Pressure Profile

There is a 10 kg10 \text{ kg} box of length, height, and width (L=2 m,H=0.5 m,W=1 m)(L = 2\text{ m}, H = 0.5\text{ m}, W = 1\text{ m}) sitting on a rough surface.

A 20 N20\text{ N} rightward horizontal force is applied at the middle of the left side of the box at a distance of H2\frac{H}{2} from the bottom of the box. There is sufficient friction that the box remains stationary.

Let the variable xx represent the horizontal position (in meters) relative to the left side of the box. With the horizontal force applied, the upward pressure (\big(in N/m2)\text{N/m}^{2}\big) associated with the normal reaction varies according to the equation P(x)=αx+βP(x) = \alpha x + \beta.

Determine βα\frac{\beta}{\alpha}, to 2 decimal places.

Details and Assumptions:

  • There is a uniform downward gravitational acceleration of 10 m/s210\text{ m/s}^{2}.
  • Assume that the box is solid and has uniform mass density.
  • Linear pressure variation is an assumption made by the problem author. See here for further discussion.

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