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 13

Chapter 14, Problem 13

A 2.00-kg, frictionless block is attached to an ideal spring with force constant 300 N/m. At t = 0 the spring is neither stretched nor compressed and the block is moving in the negative direction at 12.0 m/s. Find (a) the amplitude and (b) the phase angle. (c) Write an equation for the position as a function of time.

Verified step by step guidance

Start by identifying the type of motion involved. The block attached to the spring is undergoing simple harmonic motion (SHM). The key parameters for SHM are mass (m), spring constant (k), initial velocity, and initial position.

Use the formula for the angular frequency \( \omega \) of a spring-mass system: \( \omega = \sqrt{\frac{k}{m}} \). Substitute the given values: \( k = 300 \text{ N/m} \) and \( m = 2.00 \text{ kg} \) to find \( \omega \).

To find the amplitude (A), use the energy conservation principle. The total mechanical energy in SHM is given by \( E = \frac{1}{2} k A^2 \). At \( t = 0 \), the kinetic energy is \( \frac{1}{2} m v^2 \). Set the kinetic energy equal to the total energy to solve for \( A \).

For the phase angle (\( \phi \)), use the initial conditions. The position \( x(t) \) in SHM is given by \( x(t) = A \cos(\omega t + \phi) \). At \( t = 0 \), the spring is neither stretched nor compressed, so \( x(0) = 0 \). Use the initial velocity \( v(0) = -12.0 \text{ m/s} \) and the velocity equation \( v(t) = -A \omega \sin(\omega t + \phi) \) to solve for \( \phi \).

Write the equation for the position as a function of time using the amplitude and phase angle found: \( x(t) = A \cos(\omega t + \phi) \). Substitute the values of \( A \), \( \omega \), and \( \phi \) to express \( x(t) \) in terms of known quantities.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Simple Harmonic Motion

Simple Harmonic Motion (SHM) describes the oscillatory motion of systems like springs and pendulums, where the restoring force is proportional to the displacement. In this problem, the block attached to the spring exhibits SHM, characterized by sinusoidal functions of time for displacement, velocity, and acceleration.

Amplitude of Oscillation

The amplitude in SHM is the maximum displacement from the equilibrium position. It represents the energy stored in the system and is determined by initial conditions such as velocity and position. For this problem, the amplitude can be calculated using the initial velocity and the properties of the spring.

Phase Angle in SHM

The phase angle in SHM determines the initial position and direction of motion at t = 0. It is crucial for writing the equation of motion, as it shifts the sine or cosine function to match the initial conditions. Calculating the phase angle involves using the initial velocity and position of the block.

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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. When the amplitude of the motion is 0.090 m, it takes the block 2.70 s to travel from x = 0.090 m to x = -0.090 m. If the amplitude is doubled, to 0.180 m, how long does it take the block to travel from x = 0.180 m to x = -0.180 m?

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Textbook Question

In a physics lab, you attach a 0.200-kg air-track glider to the end of an ideal spring of negligible mass and start it oscillating. The elapsed time from when the glider first moves through the equilibrium point to the second time it moves through that point is 2.60 s. Find the spring's force constant.

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Textbook Question

A 2.40-kg ball is attached to an unknown spring and allowed to oscillate. Figure E14.7 shows a graph of the ball's position x as a function of time t. What are the oscillation's (a) period, (b) frequency, (c) angular frequency, and (d) amplitude? (e) What is the force constant of the spring?

Weighing Astronauts. This procedure has been used to 'weigh' astronauts in space: A 42.5-kg chair is attached to a spring and allowed to oscillate. When it is empty, the chair takes 1.30 s to make one complete vibration. But with an astronaut sitting in it, with her feet off the floor, the chair takes 2.54 s for one cycle. What is the mass of the astronaut?

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Textbook Question

An object is undergoing SHM with period 0.900 s and amplitude 0.320 m. At t = 0 the object is at x = 0.320 m and is instantaneously at rest. Calculate the time it takes the object to go (a) from x = 0.320 m to x = 0.160 m. (b) from x = 0.160 m to x = 0.

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