Question

An object with a mass of `1 kg` is hanging from an axle with a radius of `16 m`. If the wheel attached to the axle has a radius of `64 m`, how much force must be applied to the wheel to keep the object from falling?

Answer 1

`color(blue)(0.25Kg" or if Newtons required "2.4525 N)`

Explanation:

This is a moment arm question. It uses the principle that all moment must be equal for 'equilibrium' to exist. The values are determined by the magnitude of force time its moment arm about a point.

Example of 'equalization': Let clockwise be positive and anticlockwise be negative. Then:

`+(6Kg xx 10m)=+60Kgm` The units are very important

The apposing moment to give equilibrium could be:

`-(3Kg xx 20m)=-60Kgm`

The sum of moments is 0. If they are not then the system will be in motion.
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(blue)("Solving your question")`

The moments are:
Axil edge to centre `-> 16m` with a load of `1kg`
Wheel edge to centre `-> 64m` with an unknown load

Let the unknown load be `xKg`

Then the system is:

`1kgxx16m=xkg xx64m`

Divide both sides by `64m` Notice I am including the units

`xkg=16/64 xx (Kgcancel(m))/cancel(m)`

`color(brown)("Notice that you can manipulate the units in the same way you can")` `color(brown)("manipulate the numbers")`

`xkg=0.25kg` This is the mass

However, the question askes for 'Force'. This is quite often presented in the units of 'Newtons'

There are 9.81 Newtons for each Kg

So in Newtons we have

`0.25 Kg xx 9.81 N/(Kg)`

`0.25xx9.81 xx (Ncancel(Kg))/cancel(Kg)` Again notice the manipulation of units

`2.4525 N`

`color(green)("~~~~~~~~~~~~~~~A Maths Trick ~~~~~~~~~~~~~~~~~~~~~~~")`

The manipulation of units trick can be very useful if you know the units need in an answer but are not sure how to conduct a calculation.

What you do to the available units to obtain your objective you also do to their related numbers.