Textbook Question
The position of a 50 g oscillating mass is given by 𝓍(t) = (2.0 cm) cos (10 t - π/4), where t is in s. Determine: The velocity at t = 0.40 s.
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Ch 15: Oscillations
Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796
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Table of contents
- Ch 01: Concepts of Motion35
- Ch 02: Kinematics in One Dimension75
- Ch 03: Vectors and Coordinate Systems58
- Ch 04: Kinematics in Two Dimensions60
- Ch 05: Force and Motion23
- Ch 06: Dynamics I: Motion Along a Line53
- Ch 07: Newton's Third Law27
- Ch 08: Dynamics II: Motion in a Plane52
- Ch 09: Work and Kinetic Energy56
- Ch 10: Interactions and Potential Energy50
- Ch 11: Impulse and Momentum56
- Ch 12: Rotation of a Rigid Body71
- Ch 13: Newton's Theory of Gravity52
- Ch 14: Fluids and Elasticity44
- Ch 15: Oscillations55
- Ch 16: Traveling Waves37
- Ch 17: Superposition39
- Ch 18: A Macroscopic Description of Matter53
- Ch 19: Work, Heat, and the First Law of Thermodynamics60
- Ch 20: The Micro/Macro Connection53
- Ch 21: Heat Engines and Refrigerators34
- Ch 22: Electric Charges and Forces40
- Ch 23: The Electric Field39
- Ch 24: Gauss' Law32
- Ch 25: The Electric Potential63
- Ch 26: Potential and Field57
- Ch 27: Current and Resistance42
- Ch 28: Fundamentals of Circuits39
- Ch 29: The Magnetic Field47
- Ch 30: Electromagnetic Induction60
- Ch 31: Electromagnetic Fields and Waves32
- Ch 32: AC Circuits63
- Ch 33: Wave Optics32
- Ch 34: Ray Optics57
- Ch 35: Optical Instruments30
- Ch 36: Special Relativity38
- Ch 37: The Foundations of Modern Physics25
- Ch 38: Quantization49
- Ch 39: Wave Functions and Uncertainty31
- Ch 40: One-Dimensional Quantum Mechanics35
- Ch 41: Atomic Physics18
- Ch 42: Nuclear Physics27
Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook
Chapter 15, Problem 16h
A 200 g mass attached to a horizontal spring oscillates at a frequency of 2.0 Hz. At t = 0 s, the mass is at x = 5.0 cm and has vx = -30 cm/s. Determine: The position at t = 0.40 s.
Verified step by step guidance1
Convert all given quantities to SI units: mass (m) = 200 g = 0.2 kg, frequency (f) = 2.0 Hz, initial position (x₀) = 5.0 cm = 0.05 m, and initial velocity (vₓ₀) = -30 cm/s = -0.30 m/s.
Determine the angular frequency (ω) of the oscillation using the formula: . Substitute f = 2.0 Hz to calculate ω.
Write the general equation for the position of a mass in simple harmonic motion: , where A is the amplitude and ϕ is the phase constant. Use the initial conditions to determine A and ϕ.
Use the initial position (x₀ = 0.05 m) and initial velocity (vₓ₀ = -0.30 m/s) to solve for A and ϕ. Start by substituting x₀ into the position equation and vₓ₀ into the velocity equation: . Solve the system of equations to find A and ϕ.
Substitute the values of A, ω, and ϕ into the position equation . Evaluate the position at t = 0.40 s to find x(0.40 s).
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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 motion is characterized by a restoring force proportional to the displacement from the equilibrium, leading to sinusoidal position and velocity functions over time. In this case, the mass-spring system exhibits SHM, which can be described using parameters like amplitude, frequency, and phase.
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Frequency and Angular Frequency
Frequency is the number of oscillations per unit time, measured in Hertz (Hz). Angular frequency, denoted by ω, relates to frequency through the equation ω = 2πf, where f is the frequency. In this problem, the frequency of 2.0 Hz indicates how quickly the mass oscillates, which is crucial for determining its position at any given time.
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Position and Velocity in SHM
In SHM, the position (x) and velocity (v) of the oscillating mass can be expressed as functions of time. The position can be described by the equation x(t) = A cos(ωt + φ), where A is the amplitude, ω is the angular frequency, and φ is the phase constant. The velocity is the derivative of the position function, indicating how the position changes over time, which is essential for calculating the position at a specific time.
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Related Practice
Textbook Question
A 200 g mass attached to a horizontal spring oscillates at a frequency of 2.0 Hz. At t = 0 s, the mass is at x = 5.0 cm and has vx = -30 cm/s. Determine: The maximum speed.
Textbook Question
The position of a 50 g oscillating mass is given by 𝓍(t) = (2.0 cm) cos (10 t - π/4), where t is in s. Determine: The initial conditions.
Textbook Question
A 1.0 kg block is attached to a spring with spring constant 16 N/m. While the block is sitting at rest, a student hits it with a hammer and almost instantaneously gives it a speed of 40 cm/s. What are The block's speed at the point where 𝓍 = (½)A?
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Textbook Question
A 200 g mass attached to a horizontal spring oscillates at a frequency of 2.0 Hz. At t = 0 s, the mass is at x = 5.0 cm and has vₓ = -30 cm/s. Determine: The total energy.
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Textbook Question
A block attached to a spring with unknown spring constant oscillates with a period of 2.0 s. What is the period if The spring constant is doubled? Parts a to d are independent questions, each referring to the initial situation.