- #1

Adesh

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Let’s say we have a boat whose longitudinal axis is the y-axis (which goes into the screen in the figure below) standing upright in a still water .

##S## is the Center of Mass of the boat and ##C## is the Center of Mass of the displaced water.On ##S## lies the force ##\mathbf W## (the weight of the boat) and on ##C## acts the buoyant force and they exactly cancel each other (Archimedes’ Principle). Hence, the boat remains in the equilibrium.

It is a proven fact that the buoyant force acts on the Center of Mass of displaced water.

Now, imagine we rotate our boat (y-axis as the axis of rotation) towards right by some infinitesimal angle ##\theta## (with vertical), the new Center of Mass of displaced water is ##C’##. Rotate the boat (from equilibrium position) towards left by the same infinitesimal angle, the new Center of Mass of displaced water is ##C”##. Let ##\mathscr{M}## denote the Center of curvature of the curve ## C”~ C ~ C’##, let’s call it metacenter.

(I would like to apologise for my strange looking ##\mathscr{M}## and hand made diagrams, due to lockdown I’m away from my home and hence have no access to my Mac)

Since, buoyant force ##\mathbf B## always acts on the

Now, my doubt is that it is written that if metacenter ##\mathscr{M}## were to lie below ##S## then the net moment produced would tend to increase the disturbance (that is it would tend to increase the angle with respect to the vertical, that is it would cause a rotation to the right) and I cannot understand how and why.

How can ##\mathscr{M}## ever affect the direction of rotation? I want to why did we designated a letter to the Center of curvature of curve ##C”~C~C’##, what’s the significance of ##\mathscr{M}##?

##S## is the Center of Mass of the boat and ##C## is the Center of Mass of the displaced water.On ##S## lies the force ##\mathbf W## (the weight of the boat) and on ##C## acts the buoyant force and they exactly cancel each other (Archimedes’ Principle). Hence, the boat remains in the equilibrium.

It is a proven fact that the buoyant force acts on the Center of Mass of displaced water.

Now, imagine we rotate our boat (y-axis as the axis of rotation) towards right by some infinitesimal angle ##\theta## (with vertical), the new Center of Mass of displaced water is ##C’##. Rotate the boat (from equilibrium position) towards left by the same infinitesimal angle, the new Center of Mass of displaced water is ##C”##. Let ##\mathscr{M}## denote the Center of curvature of the curve ## C”~ C ~ C’##, let’s call it metacenter.

(I would like to apologise for my strange looking ##\mathscr{M}## and hand made diagrams, due to lockdown I’m away from my home and hence have no access to my Mac)

Since, buoyant force ##\mathbf B## always acts on the

*current*center of mass of displaced water, therefore, in the figure buoyant force acts on ##C’## vertically above. Now, ##\mathbf W## and ##\mathbf B## would acts as couple and hence produce a rotation. As far as I can think, the boat will rotate about ##S## and hence ##\mathbf W## will have no contribution in rotation of our boat, it’s our buoyant force ##\mathbf B## which would produce an anti-clockwise rotation (the vector ##\vec{r}## from ##S## to the line of action of ##\mathbf B##, and hence by right hand rule a torque will be produced which points into the paper).Now, my doubt is that it is written that if metacenter ##\mathscr{M}## were to lie below ##S## then the net moment produced would tend to increase the disturbance (that is it would tend to increase the angle with respect to the vertical, that is it would cause a rotation to the right) and I cannot understand how and why.

How can ##\mathscr{M}## ever affect the direction of rotation? I want to why did we designated a letter to the Center of curvature of curve ##C”~C~C’##, what’s the significance of ##\mathscr{M}##?

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