Ch. 14 - Oscillations

Giancoli Douglas - Physics for Scientists and Engineers 5th edition

Giancoli Douglas5th editionPhysics for Scientists and EngineersISBN: 9780137488179Not the one you use?Change textbook

Giancoli Douglas 5th edition

Ch. 14 - Oscillations

Problem 32d

Chapter 14, Problem 32d

A 0.25-kg mass at the end of a spring oscillates 3.2 times per second with an amplitude of 0.15 m. Determine the equation describing the motion of the mass, assuming that at t = 0, 𝓍 was a maximum.

Verified step by step guidance

Step 1: Understand the problem. The mass is oscillating in simple harmonic motion (SHM). The goal is to write the equation of motion for the mass. The general equation for SHM is 𝓍(t) = A cos(ωt + φ), where A is the amplitude, ω is the angular frequency, t is time, and φ is the phase constant. Since the problem states that 𝓍 is at a maximum at t = 0, the phase constant φ = 0.

Step 2: Identify the given values. The mass is 0.25 kg, the frequency of oscillation is 3.2 Hz, and the amplitude is 0.15 m. The frequency f is related to the angular frequency ω by the formula ω = 2πf.

Step 3: Calculate the angular frequency ω. Using the formula ω = 2πf, substitute f = 3.2 Hz into the equation: ω = 2π × 3.2. This gives the angular frequency in radians per second.

Step 4: Write the equation of motion. Substitute the amplitude A = 0.15 m and the angular frequency ω (calculated in Step 3) into the general SHM equation 𝓍(t) = A cos(ωt + φ). Since φ = 0, the equation simplifies to 𝓍(t) = 0.15 cos(ωt).

Step 5: Finalize the equation. Replace ω with its calculated value from Step 3 to complete the equation describing the motion of the mass: 𝓍(t) = 0.15 cos((2π × 3.2)t).

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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. In SHM, the restoring force is directly proportional to the displacement from the equilibrium and acts in the opposite direction. This motion can be described mathematically by sine or cosine functions, which represent the position of the object as a function of time.

Angular Frequency

Angular frequency, denoted by ω, is a measure of how quickly an object oscillates in SHM, expressed in radians per second. It is related to the frequency (f) of oscillation, where ω = 2πf. In this case, with a frequency of 3.2 Hz, the angular frequency can be calculated, which is essential for formulating the equation of motion.

Equation of Motion

The equation of motion for an object in SHM can be expressed as x(t) = A cos(ωt + φ), where x(t) is the displacement at time t, A is the amplitude, ω is the angular frequency, and φ is the phase constant. Given that the mass starts at maximum displacement at t = 0, the phase constant φ is 0, simplifying the equation to x(t) = A cos(ωt). This equation describes the position of the mass over time.

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Related Practice

Textbook Question

Agent Arlene devised the following method of measuring the muzzle velocity of a rifle (Fig. 14–34). She fires a bullet into a 4.148-kg wooden block resting on a smooth surface, and attached to a spring of spring constant k = 162.7 N/m. The bullet, whose mass is 7.450 g, remains embedded in the wooden block. She measures the maximum distance that the block compresses the spring to be 9.460 cm. What is the speed υ of the bullet?

Textbook Question

A mass resting on a horizontal, frictionless surface is attached to one end of a spring; the other end of the spring is fixed to a wall. It takes 3.2 J of work to compress the spring by 0.13 m. The mass is then released from rest and experiences a maximum acceleration of 12m/s². Find the value of the spring constant.

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Textbook Question

Determine the phase constant ϕ in Eq. 14–4 if, at t = 0, the oscillating mass is at 𝓍 = ― A.

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Textbook Question

Determine the phase constant ϕ in Eq. 14–4 if, at t = 0, the oscillating mass is at 𝓍 = A .

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Textbook Question

Determine the phase constant ϕ in Eq. 14–4 if, at t = 0, the oscillating mass is at 𝓍 = ― 1/2 A.

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Textbook Question

An object with mass 2.7 kg is executing simple harmonic motion, attached to a spring with spring constant k = 310 N/m. When the object is 0.020 m from its equilibrium position, it is moving with a speed of 0.60 m/s. Calculate the maximum speed attained by the object.

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