Domanda del libro di testo
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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TOC.tableOfContents
- 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: 9780137344796Non è quello che usi tu?Cambia libro di testo
Capitolo 15, Problema 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.
Guida verificata passo dopo passo1
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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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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Percorso guidato
07:52
Simple Harmonic Motion of Pendulums
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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Percorso guidato
05:08Circumference, Period, and Frequency in UCM
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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07:41Intro to Position-Time Graphs & Velocity
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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.
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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.
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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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