Ch 15: Oscillations

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 15: Oscillations

Problem 75

Chapter 15, Problem 75

A block on a frictionless table is connected as shown in FIGURE P15.75 to two springs having spring constants k₁ and k₂. Find an expression for the block’s oscillation frequency f in terms of the frequencies f₁ and f₂ at which it would oscillate if attached to spring 1 or spring 2 alone.

Verified step by step guidance

Step 1: Begin by recalling the formula for the angular frequency of a spring-mass system. The angular frequency ω is given by \( \omega = \sqrt{\frac{k}{m}} \), where \( k \) is the spring constant and \( m \) is the mass of the block. The oscillation frequency \( f \) is related to \( \omega \) by \( f = \frac{\omega}{2\pi} \).

Step 2: For the block attached to spring 1 alone, the angular frequency \( \omega_1 \) is \( \omega_1 = \sqrt{\frac{k_1}{m}} \), and the frequency \( f_1 \) is \( f_1 = \frac{\omega_1}{2\pi} \). Similarly, for spring 2 alone, \( \omega_2 = \sqrt{\frac{k_2}{m}} \) and \( f_2 = \frac{\omega_2}{2\pi} \).

Step 3: When the block is connected to both springs, the effective spring constant \( k_{\text{eff}} \) is the sum of the individual spring constants because the springs are in parallel. Thus, \( k_{\text{eff}} = k_1 + k_2 \).

Step 4: Using the effective spring constant \( k_{\text{eff}} \), the angular frequency of the block's oscillation is \( \omega = \sqrt{\frac{k_{\text{eff}}}{m}} = \sqrt{\frac{k_1 + k_2}{m}} \). The oscillation frequency \( f \) is then \( f = \frac{\omega}{2\pi} = \frac{1}{2\pi} \sqrt{\frac{k_1 + k_2}{m}} \).

Step 5: Express \( k_1 \) and \( k_2 \) in terms of \( f_1 \) and \( f_2 \) using \( k_1 = (2\pi f_1)^2 m \) and \( k_2 = (2\pi f_2)^2 m \). Substitute these into the expression for \( f \) to find \( f \) in terms of \( f_1 \) and \( f_2 \): \( f = \frac{1}{2\pi} \sqrt{(2\pi f_1)^2 + (2\pi f_2)^2} \).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Simple Harmonic Motion (SHM)

Simple Harmonic Motion is a type of periodic motion where an object oscillates around an equilibrium position. The restoring force acting on the object is directly proportional to its displacement from that position and acts in the opposite direction. This motion is characterized by a constant frequency and can be described mathematically by sinusoidal functions.

Spring Constant (k)

The spring constant, denoted as k, is a measure of a spring's stiffness. It quantifies the amount of force required to stretch or compress the spring by a unit distance. A higher spring constant indicates a stiffer spring, which results in a higher frequency of oscillation when the spring is used in a system, as described by Hooke's Law.

Frequency of Oscillation

The frequency of oscillation, denoted as f, is the number of complete cycles of motion that occur in a unit of time, typically measured in hertz (Hz). For a mass-spring system, the frequency is influenced by the mass of the object and the effective spring constant. The relationship between these variables can be expressed through the formula f = (1/2π)√(k/m), where k is the spring constant and m is the mass.

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