Question

Determine the natural period of a typical ice cube floating in water. Measure the dimensions of a typical cube from your refrigerator and calculate its natural frequency in water. (The dimensions may vary depending on your particular ice tray.) Confirm your "experiment." Place an ice cube in water, displace it slightly and release it. Make an approximate measure of the period of an oscillation with your wrist watch, or a stop watch if available. Repeat this operation several times and compare the average measured

Short answer

Answer: The natural period of a typical ice cube floating in water can be calculated by following the aforementioned steps to measure the ice cube's dimensions, calculate the buoyant force, determine the spring constant, calculate the natural frequency, and finally, find the natural period. To confirm the accuracy of the calculations, an experiment can be conducted by measuring the time it takes for an ice cube to complete one oscillation in water. The calculated natural period should be close to the average measured period from the experiment.

Step-by-step solution

Measure the Dimensions of an Ice Cube

Using a ruler or a measuring tape, measure the length (L), width (W), and height (H) of a typical ice cube in centimeters. Ensure that the measurements are accurate and taken precisely for accurate results.

Calculate the Volume of the Ice Cube

To calculate the volume of the ice cube, use the formula: \(Volume = Length \times Width \times Height\) Plug the values obtained in step 1 into the formula and calculate the volume of the ice cube.

Calculate the Buoyant Force

Now we need to determine the buoyant force on the ice cube when it is submerged in water. To do this, multiply the volume obtained in step 2 by the acceleration due to gravity (\(g \approx 9.81 m/s^2\)) and the density of water (\(\rho \approx 1000 kg/m^3\)). Ensure that the volume is converted to cubic meters. \(Buoyant\: Force = Volume \times g \times \rho\)

Calculate the Spring Constant

To calculate the spring constant (k), divide the buoyant force obtained in step 3 by the height (H) of the ice cube (convert the height to meters). \(Spring\: Constant = k = \frac{Buoyant\: Force}{Height}\)

Calculate the Natural Frequency

Now that we have the spring constant (k), we can calculate the natural frequency of the ice cube in water. The mass (m) of the ice cube can be calculated as the volume multiplied by the density of ice (\(\rho_ice \approx 917 kg/m^3\)): \(Mass = Volume \times \rho_{ice}\) Then, apply the formula for natural frequency: \(Natural\: Frequency = \omega = \sqrt{\frac{k}{m}}\)

Calculate the Natural Period

Finally, to calculate the natural period (T) of the ice cube in water, use the formula: \(Natural\: Period = T = \frac{2\pi}{\omega}\)

Confirm the Experiment

Now it's time to confirm the calculations with an experiment. Place an ice cube in water and slightly displace it. Release it and measure the time it takes for the ice cube to complete one oscillation using a stopwatch or wristwatch. Repeat the experiment several times and calculate the average measured period of oscillation. Compare the calculated natural period obtained in step 6 with the average measured period from the experiment. If the values are close to each other, this confirms the accuracy of the calculations.

Key concepts

Buoyant Force

Buoyant force is a key concept in understanding the behavior of objects in fluids, such as ice cubes in water. It's this force that keeps ships afloat and ice bobbing on the surface. Specifically, buoyant force is the upward force exerted by a fluid that opposes the weight of an object immersed in it. Physicist Archimedes famously discovered that the buoyant force on an object is equal to the weight of the fluid displaced by the object.

When you place an ice cube in water, the water pushes it up with a force equal to the weight of the water displaced by the cube. For 'engineering vibrations' exams or exercises involving objects floating in fluids, the buoyant force calculations provide essential data. In practice, after measuring an ice cube's dimensions, we can derive its volume, and using the known density of water, we calculate the buoyant force with the formula:
\(Buoyant\;Force = Volume \times g \times \rho\).

This force fundamentally influences the subsequent calculations of the system's natural frequency and oscillation period.

Spring Constant

In vibrational physics and engineering, the spring constant, denoted as 'k', measures the stiffness of a spring. It's a crucial parameter in system dynamics, often influencing how an object oscillates. For simpler comprehension, think of the spring constant as a rate at which a spring resists deformation; the higher the constant, the stiffer the spring.

In our unique case of an ice cube in water, we mimic the properties of a spring by considering the buoyant force. Analogously to a spring, when you push the ice cube down into the water and let go, it oscillates up and down. Determining the effective spring constant involves dividing the buoyant force by the displacement, here taken as the ice cube's height, which acts like the spring's deformation:
\(Spring\;Constant = k = \frac{Buoyant\;Force}{Height}\).

This calculated 'k' is pivotal for assessing the system's natural frequency, being one of the fundamental parameters for vibration analysis.

Oscillation Period

Moving onto the oscillation period, denoted as 'T', this concept tells us the time required for one complete cycle of oscillation. If you watch a pendulum swing or a plucked guitar string vibrate, you're observing periodic motion, characterized by this oscillation period. The period is inversely related to the frequency—the number of cycles per second—so as the frequency increases, the period decreases, and vice versa.

When we calculate the natural period of an oscillating system like our ice cube, we're finding out the time it takes for the ice to return to its initial position after being displaced. Using the formula found in step 6 of the solution: \(Natural\;Period = T = \frac{2\pi}{\omega}\),

we can determine how fast or slow the ice cube will bob in water. This knowledge is applied widely by engineers in designing structures and systems to ensure they can withstand various oscillations without sustaining damage.

Engineering Vibrations

Finally, let's talk about 'engineering vibrations', a vast field in engineering dealing with the analysis of systems under oscillatory conditions. This field is concerned with predicting and controlling vibrations to avoid potential failures or discomfort. Whether designing buildings to withstand earthquakes, car suspensions for a smooth ride, or simply understanding how an ice cube dances in water, mastering the principles of vibrations is crucial.

By calculating the natural frequency, we delve into the heart of engineering vibrations. The natural frequency is the rate at which a system oscillates when not subjected to any external force or damping. With the frequency and spring constant in hand, one can evaluate the dynamic response of the system. This is particularly important for ensuring the safety and durability of structures and mechanical systems under various vibration scenarios, and it illustrates the immense relevance of vibrational studies in practical engineering applications.