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Chapter 12: Problem 53

What horizontal force applied at its highest point is necessary to keep a wheel of mass \(M\) from rolling down a slope inclined at angle \(\theta\) to the horizontal?

Short Answer

Expert verified
The horizontal force necessary to keep the wheel from rolling down the slope is equal to the component of wheel's weight parallel to the slope and it is \(Mg\sin\(\theta\)\).

Step by step solution

01

Identify the Forces acting

There are three forces acting on the wheel: the weight of the wheel (\(Mg\)), the normal force (\(N\)), and the horizontal applied force (\(F\)). The weight has two components, one perpendicular and one parallel to the inclined plane. The perpendicular component is \(Mg\cos\(\theta\)\) and the parallel component is \(Mg\sin\(\theta\)\). The normal force counterbalances the perpendicular component of the weight and the frictional force (\(f\)) counterbalances the parallel component of the weight and the applied force.
02

Setting up the Equations

The wheel is in equilibrium meaning the sum of the forces equals zero. Set up the following equations based on the forces in the vertical and horizontal direction: a. Vertical direction: \(N - Mg\cos\(\theta\) = 0\), and b. Horizontal direction: \(F = Mg\sin\(\theta\) + f\).
03

Need for keeping wheel from rolling

The wheel needs to be kept from rotating. So the sum of the torques must also be zero. In this case, the friction force causes a torque that opposes that caused by the applied force. This gives the equation: \(fR - FR = 0\) where \(R\) is the radius of the wheel.
04

Solve for the Applied Force

Combine the equations from steps 2 and 3: equating \(f\) from both equations (as there's no rotation, frictional force equals the applied force times radius of the wheel), you can solve for the applied force with the equation \(F = Mg\sin\(\theta\) + FR \). Since \(R\) gets cancelled out in the equations, \(F = Mg\sin\(\theta\)\), so the force necessary is equal to the component of the wheel's weight parallel to the slope.

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