxed{n_2 = 192 \; \text{N}}$.

Is this right?

Even if it is, is there a physical explanation as to why $n_2 > n_1$? Can we say it is because $n_2$ has a smaller lever arm than $n_1$ and therefore has to be greater in order to nullify the torque produced by $n_1$ about the hinge?

 

[hutchphd]

Or that the mass of the "object" is horizontally closer to point 2 which therefore sees more force. There are a hundred ways to say it! Very nicely done.

 

[haruspex]

Another approach is to take moments about each support point in turn and dispense with the vertical force balance equation. This has the slight advantage of avoiding having to solve a pair of simultaneous equations, but it is more useful when you only need to find one reaction force.

 

[PhanthomJay]

Where did this problem come from? If there is a true pin at the Top, and if the supports at their base sit on frictionless ground, this system will instantly collapse.