Simple Harmonic motion of buoyant force

[Jake4]

Homework Statement

a cylinder is in a deep pool of liquid. it has mass 2kg, diameter 10cm, height 15cm. the density of the liquid is 1.5g/cm^3. The cylinder is submerged up to 12cm, and let go. Show that the resulting motion is simple harmonic.

Homework Equations

x=Acos(wt)

buoyant force = M(of displaced water) * g

 

[grzz]

Draw a FBD with all forces acting on system.

 

[Jake4]

i drew a fbd, and that got me F= F(buoyant)-mg

thats as far as I can get... I've never felt so lost in a physics problem, i don't even know where to start...

 

[grzz]

Try to find the position of equilibrium first.

 

[asteriatic]

The buoyant force acting on the cylinder is equal to the weight of the displaced liquid, which can be calculated by multiplying the volume of the submerged part of the cylinder by the density of the liquid and the acceleration due to gravity. In this case, the submerged volume is equal to the cross-sectional area (pi*r^2) times the height of submersion (12cm). Therefore, the buoyant force can be expressed as:

Fb = (pi*0.05^2*0.12) * 1.5g/cm^3 * 9.8m/s^2 = 0.02214N

Since the cylinder is released from rest, the only force acting on it is the buoyant force. This creates a simple harmonic motion, where the restoring force (buoyant force) is directly proportional to the displacement (depth of submersion). This can be expressed as:

Fb = -kx

Where k is the spring constant, equivalent to the ratio of the restoring force to the displacement. In this case, k = 0.02214N/0.03m = 0.738N/m.

Using Newton's Second Law, F = ma, we can set up the equation of motion for the cylinder:

ma = -kx

Rearranging and substituting in the values, we get:

m(d^2x/dt^2) = -kx

(2kg)(d^2x/dt^2) = -0.738N/m * x

Solving for x, we get:

x = Acos(wt)

Where w = sqrt(k/m) = sqrt(0.738N/m / 2kg) = 0.612 rad/s.

Therefore, the motion of the cylinder is simple harmonic, with a period of 2*pi/w = 10.26 seconds. This means that the cylinder will oscillate up and down in the liquid with a constant amplitude and a period of approximately 10 seconds. This motion is similar to that of a mass-spring system, where the restoring force is provided by the spring and the motion is simple harmonic.