Ch 14: Periodic Motion

Young & Freedman Calc - University Physics 14th Edition

Young & Freedman Calc14th EditionUniversity PhysicsISBN: 9780321973610Not the one you use?Change textbook

Young & Freedman Calc 14th Edition

Ch 14: Periodic Motion

Problem 26

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 guidance

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.

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.

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 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 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 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

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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