Normal reactions at the bases of two light supports

• brotherbobby
In summary, when applying the equations ##\Sigma \vec F = 0## and ##\Sigma \vec \tau = 0##, we can determine the normal forces ##n_1## and ##n_2## needed to maintain equilibrium in this system. We can also use trigonometry to find the relationship between these forces and determine their values to be ##n_1 = 108 \; \text{N}## and ##n_2 = 192 \; \text{N}##. This shows that ##n_2## is greater than ##n_1##, which can be explained by the fact that ##n_2## has a smaller lever arm and must therefore be larger to counter
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?

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.

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.

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.

What is a normal reaction at the bases of two light supports?

A normal reaction is the force exerted by a surface on an object in contact with it, perpendicular to the surface. In the case of two light supports, it is the force that each support exerts on the object it is supporting at its base.

How is the normal reaction calculated?

The normal reaction is calculated using Newton's Third Law of Motion, which states that for every action, there is an equal and opposite reaction. In the case of two light supports, the normal reaction at each support is equal in magnitude but opposite in direction.

What factors can affect the normal reaction at the bases of two light supports?

The normal reaction can be affected by the weight of the object being supported, the angle of the supports, and the material and strength of the supports themselves. Other external forces, such as wind or vibration, can also affect the normal reaction.

Why is the normal reaction important to consider in engineering and construction?

The normal reaction is crucial in ensuring the stability and safety of structures. Engineers must calculate and design supports that can withstand the expected normal reaction at their bases to prevent collapse or failure of the structure. It also helps in determining the distribution of weight and load on the supports.

Can the normal reaction at the bases of two light supports be greater than the weight of the object being supported?

Yes, the normal reaction can be greater than the weight of the object being supported. This is because the normal reaction is not solely dependent on the weight of the object, but also on other factors such as the angle and material of the supports. However, the normal reaction cannot be greater than the total weight and load on the supports, as this would result in instability and potential failure of the supports.

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