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Calculating point of balance (Read 1333 times)

Thearos

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Calculating point of balance
Aug 15^th, 2018 at 6:26pm

Hello

Just from reading something on spears in N. Sekunda's wonderful Osprey book on Greek hoplites. A hoplite spear is about 2.5m long, made of ash. The pole or stave tapers v. slightly from butt end to point end-- about 2cm-1.8cm diameter to 1.5cm, but let's simplify this to a 2cm uniform cylinder. I suppose that it's about 3140 cubic cm of ash, so about 2kg or just over. The spear head of iron is pretty light (modern reenactor versions are much too heavy at 500+g), so say 350g; the spearbutt is a massive bronze affair at say 500+g. (the heaviest I've seen in catalogs are 1400g).

Can someone help calculate the point of balance ?

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Sarosh

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Re: Calculating point of balance
Reply #1 - Aug 16^th, 2018 at 6:03am

http://hyperphysics.phy-astr.gsu.edu/hbase/cm.html

for 350g in front and 600g in butt end , 2.5m shaft (homogeneous mass placement) . overall CM is 1.57m from the front end (hypothesised that the spearhead's CM and buttspike CM are at the ends of the shaft not further )

tried to find the CM of the tapered shaft but i messed up with the integrals... i would like to see how to do it if someone finds it.

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Teg

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Re: Calculating point of balance
Reply #2 - Aug 16^th, 2018 at 7:06am

@ Sarosh: your calculation is not correct. You have to take into account the shaft as well. There is no need to do the integral.
I'll give you some hints: If you have a body with known center of mass you can replace this body with a point mass at its center of mass position.
The center of mass of a symmetrical body is at its symmetry center. Where does a broom handle balance?

The correct answer is: The spears "point of balance" is 1.3559 m from its front end. So roughly 10 cm behind the middle. I used: point 350 g, butt 600 g, 2.5 m shaft with 2000 g, no tapering.

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Thearos Interfector Viris Spurii Offline Take that Posts: 3410	Re: Calculating point of balance Reply #3 - Aug 16^th, 2018 at 8:41am Thanks, you guys are the best. Is there a problem to the fact that both the point and the butt are not fixed at one point on the shaft, but are socketed ? The butt can also be quite long (25 to 40 cm).
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Sarosh

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Re: Calculating point of balance
Reply #4 - Aug 16^th, 2018 at 9:45am

@Teg yes i'm wrong i didn't account for the shaft mass.
after that i wanted to do the integral to find the CM of the tapered shaft.
@Thearos the CM of the spike and butt cannot be at the same spot the shaft ends which changes some things if the CM of each attachment is not far away from the end of the wooden shaft then Teg's calculation will be accurate.

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Sarosh

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Re: Calculating point of balance
Reply #5 - Aug 16^th, 2018 at 10:35am

@Teg looking at it again i think you did something wrong.
why did you take the mass of the shaft 2000g?
i calculated 549g (density of ash 0.7g/cm^3)
and corrected the overall CM to 145.75 cm from the light end.
diameter of shaft is 2cm not radius.

@ Thearos are you sure it's 2cm diameter? and not 2cm radius?
wikipedia gives 5cm diameter for dory https://en.wikipedia.org/wiki/Dory_(spear)

lol too much confusion but i still want to see an integral Tongue

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Sarosh

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Re: Calculating point of balance
Reply #6 - Aug 16^th, 2018 at 11:28am

finally i can relax : this is all you need http://mathworld.wolfram.com/ConicalFrustum.html

for 2cm- 1.5cm diameter tapering 250cm shaft , density of ash 0.7g/cm^3, point 350g , butt 600g.

found CM is 151.38cm from light end
overall weight is 1373.7g

for 2-1,5cm radius tapering :
CM is 144.37cm from light end
overall weight is 2645.1g

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Teg

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Re: Calculating point of balance
Reply #7 - Aug 16^th, 2018 at 11:41am

Sarosh wrote on Aug 16^th, 2018 at 10:35am:

why did you take the mass of the shaft 2000g?

Because Thearos said that this is the weight and I didn't check it Cheesy

Good catch!

Looking at these numbers (which I didn't check) I guess you're right and Thearos used a radius of 2 cm, which increases the weight by a factor of 4, which would agree with the weight Thearos gave us.

Sarosh wrote on Aug 16^th, 2018 at 10:35am:

lol too much confusion but i still want to see an integral Tongue

If you're brave enough to read my handwriting I'll supply something. If you want detailed explanations alongside with it, PM me your educational background/level and I'll try to make it fit.
Going to be fun! I didn't do such a calculation for years....

Sarosh wrote on Aug 16^th, 2018 at 10:35am:

and corrected the overall CM to 145.75 cm from the light end

Using your weight I get the decimal places slightly different, but the equation must be correct. Congrats!

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Thearos Interfector Viris Spurii Offline Take that Posts: 3410	Re: Calculating point of balance Reply #8 - Aug 16^th, 2018 at 2:34pm Yes, I did use a radius of 2 cm-- which is a mistake. I do mean diameter 2 cm or so, not 4 cm (far too big). So shall we say that the volume is about 2,700 cubic cm, and hence the weight about 1.8-1.9kg ?
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