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Magazine Chapter 14: Problem 11
A 2.00 -kg, frictionless block is attached to an ideal spring with force constant 300 \(\mathrm{N} / \mathrm{m} .\) At \(t=0\) the spring is neither stretched nor compressed and the block is moving in the negative direction at 12.0 \(\mathrm{m} / \mathrm{s} .\) Find (a) the amplitude and (b) the phase angle. (c) Write an equation for the position as a function of time.
Short Answer
Expert verified
(a) Amplitude: 0.98 m
(b) Phase Angle: \(-\frac{\pi}{2}\)
(c) Position function: \(x(t) = 0.98 \cos(12.25t - \frac{\pi}{2})\)
Step by step solution
01
Identify the Known Values
The problem provides us with the following values:- Mass of the block, \( m = 2.00 \, \text{kg} \)- Spring constant, \( k = 300 \, \text{N/m} \)- Initial velocity of the block, \( v_0 = -12.0 \, \text{m/s} \) (negative direction)- Initial position, \( x_0 = 0 \) (since the spring is neither stretched nor compressed)
02
Use Conservation of Energy to Find the Amplitude
The total mechanical energy of the system is conserved. The energy in the system is the sum of kinetic energy and potential energy. At the initial condition, only kinetic energy is present:\[ E = \frac{1}{2} m v_0^2 = \frac{1}{2} k A^2 \]Plug in the known values to solve for the amplitude, \( A \):\[ \frac{1}{2} \times 2.00 \times (12.0)^2 = \frac{1}{2} \times 300 \times A^2 \]\[ 144 = 150 A^2 \]\[ A^2 = \frac{144}{150} \]\[ A = \sqrt{\frac{144}{150}} \approx \sqrt{0.96} \approx 0.98 \text{ m} \]
03
Determine the Phase Angle
At \( t = 0 \), the block is at the equilibrium position \( x_0 = 0 \) and moving in the negative direction, indicating that it is going towards the equilibrium position from the positive side. The phase angle \( \phi \) in the equation \( x(t) = A \cos(\omega t + \phi) \) must satisfy the initial condition.Thus, \( \cos(\phi) = 0 \) would not satisfy the initial velocity condition, but since it moves negatively, \( \sin(\phi) = -1 \), so \( \phi = \frac{3\pi}{2} \) or \( -\frac{\pi}{2} \).
04
Write the Equation of Position as a Function of Time
Using the formula for the position of the block as a function of time: \( x(t) = A \cos(\omega t + \phi) \), where \( \omega = \sqrt{\frac{k}{m}} \).Calculate \( \omega \):\[ \omega = \sqrt{\frac{300}{2.00}} = \sqrt{150} \approx 12.25 \, \text{rad/s} \]Substitute \( A = 0.98 \, \text{m} \), \( \omega \approx 12.25 \, \text{rad/s} \), and \( \phi = -\frac{\pi}{2} \):\[ x(t) = 0.98 \cos(12.25t - \frac{\pi}{2}) \]
05
Summary of Results
**Amplitude (a):** 0.98 m**Phase Angle (b):** \(-\frac{\pi}{2}\)**Position as Function of Time (c):** \( x(t) = 0.98 \cos(12.25t - \frac{\pi}{2}) \)
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Conservation of Energy
The principle of conservation of energy plays a crucial role in understanding simple harmonic motion, especially when dealing with springs and masses. This principle states that the total mechanical energy of a system remains constant if only conservative forces are acting. In the case of a block attached to a spring, we have two types of energy to consider: kinetic energy and potential energy stored in the spring.
At any point in time, the sum of these two energies is constant. At the initial condition in this problem, the block's potential energy is zero since it's at equilibrium. All the energy is kinetic. As the block moves, energy transitions between kinetic and potential, but the total stays the same. This understanding allows us to find the amplitude of motion by equating initial kinetic energy to the maximum potential energy (which occurs at maximum displacement).
- Kinetic Energy: This is the energy due to the mass's motion. It is given by the formula \( \frac{1}{2} m v^2 \), where \( m \) is the mass and \( v \) is its velocity.
- Potential Energy: For a spring, this is stored as \( \frac{1}{2} k x^2 \), where \( k \) is the spring constant and \( x \) is the spring's displacement from equilibrium.
At any point in time, the sum of these two energies is constant. At the initial condition in this problem, the block's potential energy is zero since it's at equilibrium. All the energy is kinetic. As the block moves, energy transitions between kinetic and potential, but the total stays the same. This understanding allows us to find the amplitude of motion by equating initial kinetic energy to the maximum potential energy (which occurs at maximum displacement).
Spring Constant
The spring constant, often denoted as \( k \), is a measure of a spring's stiffness. It quantifies the amount of force required to stretch or compress the spring by a unit length. In the context of simple harmonic motion, the spring constant determines the frequency of oscillation.
A higher spring constant means a stiffer spring, resulting in a higher natural frequency. For our given problem, the spring constant is 300 N/m, indicating a relatively stiff spring. This stiffness influences the system's oscillation characteristics, including how quickly it oscillates after being displaced.
- Hooke's Law: The force exerted by the spring is directly proportional to its displacement and is described by \( F = -kx \). Here, \( x \) is the displacement from the equilibrium position.
- Oscillation Frequency: The natural frequency, \( \omega \), of the system is calculated using \( \omega = \sqrt{\frac{k}{m}} \), where \( m \) is the mass.
A higher spring constant means a stiffer spring, resulting in a higher natural frequency. For our given problem, the spring constant is 300 N/m, indicating a relatively stiff spring. This stiffness influences the system's oscillation characteristics, including how quickly it oscillates after being displaced.
Amplitude and Phase Angle
Amplitude and phase angle are critical parameters in defining the motion of an object in simple harmonic motion.
Understanding these parameters helps in visualizing the complete cycle of motion. In our exercise, the amplitude is determined to be 0.98 meters, and the phase angle is calculated based on initial conditions, ensuring the cosine function describes the movement accurately starting from \( t=0 \).
- Amplitude (A): This represents the maximum displacement from the equilibrium position. It indicates how far the block moves away from its rest position on each side. In this problem, the amplitude is found using conservation of energy principles.
- Phase Angle (\( \phi \)): This angle specifies the initial angle at \( t = 0 \) in the cosine function used to describe the motion. It determines the starting point in the cycle of motion.
Understanding these parameters helps in visualizing the complete cycle of motion. In our exercise, the amplitude is determined to be 0.98 meters, and the phase angle is calculated based on initial conditions, ensuring the cosine function describes the movement accurately starting from \( t=0 \).
Mechanical Energy
Mechanical energy in a spring-mass system is the essence of its oscillatory behavior. It is the combination of potential energy stored in the spring and the kinetic energy of the moving mass. In the absence of non-conservative forces, like friction, the mechanical energy remains constant.
Understanding mechanical energy is crucial for solving problems involving oscillations and vibrations, providing insights into how energy shifts influence the motion of the system.
- Total Mechanical Energy (E): At any point, the total energy is constant. Initially, all energy is kinetic. But as the spring is either compressed or stretched, some of this energy converts to potential.
- Energy Transfer: As the block moves, there is a continuous exchange between kinetic and potential energy. At maximum displacement (amplitude), all energy is potential. At equilibrium, all energy is kinetic.
Understanding mechanical energy is crucial for solving problems involving oscillations and vibrations, providing insights into how energy shifts influence the motion of the system.
Cosine Function in Motion Equations
The cosine function is an integral part of the mathematical representation of simple harmonic motion. It helps describe how the displacement of the block changes over time. The general equation for the position as a function of time is given by \( x(t) = A \cos(\omega t + \phi) \).
Using the cosine function allows us to predict the block's position at any time, given its amplitude, frequency, and phase angle. This makes calculating future behavior straightforward, providing a complete picture of the block's oscillatory motion.
- Displacement Equation: This equation shows how the position changes cyclically with time.
- Angular Frequency (\( \omega \)): This determines how quickly the block oscillates back and forth. It is related to the spring constant and mass by the formula \( \omega = \sqrt{\frac{k}{m}} \).
- Phase Shift (\( \phi \)): Determines where in its cycle the block is at \( t=0 \).
Using the cosine function allows us to predict the block's position at any time, given its amplitude, frequency, and phase angle. This makes calculating future behavior straightforward, providing a complete picture of the block's oscillatory motion.
