A wheel of mass M has radius R. It is standing vertically on the floor, and we want to exert a horizontal force F at its axle so that it will climb a step against which it rests (Fig. 8–60). The step has height h, where h 6 R. What minimum force F is needed?

x2 = √[ R² - (R - h)² ] (M G) / (R - h)

Explanation:

Solution:

- Draw a picture of the wheel abutting the stair.  

- Draw a dashed line going from the wheel's midpoint horizontally toward the stair - call this unknown length "x" (this line actually stops at the vertical line that will be drawn in the next couple of steps).  

- Draw a dashed line from the midpoint to the point the wheel touches the stair - this hypotenuse length is R.  

- Connect these two lines with a vertical line to make a right-triangle - this vertical length "y" is (R - h).

- Now lets solve for x using pythagorean theorem:

                                          R² = x² + y²  

                                          x² = R² - y²  

                                          x = √[ R² - y² ]  

                                          x = √[ R² - (R - h)² ]

- Now we will compare similar triangles.  The triangle we just described relates distances.  The triangle against which we will compare the first relates forces.  Calling the two triangle's components as x1, y1, x2, y2, recall that:

                                           x1 / x2 = y1 / y2

Where, x1 = x ( Calculated above )

- Our y2 represents the force of gravity on the mass of the wheel.  Our x2 represents the horizontal force we will have to induce to match y2.

                                           y2 = m*g

- Then we can equate the two forces such that horizontal force is enough to overcome:

                                           (√[ R² - (R - h)² ]) / x2 = (R - h) / (M G)  

                                           x2 = √[ R² - (R - h)² ] (M G) / (R - h)

c. magnitude and direction