The ends of a turning fork oscillate at a frequency of 440 Hz with an amplitude of 0.50 mm. Determine (a) the maximum velocity and (b) the maximum acceleration of the ends.

To tackle the problem of a turning fork oscillating at a frequency of 440 Hz with an amplitude of 0.50 mm, we can use some fundamental concepts from harmonic motion. Let's break this down step by step to find both the maximum velocity and maximum acceleration.

Understanding the Basics of Oscillation

In simple harmonic motion, the motion can be described using the following equations:

• Maximum Velocity (V_max): This is given by the formula:
• V_max = Aω
• Maximum Acceleration (a_max): This is calculated using:
• a_max = Aω²

Where:

• A = amplitude of the oscillation
• ω = angular frequency in radians per second

Step 1: Calculate Angular Frequency

First, we need to convert the frequency (f) into angular frequency (ω). The relationship between frequency and angular frequency is:

ω = 2πf

Substituting the given frequency:

ω = 2π(440 Hz) ≈ 2764.6 rad/s

Step 2: Maximum Velocity Calculation

Now that we have ω, we can find the maximum velocity. Remember that the amplitude (A) needs to be in meters for consistency in units:

A = 0.50 mm = 0.00050 m

Using the maximum velocity formula:

V_max = Aω = (0.00050 m)(2764.6 rad/s) ≈ 1.3823 m/s

Step 3: Maximum Acceleration Calculation

Next, we will calculate the maximum acceleration using the amplitude and angular frequency:

a_max = Aω² = (0.00050 m)(2764.6 rad/s)²

Calculating ω²:

ω² ≈ (2764.6 rad/s)² ≈ 765,000 rad²/s²

Now substituting back into the acceleration formula:

a_max = (0.00050 m)(765,000 rad²/s²) ≈ 382.5 m/s²

Final Results

To summarize our findings:

• Maximum Velocity: Approximately 1.38 m/s
• Maximum Acceleration: Approximately 382.5 m/s²

This analysis illustrates how the properties of oscillation can be quantified using basic physics principles. If you have any further questions or need clarification on any of the steps, feel free to ask!