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Ch 14: Periodic Motion
Young & Freedman Calc - University Physics 14th Edition
Young & Freedman Calc14th EditionUniversity PhysicsISBN: 9780321973610Not the one you use?Change textbook
All textbooksYoung & Freedman Calc 14th EditionCh 14: Periodic MotionProblem 29e
Chapter 14, Problem 29e

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

Verified step by step guidance
1
Understand that the total mechanical energy in simple harmonic motion (SHM) is the sum of kinetic and potential energy, but it remains constant throughout the motion.
The total mechanical energy (E) in SHM can be calculated using the formula: E = (1/2) * k * A^2, where k is the spring constant and A is the amplitude of the motion.
Substitute the given values into the formula: k = 450 N/m and A = 0.040 m.
Calculate the expression: E = (1/2) * 450 * (0.040)^2.
This calculation will give you the total mechanical energy of the glider at any point in its motion.

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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. In SHM, objects oscillate around an equilibrium position, and the motion is characterized by sinusoidal patterns. Understanding SHM is crucial for analyzing the motion of the glider attached to the spring.
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Hooke's Law

Hooke's Law states that the force exerted by a spring is proportional to the displacement from its equilibrium position, expressed as F = -kx, where k is the spring constant and x is the displacement. This law is essential for calculating the potential energy stored in the spring, which contributes to the total mechanical energy of the system.
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Total Mechanical Energy in SHM

The total mechanical energy in SHM is the sum of kinetic and potential energy, which remains constant throughout the motion. For a spring-mass system, it is given by E = (1/2)kA^2, where k is the spring constant and A is the amplitude. This formula allows us to compute the total energy of the glider at any point in its oscillation.
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Related Practice
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

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?

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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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Textbook Question

A cheerleader waves her pom-pom in SHM with an amplitude of 18.0 cm and a frequency of 0.850 Hz. Find (a) the maximum magnitude of the acceleration and of the velocity; (b) the acceleration and speed when the pom-pom's coordinate is x = +9.0 cm; (c) the time required to move from the equilibrium position directly to a point 12.0 cm away. (d) Which of the quantities asked for in parts (a), (b), and (c) can be found by using the energy approach used in Section 14.3, and which cannot? Explain.

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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

A thrill-seeking cat with mass 4.00 kg is attached by a harness to an ideal spring of negligible mass and oscillates vertically in SHM. The amplitude is 0.050 m, and at the highest point of the motion the spring has its natural unstretched length. Calculate the elastic potential energy of the spring (take it to be zero for the unstretched spring), the kinetic energy of the cat, the gravitational potential energy of the system relative to the lowest point of the motion, and the sum of these three energies when the cat is at its highest point.

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