Textbook Question
A 0.500-kg glider, attached to the end of an ideal spring with force constant k = 450 N/m, undergoes SHM with an amplitude of 0.040 m. Compute the speed of the glider when it is at x = -0.015 m.
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Ch 14: Periodic Motion
Young & Freedman Calc15th EditionUniversity PhysicsISBN: 9780135159552
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Table of contents
- Ch 01: Units, Physical Quantities & Vectors37
- Ch 02: Motion Along a Straight Line57
- Ch 03: Motion in Two or Three Dimensions45
- Ch 04: Newton's Laws of Motion23
- Ch 05: Applying Newton's Laws54
- Ch 06: Work & Kinetic Energy11
- Ch 07: Potential Energy & Conservation6
- Ch 08: Momentum, Impulse, and Collisions15
- Ch 09: Rotation of Rigid Bodies37
- Ch 10: Dynamics of Rotational Motion44
- Ch 11: Equilibrium & Elasticity27
- Ch 12: Fluid Mechanics27
- Ch 13: Gravitation34
- Ch 14: Periodic Motion26
- Ch 15: Mechanical Waves22
- Ch 16: Sound & Hearing28
- Ch 17: Temperature and Heat28
- Ch 18: Thermal Properties of Matter35
- Ch 19: The First Law of Thermodynamics29
- Ch 20: The Second Law of Thermodynamics18
- Ch 21: Electric Charge and Electric Field28
- Ch 22: Gauss' Law21
- Ch 23: Electric Potential31
- Ch 24: Capacitance and Dielectrics18
- Ch 25: Current, Resistance, and EMF24
- Ch 26: Direct-Current Circuits29
- Ch 27: Magnetic Field and Magnetic Forces16
- Ch 28: Sources of Magnetic Field10
- Ch 29: Electromagnetic Induction22
- Ch 30: Inductance14
- Ch 31: Alternating Current20
- Ch 33: The Nature and Propagation of Light17
- Ch 34: Geometric Optics29
- Ch 35: Interference13
- Ch 36: Diffraction19
- Ch 37: Special Relativity19
- Ch 38: Photons: Light Waves Behaving as Particles12
- Ch 39: Particles Behaving as Waves25
- Ch 40: Quantum Mechanics I: Wave Functions20
- Ch 41: Quantum Mechanics II: Atomic Structure21
- Ch 42: Molecules and Condensed Matter17
- Ch 43: Nuclear Physics13
- Ch 44: Particle Physics and Cosmology17
Young & Freedman Calc15th EditionUniversity PhysicsISBN: 9780135159552Not the one you use?Change textbook
Chapter 14, Problem 26
A small block is attached to an ideal spring and is moving in SHM on a horizontal, frictionless surface. The amplitude of the motion is 0.250 m and the period is 3.20 s. What are the speed and acceleration of the block when x = 0.160 m?
Verified step by step guidance1
Start by understanding the problem: We have a block undergoing simple harmonic motion (SHM) with a given amplitude and period. We need to find the speed and acceleration at a specific displacement.
Use the formula for the angular frequency \( \omega \) of SHM: \( \omega = \frac{2\pi}{T} \), where \( T \) is the period. Substitute \( T = 3.20 \) s to find \( \omega \).
The velocity \( v \) in SHM at a displacement \( x \) is given by \( v = \omega \sqrt{A^2 - x^2} \), where \( A \) is the amplitude. Substitute \( A = 0.250 \) m and \( x = 0.160 \) m to find the expression for \( v \).
The acceleration \( a \) in SHM at a displacement \( x \) is given by \( a = -\omega^2 x \). Substitute \( x = 0.160 \) m and the previously calculated \( \omega \) to find the expression for \( a \).
Review the signs and units: Ensure that the acceleration is negative, indicating it is directed towards the equilibrium position, and check that all units are consistent throughout the calculations.
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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 the restoring force is directly proportional to the displacement and acts in the direction opposite to that of displacement. It is characterized by sinusoidal oscillations, with parameters such as amplitude, period, and frequency defining the motion. In this scenario, the block's motion is governed by these principles.
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Amplitude and Period
Amplitude in SHM is the maximum displacement from the equilibrium position, while the period is the time taken for one complete cycle of motion. Here, the amplitude is 0.250 m, indicating the furthest point from equilibrium, and the period is 3.20 s, which helps determine the frequency and angular frequency of the motion, crucial for calculating speed and acceleration.
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Kinematics of SHM
The kinematics of SHM involves calculating the velocity and acceleration at any point in the motion. The velocity is maximum at the equilibrium position and zero at the amplitude, while acceleration is maximum at the amplitude and zero at equilibrium. Using the displacement (x = 0.160 m), we can apply SHM equations to find the instantaneous speed and acceleration of the block.
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Related Practice
Textbook Question
A 0.500-kg glider, attached to the end of an ideal spring with force constant k = 450 N/m, undergoes SHM with an amplitude of 0.040 m. Compute the total mechanical energy of the glider at any point in its motion
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Textbook Question
A mass is oscillating with amplitude A at the end of a spring. How far (in terms of A) is this mass from the equilibrium position of the spring when the elastic potential energy equals the kinetic energy?
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Textbook Question
For the oscillating object in Fig. E14.4, what is its maximum speed?
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Textbook Question
A small block is attached to an ideal spring and is moving in SHM on a horizontal frictionless surface. The amplitude of the motion is 0.165 m. The maximum speed of the block is 3.90 m/s. What is the maximum magnitude of the acceleration of the block?
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Textbook Question
For the oscillating object in Fig. E14.4, what is its maximum acceleration?
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