Textbook Question
The position of a 50 g oscillating mass is given by π(t) = (2.0 cm) cos (10 t - Ο/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 17e
The position of a 50 g oscillating mass is given by π(t) = (2.0 cm) cos (10 t - Ο/4), where t is in s. Determine: The initial conditions.
Verified step by step guidance1
Step 1: Understand the given equation for the position of the oscillating mass: π(t) = (2.0 cm) cos(10t - Ο/4). Here, the amplitude is 2.0 cm, the angular frequency is 10 rad/s, and the phase constant is -Ο/4. The initial conditions refer to the position and velocity of the mass at t = 0.
Step 2: To find the initial position, substitute t = 0 into the position equation. This gives π(0) = (2.0 cm) cos(-Ο/4). Use the trigonometric identity cos(-ΞΈ) = cos(ΞΈ) to simplify the expression.
Step 3: To find the initial velocity, differentiate the position equation with respect to time to get the velocity equation. The derivative of π(t) = (2.0 cm) cos(10t - Ο/4) is v(t) = -AΟ sin(Οt - Ο), where A is the amplitude, Ο is the angular frequency, and Ο is the phase constant. Substituting the given values, v(t) = -(2.0 cm)(10 rad/s) sin(10t - Ο/4).
Step 4: Substitute t = 0 into the velocity equation to find the initial velocity. This gives v(0) = -(2.0 cm)(10 rad/s) sin(-Ο/4). Use the trigonometric identity sin(-ΞΈ) = -sin(ΞΈ) to simplify the expression.
Step 5: Combine the results from Step 2 and Step 4 to summarize the initial conditions: the initial position π(0) and the initial velocity v(0). These values describe the state of the oscillating mass at t = 0.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Oscillation
Oscillation refers to the repetitive variation, typically in time, of some measure about a central value or between two or more different states. In this context, it describes the motion of the mass as it moves back and forth around an equilibrium position, characterized by parameters such as amplitude, frequency, and phase.
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Oscillations in an LC Circuit
Initial Conditions
Initial conditions are the values of the variables at the start of the observation or experiment, which in this case refers to the position and velocity of the oscillating mass at time t = 0. These conditions are crucial for determining the future behavior of the system and are derived from the given position function.
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Phase Angle
The phase angle in oscillatory motion indicates the initial position of the oscillating object relative to a reference point. In the equation provided, the phase angle of -Ο/4 affects the starting position of the mass, shifting the cosine function and thus altering the initial conditions of the oscillation.
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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
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 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 π = (Β½)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 vβ = -30 cm/s. Determine: The total energy.
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Textbook Question
A 500 g air-track glider moving at 0.50 m/s collides with a horizontal spring whose opposite end is anchored to the end of the track. Measurements show that the glider is in contact with the spring for 1.5 s before it rebounds. What is the value of the spring constant?