Ch 14: Periodic Motion

Young & Freedman Calc - University Physics 15th Edition

Young & Freedman Calc15th EditionUniversity PhysicsISBN: 9780135159552Not the one you use?Change textbook

Young & Freedman Calc 15th Edition

Ch 14: Periodic Motion

Problem 29b

Chapter 14, Problem 29b

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.

Verified step by step guidance

Start by understanding the concept of Simple Harmonic Motion (SHM). In SHM, the motion of the glider can be described by the equation:

x = A cos (ω t + φ)

, where

A

is the amplitude,

ω

is the angular frequency, and

φ

is the phase constant.

Calculate the angular frequency

ω

using the formula:

ω = \sqrt{\frac{k}{m}}

, where

k

is the spring constant and

m

is the mass of the glider.

Use the energy conservation principle in SHM, which states that the total mechanical energy is constant and is given by:

E = \frac{1}{2} k A^{2}

. At any position

x

, the energy is the sum of kinetic and potential energy:

E = \frac{1}{2} k x^{2} + \frac{1}{2} m v^{2}

Rearrange the energy equation to solve for the speed

v

of the glider at position

x

v = \sqrt{\frac{k}{m} (A^{2} - x^{2})}

Substitute the given values into the equation:

k = 450

N/m,

m = 0.500

kg,

A = 0.040

m, and

x = - 0.015

m, to find the speed

v

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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 can be described using sine or cosine functions. Understanding SHM is crucial for analyzing the motion of the glider attached to the spring.

Energy Conservation in SHM

In SHM, the total mechanical energy of the system is conserved and is the sum of kinetic and potential energy. At any point in the motion, the potential energy stored in the spring and the kinetic energy of the glider can be calculated. This principle allows us to determine the speed of the glider at a given position by equating the total energy at different points in the motion.

Spring Force Constant

The spring force constant, denoted as k, measures the stiffness of the spring and determines the force exerted by the spring when compressed or stretched. It is a crucial parameter in Hooke's Law, which states that the force exerted by the spring is proportional to the displacement from its equilibrium position. In this problem, k is used to calculate the potential energy stored in the spring at different displacements.

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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 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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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 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

For the oscillating object in Fig. E14.4, what is its maximum acceleration?

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