91Ó°ÊÓ

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 the cross- sectional area A of the object. You can ignore the damping due to fluid friction (see Section 14.7).

Step by step solution

01

Understand the Problem

We have an object floating upright in a liquid. Initially, it is at equilibrium due to buoyancy. We're tasked with finding the submerged part of the object when subjected to a force and discovering the period of oscillation in SHM (Simple Harmonic Motion) when the object is disturbed.

02

Calculate Equilibrium Submersion (Part a)

At equilibrium, the buoyant force equals the weight of the object. Thus:\[ \text{Buoyant force} = \text{Weight of object} \]\[ \rho V g = Mg \]Here, \(V\) is the volume of the object submerged, which is \(A \cdot x\), where \(x\) is the vertical distance from the surface to the bottom of the object:\[ \rho (A \cdot x) g = Mg \]Solving for \(x\):\[ x = \frac{M}{\rho A} \]

03

Calculate Additional Submersion Due to Force (Part b)

A force \(F\) is applied, resulting in a new equilibrium. The additional force changes the equilibrium position:\[ \text{New Buoyant Force} = Mg + F \]Thus, the new submerged volume relation is:\[ \rho A (x + \Delta x) g = Mg + F \]Rearrange to find \(\Delta x\):\[ \rho A \Delta x g = F \]\[ \Delta x = \frac{F}{\rho A g} \]

04

Calculate Period of SHM (Part c)

When the force is removed, the object oscillates in SHM. The restoring force, according to Hooke's law is:\[ F = -k x \]Where \(k\) is the effective spring constant:\[ k = \frac{\rho A g}{L} \](\(L\) is a characteristic length related to displacement)The period \(T\) for SHM is given by:\[ T = 2\pi \sqrt{\frac{m}{k}} \]We replace \(k\) and solve:\[ T = 2\pi \sqrt{\frac{M}{\rho A g/L}} = 2\pi \sqrt{\frac{M}{\rho A g}} \]

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Key Concepts

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

Equilibrium Submersion

When an object is floating in a liquid and is at rest, it is said to be in equilibrium submersion. At this point, the buoyant force, which is exerted by the liquid, balances the gravitational force, or weight, of the object itself. This state is governed by the principle of buoyancy discovered by Archimedes.

In practice, we can calculate the equilibrium submersion depth \(x\) using:

• The density of the liquid \(\rho\)
• The gravitational acceleration \(g\)
• The object's mass \(M\) and cross-sectional area \(A\)

The balance of forces can be expressed as:\[ \rho A x g = Mg \]Solving for \(x\) gives us:\[ x = \frac{M}{\rho A} \]
This shows that the submersion depth depends directly on the object's mass and inversely on the liquid's density and the object's cross-section.

Additional Submersion due to Force

When an external downward force \(F\) is applied to a floating object, it results in additional submersion. This happens because the object must displace more liquid to balance the added force.

The new equilibrium is defined by the additional force being balanced by an increase in the buoyant force:\[ \rho A (x + \Delta x) g = Mg + F \]
Where \(\Delta x\) is the additional submersion depth caused by \(F\). Solving for \(\Delta x\) gives us:\[ \Delta x = \frac{F}{\rho A g} \]
This equation shows that the extra submersion \(\Delta x\) is directly proportional to the applied force \(F\) and inversely proportional to the liquid's density, the object's cross-sectional area, and the gravitational force.

Simple Harmonic Motion

Simple Harmonic Motion (SHM) is a type of periodic motion where the restoring force is directly proportional to the displacement from equilibrium. When the applied force on the submerged object is suddenly removed, the object will undergo SHM. In this context, SHM describes the object oscillating up and down in the liquid.

Key characteristics of SHM include:

• The motion is sinusoidal in time.
• It is characterized by its amplitude, frequency, and period.
• The restoring force is pivotal for SHM, as defined by Hooke's Law: \(F = -kx\).

In the case of a submerged object, this restoring force returns it to its equilibrium position, enabling oscillation similar to a mass-spring system.

Buoyant Force

The buoyant force is the upward force exerted on an object when it is submerged in a fluid, partially or completely. This force results from the pressure difference of the fluid exerted at different depths and is mathematically given by:\[ F_{\text{buoyant}} = \rho V g \]
Where:

• \(\rho\) is the fluid density.
• \(V\) is the volume of fluid displaced by the object.
• \(g\) is the gravitational acceleration.

In the floating object scenario, the buoyant force counteracts the object's weight. At equilibrium, the buoyant force matches exactly with the weight, holding the object in stable position in the fluid. This principle is key to understanding buoyancy and floatation.

In effect, the buoyant force determines how much of the object remains submerged.

Oscillation Period

In the context of SHM, the oscillation period \(T\) is the time taken for one complete cycle of the motion. For the floating object, when the external force is removed, the object oscillates vertically in the liquid.

The period of oscillation can be derived from the balance of forces and is expressed as:\[ T = 2\pi \sqrt{\frac{M}{\rho A g}} \]
This formulation shows that the period depends directly on the object's mass and inversely on the liquid's density, the object's cross-sectional area, and gravitational acceleration.

Key points to remember about the oscillation period include:

• It's independent of the amplitude of oscillation for harmonic systems.
• It's determined by the physical characteristics of the system (mass and geometry).

Understanding the period is crucial for predicting how and when the object returns to the equilibrium position during oscillation.