What forces act on the supports of a rectangle?

[hyperddude]

To anyone who saw my previous thread, yes, this is quite similar to it .

Homework Statement

Given a rectangle, say a painting, with with mass $m$, height $h$, and width $w$ with two point supports to a wall at its two upper corners, what force does each support exert?

Homework Equations

Moment of inertia of a rectangle about its center: $\frac{m(h^2+w^2)}{12}$
Moment of inertia of a rectangle about a corner: $\frac{m(h^2+w^2)}{3}$
^Not sure if those equations will be relevant

The Attempt at a Solution

Common sense and intuition tells us that the vertical component from each support will be $mg/2$. But is there a horizontal component? That's what I'm trying to find out. One solution I had in mind was to break the $mg$ downward force into components and try setting them as the forces by the pivots, but I ended up just going in circles.

 

[Dick]

To anyone who saw my previous thread, yes, this is quite similar to it .

Homework Statement

Given a rectangle, say a painting, with with mass $m$, height $h$, and width $w$ with two point supports to a wall at its two upper corners, what force does each support exert?

Homework Equations

Moment of inertia of a rectangle about its center: $\frac{m(h^2+w^2)}{12}$
Moment of inertia of a rectangle about a corner: $\frac{m(h^2+w^2)}{3}$
^Not sure if those equations will be relevant

The Attempt at a Solution

Common sense and intuition tells us that the vertical component from each support will be $mg/2$. But is there a horizontal component? That's what I'm trying to find out. One solution I had in mind was to break the $mg$ downward force into components and try setting them as the forces by the pivots, but I ended up just going in circles.

Sure there could be horizontal forces. But you know the horizontal forces must sum to zero. Otherwise the picture will accelerate in the horizontal direction.

 

[hyperddude]

You know the horizontal forces must sum to zero. Otherwise the picture will accelerate in the horizontal direction.

Yes, but I'm interested in finding what the horizontal force for one of the supports is.

 

[Dick]

Yes, but I'm interested in finding what the horizontal force for one of the supports is.

You can't find it. The horizontal forces can be anything as long as they cancel. How could you find it? If you know what direction the total force acts in, like if it's the tension supported by a string nailed to the wall, you might.

 

[Nickie]

As a scientist, it is important to approach problems like this using a combination of intuition and mathematical analysis. In this case, we first need to consider the forces acting on the rectangle as a whole. The weight of the rectangle, mg, will act downward towards the ground. Since the rectangle is supported at two points, there will be a reaction force from the supports acting upward to balance the weight.

Now, let's consider the forces acting on each support individually. Since the supports are at the upper corners of the rectangle, they will experience a force from the weight of the rectangle acting downwards. This force can be broken into two components - a vertical component, mg/2, and a horizontal component, 0. The vertical component is the force that we intuitively expect, as it balances the weight of the rectangle. The horizontal component, however, is canceled out by the reaction force from the other support.

In terms of moments, the supports will also experience a moment due to the weight of the rectangle. This moment can be calculated using the equations provided, and it will be the same for both supports since the weight of the rectangle is symmetrically distributed. However, the supports will also experience an additional moment due to the reaction force from the other support. This moment will be equal in magnitude but opposite in direction, resulting in a net moment of 0 for each support.

In summary, the forces acting on the supports of a rectangle are a vertical component, mg/2, and a moment due to the weight of the rectangle. There is no horizontal component, as it is canceled out by the reaction force from the other support.