Resonance of a cube floating on water under external force

AI Thread Summary
The discussion focuses on modeling the dynamics of a cube floating on water under an external force using a mass-spring analogy, where buoyancy is treated as a spring force. The participants explore how to express the buoyancy force as a function of displacement, leading to a mass-spring equation that incorporates external forces. They derive equations for equilibrium and oscillation, noting that the natural frequency of the system is higher than the frequency of the external force, suggesting the cube's motion will closely follow the external force. The final equations indicate a steady-state oscillation pattern, with specific values calculated for amplitude and frequency. The conversation emphasizes the importance of correctly defining reference systems and forces in the analysis.
tanhanhbi
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Homework Statement
The problem discusses a scenario involving a rigid plastic cube floating on water, with specific assumptions made:

-The cube's dimensions are 1m x 1m x 1m, and its mass is 500kg.
-Only vertical motion is considered.
-The buoyancy force is determined by the cube's submerged volume and gravity (g = 10 m/s^2).
-The water pool is assumed to be infinite, so the water surface remains level as the cube moves.
-Friction is neglected.
-Displacement (a) is measured relative to the neutral position, where a = 0 indicates the cube is at -equilibrium (zero net force).

Find the displacement as a function of time when subjected to an external force of F(t)=100cos(0.1t) N in the vertical direction.
Relevant Equations
π‘š (𝑑^2 π‘₯)/(𝑑𝑑^2 )+mg=𝐹_𝑒π‘₯𝑑
Fbuoyancy = -pgV
My attempt is approaching this problem like the mass spring model. Considering the buoyancy force as spring force. By doing so, we can have the typical mass-spring equation

π‘š (𝑑^2 π‘₯)/(𝑑𝑑^2 )+Fbuoyancy = 𝐹_𝑒π‘₯𝑑

Then I can assuming the displacement a will be the sinusoidal function

a=a_𝑀cos⁑(πœ”π‘‘+πœ‘)

The only problem that I am confusing how to represent Fbuoyancy as a function of displacement a. If I can do so, I think I can finish the problem.

Hope to hear from you guys soon.
 
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tanhanhbi said:
The only problem that I am confusing how to represent Fbuoyancy as a function of displacement a.
Archimedes
 
What did Archimedes say?
 
Hill said:

Gordianus said:
What did Archimedes say?
Thank you for quick reply. This is my attempt to represent Fb as a function of displacement a
1710766123475.png

But I am not sure is it right or not. Cause If I substitute Fb onto mass-spring equation. Due to additional mg, the result is not as I expected.
Hope to hear from you guys opinion.
Thank you.
 
k makes things worse. Choose a reference system such that "a" carries the right sign
 
Gordianus said:
k makes things worse. Choose a reference system such that "a" carries the right sign
Thank you, this is my attempt. Not sure it make sense or not ...
1710767524366.png
 
Assuming the force ##F_{ext}## is upward, the force balance is $$F_{ext}-500g+(1)(1000) dg=500\frac{d^2x}{dt^2}$$where d is the submerged depth and x=-d. At equilibrium, without the external force present, we have: $$-500g+(1)(1000) d_eg=0$$or $$d_e=0.5$$Letting ##y=d-d_e##, the equation reduces to $$F_{ext}+1000gy=-500\frac{d^2y}{dt^2}$$where y is the downward displacement relative to the equilibrium depth.
 
Pedantry note: I doubt the cube would float upright in its equilibrium position without some horizontal constraints.
 
Last edited:
Chestermiller said:
Assuming the force ##F_{ext}## is upward, the force balance is $$F_{ext}-500g+(1)(1000) dg=500\frac{d^2x}{dt^2}$$where d is the submerged depth and x=-d. At equilibrium, without the external force present, we have: $$-500g+(1)(1000) d_eg=0$$or $$d_e=0.5$$Letting ##y=d-d_e##, the equation reduces to $$F_{ext}+1000gy=-500\frac{d^2y}{dt^2}$$where y is the downward displacement relative to the equilibrium depth.
Following up on this derivation, if we substitute the given values for F_{ext} and g into the final equation above, we obtain $$\frac{d^2y}{dt^2}+20 y=-0.2\cos{(0.1t)}$$According to this, the natural frequency of oscillation would be ##\sqrt{20}=4.48/ radians/sec## while the forced frequency is only 0.1 radians/second. This tells us that the forcing in changing very slowly, and the cube motion is going to be nearly in phase with the external force and very close to the quasi steady state displacement ##y=0.01 \cos{(0.1t)}##. Let's see how this plays out. After a very short transient, the oscillatory steady state can be represented by $$y=A\cos{(0.1t)}+B\sin{(0.1t)}$$Substituting this into the differential equation, we obtain: $$-0.01 A\cos{(0.1t)}-0.01 B\sin{(0.1t)}+20 A\cos{(0.1t)}+20B\sin{(0.1t)}=-0.2\cos{(0.1t)}$$This gives us B+0 and $$A=-\frac{0.2}{20-0.01}=-0.01$$
 
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