Problem 25 - University Physics with Modern Physics

Question

A tuning fork labeled 392 Hz has the tip of each of its two prongs vibrating with an amplitude of 0.600 \(\mathrm{mm} .\) (a) What is the maximum speed of the tip of a prong? (b) A housefly (Musca domestica) with mass 0.0270 \(\mathrm{g}\) is holding onto the tip of one of the prongs. As the prong vibrates, what is the fly's maximum kinetic energy? Assume that the fly's mass has a negligible effect on the frequency of oscillation.

Step-by-Step Solution

Answer

The maximum speed is 1.478 m/s; maximum kinetic energy is \( 2.94 \times 10^{-5} \) J.

Step 1: Understand Simple Harmonic Motion

The prongs of the tuning fork vibrate in simple harmonic motion (SHM). The maximum speed of an object in SHM is given by the formula: \( v_{max} = A \cdot \omega \)where \(A\) is the amplitude and \(\omega\) is the angular frequency.

Step 2: Calculate Angular Frequency

The angular frequency \(\omega\) is related to the frequency \(f\) by the equation: \( \omega = 2 \pi f \)Given the frequency \(f = 392\, \text{Hz}\), we calculate:\[ \omega = 2 \pi \times 392 = 2463.01 \, \text{rad/s}.\]

Step 3: Calculate Maximum Speed of the Prong Tip

We can now calculate the maximum speed using the amplitude \(A = 0.600 \, \text{mm} = 0.0006 \, \text{m}\):\[ v_{max} = 0.0006 \, \text{m} \times 2463.01 \, \text{rad/s} = 1.4778 \, \text{m/s}.\]

Step 4: Understand Kinetic Energy in SHM

The kinetic energy (KE) of the fly, which moves with the prong, can be found using: \( KE = \frac{1}{2} m v^2 \)Where \(m\) is the mass of the fly and \(v\) is the speed of the vibration.

Step 5: Convert Mass of the Fly

Convert the mass of the fly from grams to kilograms: \( m = 0.0270 \, \text{g} = 0.000027 \, \text{kg}.\)

Step 6: Calculate Maximum Kinetic Energy of the Fly

Substitute the values for mass and maximum speed into the kinetic energy formula:\[ KE = \frac{1}{2} \cdot 0.000027 \, \text{kg} \cdot (1.4778 \, \text{m/s})^2 = 2.94 \times 10^{-5} \, \text{J}.\]

Key Concepts

Frequency

Frequency is a crucial concept in understanding oscillations and waves. It refers to the number of cycles or vibrations that an oscillating object completes in one second. The unit of frequency is hertz (Hz), where 1 Hz equals one cycle per second.
Frequency is vital in various applications, such as engineering, music, and everyday technology.
For instance:

• Radio stations are identified by their frequency.
• Musical notes have specific frequencies corresponding to different pitches.
• Everyday gadgets like microwaves use frequency to heat food effectively.

Understanding frequency helps in grasping how sound and other waves behave, making it easier to comprehend simple harmonic motion (SHM) as seen in vibrating tuning forks.

Angular Frequency

Angular frequency, denoted by the Greek letter omega (\(\omega\)), is another way to describe the rate at which an object oscillates in simple harmonic motion. While regular frequency measures vibrations per second in linear terms, angular frequency expresses how many radians an object travels per second.
Mathematically, it is defined as:\[\omega = 2 \pi f\]where \(f\) is the ordinary frequency in hertz (Hz).
This conversion is important because SHM often occurs in rotational systems where describing motion in radians is more useful.

• Angular frequency links the circular aspect of motion with linear vibration frequency.
• It is measured in radians per second (rad/s).

Utilizing angular frequency allows for a deeper understanding of dynamic systems, providing clarity on how rapidly oscillations occur.

Kinetic Energy

Kinetic energy is a form of energy that an object possesses due to its motion. In the context of simple harmonic motion, such as a vibrating tuning fork, it plays a pivotal role in describing how energy is transferred and transformed.
The formula to calculate kinetic energy is:\[KE = \frac{1}{2} m v^2\]where \(m\) is the mass of the object and \(v\) is the velocity.
In our tuning fork example, the housefly experiences kinetic energy as it vibrates with the prong.

• This energy equals one half of the fly's mass multiplied by the square of its velocity.
• As the speed changes during oscillation, so does the kinetic energy, peaking at maximum speed.

Understanding kinetic energy enriches comprehension of movement and physics, showing how moving objects store and use energy.

Tuning Fork

A tuning fork is a slender, Y-shaped metal device, typically used to produce a precise pitch by emitting sound waves at a constant frequency when struck. The prongs of the fork vibrate back and forth, pushing surrounding air molecules, which creates sound waves.
Tuning forks commonly come into play in music and scientific experiments due to their reliable frequency.

• They help musicians tune their instruments accurately.
• Scientists use them to study sound waves and resonance.
• They can serve as teaching tools for demonstrating physics concepts like SHM.

Tuning forks are superb examples of simple harmonic motion, as they oscillate with a known frequency, making them excellent for educational demonstrations and experiments involving wave properties and acoustics.