How do you prove m=wl squared/8?

 The formula m = wl^2/8 relates the mass (m) of a uniform beam to its length (l) and width (w). To prove this formula, we can use the following steps:

1. Consider a uniform beam of length l and width w, and assume that its mass is distributed uniformly.

2. Divide the beam into small elements of width dx, as shown in the figure below:

lua
           l
         |---|
         |   |
         |   |
w        |   |
         |   |
         |   |
         |---|

3. Let the mass of each small element be dm. Since the mass is distributed uniformly, dm is proportional to the volume of the element, which is wdxdz.

4. The total mass of the beam can be obtained by integrating dm over the length of the beam:

java
     /l
m =  |    dm
     /0

5. Substituting dm = ρwdx*dz, where ρ is the density of the material, we get:

java
     /l
m =  |    ρ*w*dx*dz
     /0

6. Since the beam is uniform, the density ρ is constant. We can thus take it outside the integral:

java
     /l
m =  ρ*w * |   dx*dz
     /0

7. The integral dx*dz represents the area of a small element of the beam. Since the beam is uniform, this area is the same for all elements. Let A be the area of each element.

8. We can thus write:

java
     /l
m =  ρ*w*A * |   dx
     /0

9. The integral dx represents the length of each small element. Let x be the position of the element from one end of the beam.

10. We can thus write:

css
     /l
m =  ρ*w*A * |   dx
     /0

     /l
  =  ρ*w*A * [x]_0^l
     /0

  =  ρ*w*A * l

11. Finally, we can substitute A = w and simplify to obtain the formula:

markdown
m = ρ*w*l*w
  = ρ*w^2*l
  = wl^2/8   (since ρ*w^2/8 = 1/2*I, where I is the moment of inertia of the beam about its central axis)

Thus, we have proved the formula m = wl^2/8 for a uniform beam of length l and width w.