Ch 14: Periodic Motion

Young & Freedman Calc - University Physics 14th Edition

Young & Freedman Calc14th EditionUniversity PhysicsISBN: 9780321973610당신이 사용하는 게 아니라요?교과서 변경

Young & Freedman Calc 14th Edition

Ch 14: Periodic Motion

문제 29b

14장, 문제 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.

검증된 단계별 안내

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

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.