Question

The figure shows a heavy wheel of mass w that can rotate freely about the axis passing through the point O. The wheel is to be raised on the pavement AB by applying a minimum force. Show the direction of application of force and calculate its magnitude.

Answer 1

Given Data:

Weight of the wheel = w (acting downwards from center O)

Radius of the wheel = r

Height of the pavement = `r/2`

Pivot Point (Fulcrum) = Point A

1. Direction of Force:

To minimise the required force F, it must be applied at the maximum possible perpendicular distance from pivot A. Therefore, force F must be applied perpendicular to line OA in the upward direction.

Perpendicular distance for force F (d₁) = r

2. Perpendicular Distance for Weight (d₂) from Pivot A:

Using Pythagoras' theorem in the right-angled triangle formed between center O and pivot A:

Hypotenuse (OA) = r

Vertical side = `r - r/2 = r/2`

Horizontal distance (d₂) = `sqrt(r^2 - (r/2)^2)`

= `sqrt(r^2 - r^2/4)`

= `sqrt((3r^2)/4)`

= `sqrt3/2` r

Calculation (Principle of Moments about Point A):

Moment of Force F = Moment of Weight w

F × d₁ = w × d₂

F × r = `w xx (sqrt3/2)r`

F = `(sqrt(3)w)/2`