Determine the period of the oscillations

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The discussion centers on demonstrating that a weighted cylindrical rod floating in a fluid will undergo simple harmonic motion (SHM) when displaced from its equilibrium position. The key forces at play are the weight of the rod (mg) and the buoyant force, which varies with displacement. By establishing that the net force acting on the rod can be expressed as mg minus the buoyant force, it can be shown that this force is proportional to the negative of the displacement, leading to the equation F = -kx. This relationship confirms that the motion is harmonic, and from this, the period of oscillation can be derived using the formula T = 2π√(m/k). Ultimately, the analysis confirms that the rod will indeed execute SHM under the specified conditions.
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1) A long cylindrical rod of radius, r is weighted on one end so that it floats upright in a fluid having a density. It is pushed down a distance x from its equilibrium position and released. Show that the rod will execute simple harmonic motion if the resistive effects of the fluid are negligible and determine the period of the oscillations>>

Start with the forces: mg and the "restoring" force, bouyancy. Bouyance is dependent on displacement.

mg- restoring force = mass*acceleration

where acceleration is d"x/dx".

BUT, how does that lead me to showing the rod will execute simple harmonic motion? and how do i go about determining period of oscillations from there?
 
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show that acceleration is proportional to the negative of the displacement
 
?

i'm not sure how to go about that.. show it is proportional to negative displacement? Are there are any steps for this type of mathematical writeup? any ideas or help would be appreciated. thanks again.
 
If you can prove the thatF= -kx
where F is the resaultant force on the rod x is displacement and k is a constant, then this is a SHM.
 
Thread 'Variable mass system : water sprayed into a moving container'

Thread 'Variable mass system : water sprayed into a moving container'

Starting with the mass considerations #m(t)# is mass of water #M_{c}# mass of container and #M(t)# mass of total system $$M(t) = M_{C} + m(t)$$ $$\Rightarrow \frac{dM(t)}{dt} = \frac{dm(t)}{dt}$$ $$P_i = Mv + u \, dm$$ $$P_f = (M + dm)(v + dv)$$ $$\Delta P = M \, dv + (v - u) \, dm$$ $$F = \frac{dP}{dt} = M \frac{dv}{dt} + (v - u) \frac{dm}{dt}$$ $$F = u \frac{dm}{dt} = \rho A u^2$$ from conservation of momentum , the cannon recoils with the same force which it applies. $$\quad \frac{dm}{dt}...
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