Question

A 3000 pound cylindrical pontoon having a radius of 6 feet floats in a body of fluid. A driver exerts a harmonic force of magnitude \(500 \mathrm{lb}\) at a rate of \(200 \mathrm{cy}-\) cles per minute at the center of the up- per surface of the float as indicated. \((a)\) Determine the density of the fluid if the pontoon is observed to bob with an amplitude of 1 foot. (b) What is the magnitude of the bobbing motion of the pontoon when the excitation fre- quency is reduced to \(5 \mathrm{rad} / \mathrm{sec}\) ?

Short answer

Answer: To find the density of the fluid, use the following formula: $$ ρ_\text{fluid} = \frac{3000~\text{lb} /(32.2~\text{ft} / \text{s}^2)}{\pi(6~\text{ft})^2(2~\text{ft})(32.2~\text{ft}/\text{s}²)} $$ Plugging in the given values and calculating the fluid density will give you the answer. Question: Can we find the magnitude of the bobbing motion of the pontoon when the excitation frequency is reduced to 5 rad/sec? Answer: No, we cannot find the new amplitude at the reduced excitation frequency, as we do not have enough information about the mass-spring system parameters, such as the spring constant (k) and damping coefficient (b).

Step-by-step solution

Calculate the Volume of the Pontoon

First, we need to find the volume of the cylindrical pontoon. The formula for the volume of a cylinder is: $$ V_\text{cylinder} = \pi r^2 h $$ Since we do not have height (h), we have to assume it as a variable ‘h’ for now.

Calculate the Buoyant Force

The buoyant force on the pontoon is equal to the weight of the fluid displaced by the pontoon. The formula for buoyant force is: $$ F_\text{buoyant} = V_\text{displaced} \times \rho_\text{fluid} \times g $$ Where \(V_\text{displaced}\) is the volume of fluid displaced, \(\rho_\text{fluid}\) is the density of the fluid, and \(g\) is the acceleration due to gravity. Since the submerged volume of the pontoon is the same as the volume of the fluid displaced, we can write: $$ F_\text{buoyant} = V_\text{cylinder} \times \rho_\text{fluid} \times g = \pi r^2 h \rho_\text{fluid} g $$

Establish Equilibrium of Forces

At equilibrium, the weight of the pontoon will be balanced by the buoyant force: $$ W_\text{pontoon} = F_\text{buoyant} $$ $$ 3000~\text{lb} = \pi r^2 h \rho_\text{fluid} g $$ We can use this equation to find the density of the fluid (\(\rho_\text{fluid}\)).

Solve for Fluid Density

Rearrange the equation to solve for the fluid density: $$ \rho_\text{fluid} = \frac{3000~\text{lb}}{\pi (6~\text{ft})^2 h g} $$ Since we are given that the pontoon bobs with an amplitude of 1 foot, it means that \(\frac{1}{2} h = 1~\text{ft}\). Therefore, \(h=2~\text{ft}\). Now, we can plug the value of \(h\) and convert lb to slug (~32.2 ft/s²) to get the fluid density: $$ \rho_\text{fluid} = \frac{3000~\text{lb} /(32.2~\text{ft} / \text{s}^2)}{\pi(6~\text{ft})^2(2~\text{ft})(32.2~\text{ft}/\text{s}²)} $$ Calculate the value of the fluid density.

Calculate the Amplitude at Reduced Excitation Frequency

We are given that the excitation frequency is reduced to 5 rad/sec. We know that amplitude and excitation frequency are related through the equation: $$ A = \frac{F_0}{\sqrt{(k - m\omega^2)^2 + (b\omega)^2}} $$ Where \(A\) is the amplitude, \(F_0\) is the force, \(k\) is the stiffness, \(m\) is the mass, \(\omega\) is the frequency, and \(b\) is the damping coefficient. In this case, we have to assume that the whole mass of the pontoon is acting as a mass-spring system with a spring constant \(k\) and damping coefficient \(b\). Since we do not know these parameters and only have the fluid density, we cannot find the amplitude at the new excitation frequency.

Key concepts

Harmonic Motion

Harmonic motion describes a type of periodic motion where the restoring force is directly proportional to the displacement. This is typically seen in systems like springs and pendulums, but it also applies to floating objects like our cylindrical pontoon. In such cases, when the pontoon is displaced from its equilibrium position, the buoyant force works to restore it to its original position. This results in a simple harmonic motion.

For the pontoon in question, the driver applies a harmonic force—one that repeats itself in regular cycles. In this exercise, this concept is used to determine how the pontoon moves vertically when influenced by external forces. The key factor here is the frequency of the applied force, as it will dictate the response amplitude due to the harmonic nature of the resulting motion.

Important aspects of harmonic motion include:

• Oscillation Frequency: Determines how fast the pontoon bobs in the fluid.
• Amplitude: The maximum distance from the equilibrium position the pontoon moves during its bobbing action.
• Phase: The starting point of the cycle, not specifically crucial in this exercise, but part of harmonic analysis.

Fluid Density

Fluid density is a measure of the mass per unit volume of a fluid, typically expressed in kilograms per cubic meter (kg/m³) in the metric system or slugs per cubic foot (slug/ft³) in the imperial system. In our context, it determines how easily the pontoon can float and respond to external forces like the harmonic force applied by the driver.

To solve the exercise, fluid density plays a crucial role in calculating the buoyant force exerted upon the pontoon by the surrounding fluid. This is directly used in the formula:\[ F_{\text{buoyant}} = V_{\text{displaced}} \times \rho_{\text{fluid}} \times g \]Here, \( \rho_{\text{fluid}} \) is the density of the fluid, and this expression helps us equate the weight of the pontoon to the buoyant force. This is necessary for determining the fluid density needed to maintain equilibrium when external forces act upon the pontoon.

Cylindrical Pontoon

A cylindrical pontoon is essentially a floating device shaped as a cylinder, used commonly in marine or floating constructions. Its buoyancy properties make it an interesting object for study in fluid dynamics and harmonic motion problems. The pontoon's geometry, namely its radius and submerged height, influence the volume of fluid it displaces, which in turn determines the buoyant force.

The formula to find its volume is:\[ V_{\text{cylinder}} = \pi r^2 h \]where \( r \) is the radius, and \( h \) is the height of the pontoon. This calculation is a starting point for solving the equilibrium conditions in our exercise, linking the weight of the cylinder to the displaced fluid volume through these dimensions. By understanding these relationships, we can derive other parameters important for the stability and motion of the pontoon.

Equilibrium of Forces

The equilibrium of forces is an essential concept in physics, often dealing with situations where multiple forces acting on an object reach a state of balance. In relation to our pontoon, achieving equilibrium means the weight of the pontoon is perfectly counterbalanced by the buoyant force it experiences.

Mathematically, this is expressed as:\[ W_{\text{pontoon}} = F_{\text{buoyant}} \]Given that the pontoon weighs 3000 pounds, this weight must equal the force lifting it due to the fluid displacement, ensuring it floats naturally. This equilibrium allows us to deduce the fluid density required to support the pontoon or predict how it will react when influenced by forces such as those detailed in harmonic motion.

Understanding this balance supports the broader analysis of how variations in external conditions, like changes in applied frequency, influence the floating and stability of the pontoon in its fluid environment. This concept is foundational for efficiently designing and analyzing marine vessels and structures.