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 cycles per minute at the center of the upper 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 frequency is reduced to \(5 \mathrm{rad} / \mathrm{sec}\) ?

Short answer

(b) What is the bobbing amplitude when the excitation frequency is reduced to 5 rad/s? Solution: (a) The density of the fluid can be calculated using the derived equation \(\rho_f = \frac{F_b}{Vg}\), where \(F_b\) is the buoyant force, \(V\) is the volume of the pontoon, and \(g\) is the gravitational constant. (b) The new amplitude \(A'\) can be found using the proportion between the forces and the relationship \(F_h \propto A\).

Step-by-step solution

Analyze the forces on the pontoon

To determine the density of the fluid in part (a), we need to take into account the weight of the pontoon, the buoyant force, and the harmonic force. We'll use the equilibrium condition to do so.

Find the buoyant force acting on the pontoon

The buoyant force, \(F_b\), is given by \(F_b = V \rho_f g\), where \(V\) is the volume of the pontoon, \(\rho_f\) is the density of the fluid, and \(g\) is the gravitational constant (\(g=32.2\,\text{ft/s}^2\)). The volume of the cylindrical pontoon can be calculated as \(V = \pi r^2 h\), where \(r\) is the radius and \(h\) is the height of the pontoon.

Calculate the volume of the pontoon

Given the radius \(r = 6\,\text{ft}\), and knowing that the pontoon has a weight of \(W = 3,\!000\,\text{lb}\), we can determine the height of the pontoon, as it is equal to \(h = \frac{W}{\pi r^2 g}\).

Determine the density of the fluid (part a)

We know that the pontoon is in equilibrium at the maximum displacement of 1 foot, so the buoyant force equals the combined weight, which means \(F_b = W + F_h\). The harmonic force \(F_h = 500\,\text{lb}\) is given. Solving for the density of the fluid, \(\rho_f = \frac{F_b}{Vg}\).

Determine the new amplitude (part b)

To find the amplitude when the excitation frequency is reduced to \(5 \,\text{rad/s}\), we will use the equation of motion for a system with simple harmonic motion, \(x(t) = A\cos(\omega t)\), where \(A\) is the amplitude and \(\omega\) is the angular frequency in rad/s (different from the initial given 200 cycles per minute). Since we know that the excitation force is \(F_h = 500\,\text{lb}\), we can use the relationship between force and amplitude \(F_h \propto A\), and calculate the new amplitude \(A'\) as the proportion of the forces when the frequency changes. The final answers are the density of the fluid in part (a) and the new amplitude in part (b).

Key concepts

Buoyant Force Calculation

Understanding the concept of buoyant force is essential when studying the dynamics of floating bodies like pontoons. The buoyant force is an upward force exerted by a fluid that opposes the weight of an object immersed in it. It is based on Archimedes' principle, which states that the buoyant force on a submerged object is equal to the weight of the fluid that the object displaces.

To calculate the buoyant force acting on a cylindrical pontoon, we need to find the volume of the pontoon submerged in the fluid. This volume can be computed using the formula for the volume of a cylinder, \(V = \pi r^2 h\), where \(r\) is the radius, and \(h\) is the height. In our scenario, the height of the pontoon will depend on its weight and density such that \( h = \frac{W}{\pi r^2 g} \). Once we have the volume, we multiply it by the fluid's density \(\rho_f\) and the acceleration due to gravity \(g\) to obtain the buoyant force: \(F_b = V \rho_f g\).

When a pontoon is perfectly afloat and at rest, the weight of the pontoon is balanced out by the buoyant force. However, if additional forces are introduced, such as the harmonic force in our exercise, these forces must be taken into account to maintain equilibrium and ascertain the fluid density.

Cylindrical Pontoon Dynamics

The dynamic behavior of cylindrical pontoons when they are subjected to external forces is fascinating and can be described using principles of physics. A pontoon floating in a fluid will experience a restoring force that tries to bring it back to its equilibrium position when it is disturbed. This is closely related to the buoyant force that we've discussed earlier.

When the pontoon is pushed down into the water, it displaces more fluid, leading to an increase in the buoyant force, which in turn pushes the pontoon back up. This dynamic interplay results in the pontoon bobbing up and down when an external harmonic force is applied, as seen in our exercise. The magnitude and frequency of the harmonic force govern how the pontoon responds to this input.

Equilibrium and Excitation

The equilibrium condition is used to determine the fluid's density affecting the pontoon. We consider the pontoon's weight, the buoyant force, and the applied harmonic force. The problem statement mentions the pontoon bobbing with a certain amplitude, which is a direct result of the harmonic force applied at a specific frequency.

Simple Harmonic Motion

Simple harmonic motion (SHM) is a type of periodic motion where the restoring force is directly proportional to the displacement and acts in the direction opposite to that displacement. This concept applies to various physical systems, including springs, pendulums, and in our case, floating bodies like pontoons.

The motion of the pontoon induced by the external force can be described by the equation of motion for SHM, \(x(t) = A\cos(\omega t)\), where \(A\) stands for amplitude, \(\omega\) is the angular frequency, and \(x(t)\) represents the displacement of the pontoon from its equilibrium position over time \(t\).

Frequency and Amplitude

Amplitude \(A\) is a crucial factor as it measures the maximum displacement of the pontoon from its rest position. The exercise demonstrates that by altering the excitation frequency, the amplitude of the pontoon's motion can change. Understanding this relationship is critical when analyzing floating body responses to vibrational forces, which is vital in designing stable marine structures.