# Measuring the distribution of mass in a solid object

#### Equality

I am interested in measuring the distribution of mass in a solid object, specifically a tennis racket. Is there a simple way to do this?

I only need to know the approximate distribution of mass along the length of the racket, which is 68.5 cm. So, for example, I could cut the racket into sections and then weigh the sections, but I don't want to destroy the tennis racket.

I thought that there might be a way to do it using a kitchen scale and a pivot, by resting one end of the racket on the scale and taking weight measurements with the pivot at different distances from the other end of the racket, as shown below (the handle of the racket is shown as .......)

________________.........
==== ^

________________.........
==== ^

I tried this. The weight was close to 0 grams when the pivot was at the racket's balance point (32 cm from the end of the handle) and of course the weight went up as I moved the pivot closer to the end of the handle. But if I move the pivot 5 cm, and the weight goes up, say, 10 grams, what does that actually mean in terms of weight distribution? What part of the racket do those 10 grams represent?

I also noticed that when I take two weight measurements, one with the top of the racket on the scale, and one with the handle on the scale, and both with the pivot at the other end of the racket (as shown below), the sum of the weight measurements is approximately equal to the total weight of the racket.

________________.........
==== ^ 165.6 g

.........________________
==== ^ 193.6 g

165.6 + 193.6 = 359.2

Actual weight of racket = 355 g

It makes sense to me that the weight would be higher when the handle is placed on the scale, since the racket's balance point is at 32 cm from the end of the handle, and the racket length is 68.5 cm. In other words, if the racket were cut in half, the half with the handle would be heavier.

(EDIT: unfortunately my diagrams are not displaying correctly. Successive spaces are automatically removed, apparently. Hopefully the text explains what I mean)

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#### Woody

There are ways of determining mass distributions by spinning the object, about different points, and measuring how out of balance it is.

However the easiest way to determine the mass distribution is probably by determining the densities of the various materials used to make the racket,
and thus working out the mass contributed to the various parts by these different materials.

For example the racket strings,
I am guessing that these could be obtained separately from the racket and measured and weighed.
Similarly the fabric of the grip.

If we assume that the frame is all that is left, and that it is of the same material throughout,
then subtract the strings and the grip from the total...

#### Equality

There are ways of determining mass distributions by spinning the object, about different points, and measuring how out of balance it is.

However the easiest way to determine the mass distribution is probably by determining the densities of the various materials used to make the racket,
and thus working out the mass contributed to the various parts by these different materials.

For example the racket strings,
I am guessing that these could be obtained separately from the racket and measured and weighed.
Similarly the fabric of the grip.

If we assume that the frame is all that is left, and that it is of the same material throughout,
then subtract the strings and the grip from the total...
Thanks for your suggestion. You are right that I can weigh the frame with the strings and grip removed. But all that would tell me is the total weight of the frame. I am wanting to know what is the distribution of mass along the length of the frame. I used the example of cutting the frame into sections and then weighing the sections. I want to obtain that same data but without destroying the frame.

#### Woody

Balance Point

Find the Balance Point, where the racket will balance with the head over one side of a round bar and the handle on the other.

Move the racket a couple of centimetres to one side of the balance point.
What load has to be applied to restore the balance?
Move it a couple more centimetres, and again find the load to restore the balance.
Repeat as required.

The mass of the racket you have moved from one side of the balance pole to the other has to be matched by the load required to restore the balance.

Be aware of the law of levers (a load close to the balance has less effect than a load far from the balance).

#### oz93666

If you knew the density of the racket was the same throughout (it probably is) , then by measuring it's volume distribution you could achieve your goal.

Weigh the racket , then weigh again immersed in water ,this will give you the average density ( watch out for ,and eliminate any air pockets in handle etc).

Then weigh again many times as you slowly immerse in water , noteing the level of the water on the racket ..

In this way you get the volume distribution and so can find the mass distribution.

#### Equality

Find the Balance Point, where the racket will balance with the head over one side of a round bar and the handle on the other.

Move the racket a couple of centimetres to one side of the balance point.
What load has to be applied to restore the balance?
Move it a couple more centimetres, and again find the load to restore the balance.
Repeat as required.

The mass of the racket you have moved from one side of the balance pole to the other has to be matched by the load required to restore the balance.

Be aware of the law of levers (a load close to the balance has less effect than a load far from the balance).
Hmm. I would have to figure out a way to gradually apply increasing loads. That might be difficult. Do you think that it can't be done with a kitchen scale and pivot point?

#### Equality

If you knew the density of the racket was the same throughout (it probably is) , then by measuring it's volume distribution you could achieve your goal.

Weigh the racket , then weigh again immersed in water ,this will give you the average density ( watch out for ,and eliminate any air pockets in handle etc).

Then weigh again many times as you slowly immerse in water , noteing the level of the water on the racket ..

In this way you get the volume distribution and so can find the mass distribution.
Thanks for your suggestion. However, the density of the racket is not the same throughout. It has different materials in different spots and mass is concentrated at different places in the frame, but I don't know where.

#### Woody

Levers and Balances

Do you think that it can't be done with a kitchen scale and pivot point?
Yes, but...
You have to be careful to distinguish between balancing the differences in mass
and balancing the moments trying to rotate the racket about the pivot.
see: <Wikipedia:Levers>

As you shift mass to one side of the pivot, you will have to imagine the racket split at the pivot point
and consider how the centres of mass of the 2 parts change as you move the pivot as well as how the mass of the 2 parts change.

So the problem is that you always have 2 parameters changing at the same time (the mass and the distance of the centre of mass from the pivot point).