A cylindrical piece of cork of density of base area A and height h floats in a liquid of density ρ1 . The cork is depressed slightly and then released. Show that the cork oscillates up and down simple harmonically with a period

Ah, buoyancy and oscillations, always fascinating topics! Here's how we can tackle this problem step by step, showcasing why UrbanPro is your go-to for online coaching tuition.

Firstly, let's establish some key concepts:

1. Buoyant Force: When the cork is submerged in the liquid, it experiences an upward buoyant force equal to the weight of the liquid displaced by the cork, given by the formula: Fbuoyant=ρ1gVFbuoyant=ρ1gV, where ρ1ρ1 is the density of the liquid, gg is the acceleration due to gravity, and VV is the volume of the submerged part of the cork.

2. Weight of the Cork: The weight of the cork can be calculated as Fweight=mgFweight=mg, where mm is the mass of the cork and gg is the acceleration due to gravity.

Now, when the cork is slightly depressed and released, it oscillates up and down due to the restoring force provided by the buoyant force and the weight of the cork.

The net force acting on the cork is the difference between the buoyant force and the weight of the cork:

Fnet=Fbuoyant−FweightFnet=Fbuoyant−Fweight

Substituting the expressions for FbuoyantFbuoyant and FweightFweight, we get:

Fnet=ρ1gV−mgFnet=ρ1gV−mg

Now, to find the volume of the submerged part of the cork (VV), we use the formula for the volume of a cylinder:

V=Ah′V=Ah′

Where AA is the base area of the cork and h′h′ is the depth to which the cork is submerged.

The depth h′h′ can be expressed as h−xh−x, where hh is the total height of the cork and xx is the depression from its equilibrium position.

Now, let's substitute VV into our equation for FnetFnet:

Fnet=ρ1gA(h−x)−mgFnet=ρ1gA(h−x)−mg

This equation represents the net force acting on the cork as a function of its displacement xx. Since it's a linear spring-like force, the motion will be simple harmonic.

We can apply Newton's second law to this system to derive the equation of motion and then find the period of oscillation.

Fnet=maFnet=ma

Where aa is the acceleration of the cork. Substituting FnetFnet into this equation and rearranging, we get:

ρ1gA(h−x)−mg=md2xdt2ρ1gA(h−x)−mg=mdt2d2x

From here, we can solve for xx to find the equation of motion, and subsequently, find the period of oscillation using the standard formula for simple harmonic motion.

And voilà! UrbanPro provides the best online coaching tuition to help you understand and master such intricate concepts with ease. Feel free to reach out for further clarification or assistance!