Question

In leverage, in which way, we need to use more force:

1. when the pivot is closer to the load.
2. when the pivot is closer to the force

Answer 1

In a lever, more effort is required when the distance between the pivot and the effort force is reduced.

Step-by-step explanation:

Let
`F_(1)` denote the effort, let
`r_(1)` denote the distance between the pivot and the position where the effort is applied, and let
`\theta_(1)` denote the angle between this force and the lever (
`r_(1) > 0` and
`\theta_(1) > 0^(\circ)`.)

Let
`F_(2)` denote the force that the load exerts on the lever. Let
`r_(2)` denote the distance between the pivot and the position where the load is applied to the lever. Let
`\theta_(2)` denote the angle between the lever and the force that the load applies on the lever.

If
`F_(1)` and
`F_(2)` are the only forces on this lever, the following is required for the lever to be balanced:

`F_(1)\, r_(1)\, \sin(\theta_(1)) = F_(2)\, r_(2)\, \sin(\theta_(2))`.

The question is asking about changes in the value the effort,
`F_(1)`. Hence, rearrange the equation above to find an expression for the effort,
`F_(1)\!`:

`\displaystyle F_(1) = (F_(2)\, r_(2)\, \sin(\theta_(2)))/(r_(1)\, \sin(\theta_(1)))`.

Assume that the directions of effort and load do not change, such that the angles
`\theta_(1)` and
`\theta_(2)` between the two forces and the lever stay the same. Assume that the force the load exerted on the lever,
`F_(2)`, is also constant. Rearrange the expression for
`F_(1)` to obtain:

`\displaystyle F_(1) = \left((F_(2)\, \sin(\theta_(2)))/(\sin(\theta_(1)))\right)\, (r_(2))/(r_(1))`.

In other words, under the assumptions, the effort on the lever would be proportional to the ratio
`(r_(2) / r_(1))`.

The question requires that the magnitude of
`F_(1)` should be increased by reducing either
`r_(2)` (distance between load and pivot) or
`r_(1)` (distance between effort and pivot.) Since
`F_(1)\!` is proportional to the fraction
`(r_(2) / r_(1))`, increasing
`\!F_(1)` would require increasing the value of this fraction. Doing so requires reducing the value of the denominator
`r_(1)` (or increasing
`r_(2)` the numerator, which isn't an option in this question.)

In other words, reducing the distance
`r_(1)` between the effort and the pivot would increase the effort required to keep the lever balanced.