Textbook Question
An air-track glider attached to a spring oscillates with a period of 1.5 s. At t = 0 s the glider is 5.00 cm left of the equilibrium position and moving to the right at 36.3 cm/s. What is the phase constant?
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 14
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?
Verified step by step guidance1
Convert the mass of the glider from grams to kilograms. Since 1 g = 0.001 kg, the mass \( m \) is \( 200 \times 0.001 = 0.2 \, \text{kg} \).
Determine the period of one oscillation. The period \( T \) is the time for one complete oscillation, which can be calculated by dividing the total time for 10 oscillations by 10: \( T = \frac{12.0}{10} \, \text{s} \).
Use the formula for the period of a mass-spring system: \( T = 2\pi \sqrt{\frac{m}{k}} \), where \( T \) is the period, \( m \) is the mass, and \( k \) is the spring constant. Rearrange this formula to solve for \( k \): \( k = \frac{4\pi^2 m}{T^2} \).
Substitute the known values into the formula: \( m = 0.2 \, \text{kg} \) and \( T \) (calculated in step 2). Ensure that \( T \) is squared in the denominator.
Simplify the expression to find the spring constant \( k \). The units of \( k \) will be in \( \text{N/m} \), as it represents the stiffness of the spring.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Hooke's Law
Hooke's Law states that the force exerted by a spring is directly proportional to the displacement from its equilibrium position, expressed as F = -kx, where F is the force, k is the spring constant, and x is the displacement. This principle is fundamental in understanding how springs behave when compressed or stretched.
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Spring Force (Hooke's Law)
Simple Harmonic Motion (SHM)
Simple Harmonic Motion is a type of periodic motion where an object oscillates around an equilibrium position. In the case of a mass-spring system, the motion is characterized by a restoring force proportional to the displacement, leading to a sinusoidal motion. The period of oscillation depends on the mass and the spring constant.
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Period of Oscillation
The period of oscillation is the time taken for one complete cycle of motion in a harmonic system. For a mass-spring system, the period T can be calculated using the formula T = 2π√(m/k), where m is the mass attached to the spring and k is the spring constant. This relationship is crucial for determining the spring constant from the observed oscillation 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
FIGURE EX15.7 is the position-versus-time graph of a particle in simple harmonic motion. What is vmax?
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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 amplitude is doubled?
Textbook Question
An object in simple harmonic motion has an amplitude of 8.0 cm, n angular frequency of 0.25 rad/s, and a phase constant of π rad. Draw a velocity graph showing two cycles of the motion.
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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.