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SHM of a Floating Object. An object with height \(h\) , mass \(M,\) and a uniform cross-sectional area \(A\) floats upright in a liquid with density \(\rho\) . (a) Calculate the vertical distance from the surface of the liquid to the bottom of the floating object at equilibrium. (b) A downward force with magnitude \(F\) is applied to the top of the object. At the new equilibrium position, how much farther below the surface of the liquid is the bottom of the object than it was in part (a)? (Assume that some of the object remains above the surface of the liquid.) (c) Your result in part (b) shows that if the force is suddenly removed, the object will oscillate up and down in SHM. Calculate the period of this motion in terms of the density \(\rho\) of the liquid, the mass \(M,\) and cross-sectional area \(A\) of the object. You can ignore the damping due to fluid friction (see Section \(13.7 ) .\)

Step by step solution

01

Calculate initial equilibrium depth

The buoyant force acting on the object is equal to the weight of the object at equilibrium. So, we equate the buoyant force to the gravitational force:\[ \rho g A x = Mg \]Here, \(x\) is the depth of the submerged part of the object. Solving for \(x\), we find:\[ x = \frac{M}{\rho A} \]

02

Calculate the new equilibrium depth with force

With an additional downward force \(F\), the total force acting downwards becomes \(Mg + F\). The new buoyant force is:\[ \rho g A (x + \Delta x) = Mg + F \]Where \(x + \Delta x\) is the new submerged depth. Solving for \(\Delta x\), we substitute \( x = \frac{M}{\rho A} \) and solve the equation:\[ \Delta x = \frac{F}{\rho g A} \]

03

Determine the period of SHM

The restoring force when the object is displaced from equilibrium is provided by the buoyant force, which leads to a simple harmonic motion. We equate the restoring force to Hooke's law:\[ F_{\text{restoring}} = \rho g A \Delta x = k \Delta x \]Comparing with Hooke's law:\[ k = \rho g A \]The period \(T\) of the SHM is given by:\[ T = 2\pi \sqrt{\frac{m}{k}} \]Substituting the values, we find:\[ T = 2\pi \sqrt{\frac{M}{\rho g A}} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Buoyant Force

Buoyant force is a fascinating concept that explains why some objects float in fluids while others sink. It relies on the principle that when an object is submerged in a fluid, it experiences an upward force from the fluid. This upward force is the buoyant force. It is equal to the weight of the fluid displaced by the submerged part of the object, expressed in the simple equation:\[ F_{\text{buoyant}} = \rho g V \]where

• \( \rho \) is the fluid's density,
• \( g \) is the acceleration due to gravity, and
• \( V \) is the volume of the displaced fluid.

For a floating object, the buoyant force equals the gravitational force on the object: \[ \rho g A x = Mg \]This equality ensures that the object remains afloat. Hence, the object's weight is balanced by the force of displaced fluid.

Equilibrium Position

The equilibrium position of a floating object is the point at which the object neither sinks nor rises, as the forces acting on it are balanced. It is crucial in understanding buoyancy and stability in fluids. At equilibrium, the buoyant force equals the weight of the object.In mathematical terms, for our floating object:\[ x = \frac{M}{\rho A} \]where

• \( x \) is the submerged depth,
• \( M \) is the mass of the object,
• \( \rho \) is the liquid's density,
• \( A \) is the object's cross-sectional area.

In other words, the object settles at a depth where the buoyant force balances its weight. If additional forces are applied, the equilibrium may shift, allowing for analysis of dynamic systems and oscillations.

Periodic Motion

Periodic motion refers to motion that repeats at regular intervals, an essential feature of systems in simple harmonic motion (SHM). In our context, it's the up and down oscillation of a floating object after a force initially disturbs its equilibrium position.Once the force is removed, the object returns to its original position, driven by the buoyant force which acts as a restoring force. This creates an oscillatory motion around the equilibrium point, at a constant period given by:\[ T = 2\pi \sqrt{\frac{M}{\rho g A}} \]Understanding the role of the displaced liquid in SHM allows insights into phenomena in physics and engineering involving buoyancy and harmonic systems.

Restoring Force

A restoring force is a key concept behind simple harmonic motion, responsible for bringing an object back to its equilibrium position when it's displaced. For a floating object, any displacement from equilibrium is countered by changes in the buoyant force.Consider the relation for the restoring force given by:\[ F_{\text{restoring}} = \rho g A \Delta x \]This force counteracts the displacement \( \Delta x \), ensuring the return to equilibrium. In our scenario, it's the changes in the submerged volume due to displacement that adjust the force:The force aligns with Hooke's law for springs, where the spring constant \( k \) is:\[ k = \rho g A \]It is fascinating how the principles of buoyancy translate to SHM, illustrating the universality of physics principles across different systems.