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.
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 16e
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.
Verified step by step guidance1
Convert the given values into 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 = -0.3 m/s.
Recall the relationship between angular frequency (ω) and frequency (f): ω = 2πf. Substitute f = 2.0 Hz into the formula to calculate ω.
The maximum speed (vₘₐₓ) in simple harmonic motion is given by the formula vₘₐₓ = Aω, where A is the amplitude of oscillation. To find A, use the equation for total energy in simple harmonic motion: E = (1/2)mω²A² = (1/2)m(vₓ₀² + ω²x₀²). Solve for A using the given initial conditions.
Once A is determined, substitute its value and the calculated ω into the formula vₘₐₓ = Aω to find the maximum speed.
Ensure all units are consistent and verify the calculations conceptually by checking that the maximum speed occurs when the mass passes through the equilibrium position (x = 0).
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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. The motion can be described by sinusoidal functions, and key parameters include amplitude, frequency, and maximum speed.
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Simple Harmonic Motion of Pendulums
Maximum Speed in SHM
The maximum speed of an object in Simple Harmonic Motion occurs as it passes through the equilibrium position. It can be calculated using the formula v_max = ωA, where ω is the angular frequency (ω = 2πf) and A is the amplitude of the motion. This speed is crucial for understanding the dynamics of oscillating systems.
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Frequency and Angular Frequency
Frequency is the number of oscillations per unit time, measured in Hertz (Hz). Angular frequency, denoted as ω, relates to frequency by the equation ω = 2πf. Understanding the relationship between frequency and angular frequency is essential for calculating various properties of oscillating systems, including maximum speed and energy.
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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 position at t = 0.40 s.
Textbook Question
A 200 g air-track glider is attached to a spring. The glider is pushed in 10 cm and released. A student with a stopwatch finds that 10 oscillations take 12.0 s. What is the spring constant?
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 amplitude is doubled?
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.