. A weight W is to be raised by means of a lever with the force F at one end and the point of support at the other. If the weight is suspended from a point at a distance a from the point of support, and the weight of the beam is w pounds per linear foot, what should be the length of the lever in order that the force required to lift it shall be a minimum?
To find the length of the lever that minimizes the force required to lift the weight, we need to consider the principle of moments.
The principle of moments states that for an object to be in equilibrium, the sum of the clockwise moments must be equal to the sum of the anticlockwise moments about any point.
Let's consider a point P on the lever at a distance x from the point of support. The moment due to the weight W about point P is given by the equation:
M1 = W * (a - x)
The moment due to the force F about point P is given by the equation:
M2 = F * x
Since we want to minimize the force F, we need to find the value of x that minimizes M2.
By applying the principle of moments, we can set M1 equal to M2:
W * (a - x) = F * x
Simplifying this equation will help us find the relationship between x and other variables:
W * a - W * x = F * x
Now, we can isolate x:
W * a = F * x + W * x
W * a = x * (F + W)
Divide both sides of the equation by (F + W):
x = (W * a) / (F + W)
Therefore, the length of the lever (x) that minimizes the force required to lift the weight is given by the equation:
x = (W * a) / (F + W)