What is the best unit to measure a rulers mass?

Answer 1

The solution requires some props, such as a pivot, a standard mass and another ruler. Assume the rulers have accurate measures and the material is uniformly distributed -- its density is a constant across the whole bar.

Let us assume the ruler in question (with end points A and B) is L [m] in length, measured independently with the second ruler, and M [g] in mass (unknown).

Find a standard mass whose weight is about half of the weight of the ruler (let us say we pick a standard of 30 g). Place the standard weight at the end point A. Position the pivot beneath the ruler until the ruler is balanced horizontally -- let us say the pivot ends up at point C, between points A and B. AC can be measured with the second ruler. Note that if the standard weight is much higher than the ruler's weight, the pivot point will be very close to point A, and an accurate measurement of length (AC) will be very problematic. Hence, I prefer a standard mass whose weight is about half the weight of the ruler.

From the Principle of moments,

The moment on the LHS = (30 g)*(AC)+(AC/AB)*M*(AC/2)

* note that the earth's gravitational constants cancel out, so we can deal with masses, rather than weights

* two masses on the left-hand side: the standard mass (30 g) acting on the whole left section of the rule, and the mass of the section of the ruler to the left of the pivot acting at the center-of-gravity point of AC, which is half of AC.

The moment on the RHS = (CB/AB)*M*(CB/2)

* the mass on the right-hand side is proportional to the section of the ruler to the right of the pivot = (CB/AB)*M, acting at the center-of-gravity point of CB, which is half of CB.

Equate the two expressions.

Since AB and AC are measured, the only unknown left is M.

Let us do an example below. Let us say we have measured AB to be 0.3 [m] and AC, 0.1 m. Therefore, CB = 0.2 [m].

LHS = (30 g) * (0.1 m) + (0.1/0.3) * M * (0.05 m) = 3 + M / 60

RHS = (0.2/0.3) * M * (0.1 m) = M / 15

Equating the sides, 3 + M / 60 = M / 15

We get 3 * M = 180 or M = 60 [g]

Q.E.D.

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Answer 2

I would use grams or kilograms.

Answer 3

yes