How to calculate how a weight added to the top of a sailboat mast affects COG

[Greg2]

I'm in the process of purchasing a sailboat for open ocean cruising and would like to understanding how a mass (say 1kg) mounted at the top of the mast affects the center of gravity, assuming that the COG is at a known location relative to the waterline.

 

[Dale]

Are you familiar with the concept of a weighted average?

 

[Greg2]

I understand the concept, but wouldn't say that I'm terribly familiar with it.

 

[sophiecentaur]

I'm in the process of purchasing a sailboat for open ocean cruising and would like to understanding how a mass (say 1kg) mounted at the top of the mast affects the center of gravity, assuming that the COG is at a known location relative to the waterline.

I imagine you have 'looked it up' and that what you have found didn't seem relevant to your particular situation.
This link tells you what you need to know about the basics. ~The COM of an object is the point about which the object would balance. It's the point about which the turning effect of all the parts on one side is balanced (equilibrium) by the turning effect of all the parts on the other. It is best to start thinking in terms of a 1D object (a rod with masses on it) and then work towards 2D (a plate) and then 3D (a solid object)
If you take a point to one side of the COM, there is a net turning effect (no equilibrium) which can be eliminated by putting another mass somewhere on the other side of the new point and then the new point becomes the new centre of mass. Your boat problem can use the COM_x formula with just two masses; the mass of the boat and the added 1kg. Use the bottom of the keel as the reference for distances. The new COM_X position will be
COM_new = (Mass_boat . COM_boat + Mass_1kg. Height above keel)/(Mass_boat +Mass_1kg)
An in your head calculation will tell you that the shift will be about Mast height.(1/Mass_boat)

 

[Dale]

I understand the concept, but wouldn't say that I'm terribly familiar with it.

OK, that should be enough. The COM of a system of multiple objects is the weighted average of the COM of each individual object. So if x is the position of the COM and m is the object mass then

$$x_{total}=\frac{m_{boat}x_{boat}+m_{weight}x_{weight}}{m_{boat}+m_{weight}}$$

 

[Greg2]

OK, that should be enough. The COM of a system of multiple objects is the weighted average of the COM of each individual object. So if x is the position of the COM and m is the object mass then

$$x_{total}=\frac{m_{boat}x_{boat}+m_{weight}x_{weight}}{m_{boat}+m_{weight}}$$

Thanks for the help (with clarity).

For the record, given an average 8000 kg sailing vessel with a 17 meter mast, we're talking a movement of the location of the COM upwards from the keel of less than 1 cm. Upon reflecting as to how I might have tested this with a scale model, it's obvious now that a 1 kg weight at the top is nearly inconsequential given the weight of the boat.

Next time, perhaps I'll start from that end of the problem.

 

[Dale]

No problem! At least you had thought about it enough to clearly specify the problem, and it was a problem that has a fairly direct and clear answer.

 

[sandy stone]

If you consider that the weight of the keel is probably only a meter or so from the CG of the boat, you can see that nearly 17 kg of keel weight is needed to compensate for 1 kg at the top of the mast, so not exactly inconsequential. This is the reason manufacturers of racing sailboats go to all the trouble (and expense) of, for instance, tapering the top of the mast.

 

[Greg2]

If you consider that the weight of the keel is probably only a meter or so from the CG of the boat, you can see that nearly 17 kg of keel weight is needed to compensate for 1 kg at the top of the mast, so not exactly inconsequential. This is the reason manufacturers of racing sailboats go to all the trouble (and expense) of, for instance, tapering the top of the mast.

Good point. I had not considered the racers point of view. I approached the problem from the conservative cruiser point of view - sacrificing some speed for stability.

 

[sandy stone]

Yes, for that purpose 1 kg is probably not significant. Note that adding something as substantial as a roller-furling sail can reduce the Angle of Vanishing Stability as much as 10 degrees from the original design.

 

[Greg2]

Yes, for that purpose 1 kg is probably not significant. Note that adding something as substantial as a roller-furling sail can reduce the Angle of Vanishing Stability as much as 10 degrees from the original design.

Fortunately, we're quite happy to sacrifice a bit of convenience for better stability, so no roller-furling sail for us.

 

[sophiecentaur]

Fortunately, we're quite happy to sacrifice a bit of convenience for better stability, so no roller-furling sail for us.

For many people it's more than just a bit of convenience. Getting up on the foredeck and changing foresail frequently is a lot to ask of many cruising sailors especially single handers. The compromise to suit everyone is to have a lower mast, I think. But that's sailing and old age and not Physics.

 

[Greg2]

For many people it's more than just a bit of convenience. Getting up on the foredeck and changing foresail frequently is a lot to ask of many cruising sailors especially single handers. The compromise to suit everyone is to have a lower mast, I think. But that's sailing and old age and not Physics.

Point taken. We'll likely reach that point once we get more experience under our belts. We're just getting started.

 

[sophiecentaur]

Point taken. We'll likely reach that point once we get more experience under our belts. We're just getting started.

I don't know how long it takes before a newbie sailor stops being chicken long before the boat does. I hope you never get to the stage where the Angle of Vanishing Stability is a serious issue for you. It won't be until you get to be one of those nutty gung ho racing types.

 

[sandy stone]

It won't be until you get to be one of those nutty gung ho racing types.

Hey! I resemble that remark! But the OP did mention open ocean cruising, so there is the faint possibility of being caught broadside by a big wave. However, I agree with you, on a 17 meter mast I really want roller furling (at least on the headsail.)

 

[sophiecentaur]

Hey! I resemble that remark!

Nothin personal but you must have come across racers who seem to put health and safety behind them in the interests of getting a win. It's mostly testosterone driven and few of them would expose their partners and families to the same risks. They are just wealthy versions of the (mostly) young men who cycle against traffic lights and up one way streets with their personal force field switched on.
Roller furling gives you the chance to reef when you like and however much is needed. This protects you against the not-so-freak conditions that can hit you with a too-massive headsail up there. I'm all for an easy life, personally - particularly on a long cruise.