Calculating the Period of Small Oscillations for a Floating Object

In summary, the conversation discusses a problem involving a floating object with given dimensions and densities and finding the period of small oscillations about its equilibrium position. The solution involves considering the forces acting on the object at equilibrium and finding the restoring force for small displacements. Ultimately, the period is given by t = 2*pi*(v/gA)^.5, where g is the gravitational field strength. The value of t can be determined by plugging in the given values.
  • #1
Varnson
23
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[SOLVED] Mechnics - Small Oscillations

Homework Statement


A body of uniform cross-sectional are A= 1cm^2 and a mass of density p= 0.8g/cm^3floats in a liquid of density po=1g/cm^3 and at equilibrium displaces a volume of V=0.8cm^3. Show that the period of small oscillations about the equilibrium position is given by t = 2*pi*(v/gA)^.5 where g is the gravitational field strength. Determine the value of t.


Homework Equations





The Attempt at a Solution


I am totally lost on this one. I am thinking that I would need to find the mass of the object, which is simple because the density and volume are given. What do I do after that?
 
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  • #2
i) At equilibrium, what are the forces acting on the object?
ii) If you displace the object by a small amount from the equilibrium position, what's the restoring force?
iii) Is this a type of Simple harmonic motion for small displacements?
 
  • #3
I figured it out, thanks for the pointers!
 

1. What is the definition of small oscillations in mechanics?

Small oscillations refer to a type of harmonic motion in which the displacement from equilibrium is small compared to the equilibrium position. This means that the restoring force is approximately proportional to the displacement.

2. What is the difference between small oscillations and large oscillations?

The main difference between small and large oscillations is the amplitude of the oscillation. Small oscillations have a small amplitude and are closer to the equilibrium position, while large oscillations have a larger amplitude and involve more energy.

3. What are the factors that affect the frequency of small oscillations?

The frequency of small oscillations is affected by the mass of the object, the spring constant, and the amplitude of the oscillation. It is also inversely proportional to the square root of the mass and directly proportional to the square root of the spring constant.

4. How do small oscillations relate to the concept of simple harmonic motion?

Small oscillations are a type of simple harmonic motion, which is a type of periodic motion that follows a sinusoidal pattern. Simple harmonic motion occurs when the restoring force is directly proportional to the displacement from equilibrium, which is the case in small oscillations.

5. What are some real-life examples of small oscillations?

Some examples of small oscillations in everyday life include the motion of a pendulum, the vibration of a guitar string, and the motion of a mass on a spring. These systems exhibit small oscillations because the displacement from equilibrium is small and the restoring force is proportional to the displacement.

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