Show that the minimum force required to pull a wheel if Radius R and mass M over a step of height h, given R> h when the force is applied at the top of the wheel is F = Mg sqrt(h/2R-h)
Thank you so much
Dont need help thanks though
To show that the minimum force required to pull a wheel over a step of height h, given the radius R and mass M, and assuming R > h, when the force is applied at the top of the wheel is F = Mg * sqrt(h / (2R - h)), we can use the principles of mechanics and work.
Here's an explanation of how to derive this equation:
1. Start by considering the forces acting on the wheel. The weight of the wheel, given by Mg, is acting vertically downward. The minimum force required to pull the wheel over the step is also acting vertically upward.
2. When the force is applied at the top of the wheel, it creates a torque about the point where the wheel touches the step. This torque helps the wheel overcome the step.
3. To analyze the torque, consider the perpendicular distance between the point of application of the force and the point where the wheel touches the step. Since the force is applied at the top of the wheel, this distance is equal to the radius R.
4. The torque created by the force is given by Torque = Force * Distance. Therefore, the torque about the point of contact with the step is F * R.
5. To overcome the step, the torque created by the force must be equal to or greater than the torque created by the wheel's weight. The torque due to the weight is given by Weight * Distance, which is Mg * h.
6. Setting up the torque equation: F * R = Mg * h.
7. Solving for F: F = (Mg * h) / R.
8. However, to simplify this equation, we need to express h in terms of R.
9. Looking at the situation, we notice that the step cannot be higher than the radius of the wheel, so h < R.
10. Rearrange the equation from step 7 by multiplying numerator and denominator by √(2R - h): F = (Mg * h) * √(2R - h) / R * √(2R - h).
11. Simplify the equation by canceling out common terms: F = Mg * √(h / (2R - h)).
Therefore, the minimum force required to pull the wheel over a step, given R > h when the force is applied at the top of the wheel, is F = Mg * √(h / (2R - h)).