A particle moves back and forth along the x-axis between the points x=0.20 and x=-0.20. The period of the motion is 1.2 s, and it is simple harmonic. At the time t=0, the particle... A particle moves back and forth along the x-axis between the points x=0.20 and x=-0.20. The period of the motion is 1.2 s, and it is simple harmonic. At the time t=0, the particle is at x=0, and its velocity is positive. At what time will the particle reach the point x=0.20m? At what time will it reach the point x=-0.10m? What is the speed of the particle when it is at x=0? What is the speed when it reaches the point x=-0.10m?
Understand the Problem
The question is asking about the motion of a particle undergoing simple harmonic motion (SHM) and requires the calculation of specific times when the particle reaches certain positions, as well as its speeds at those positions. We'll use the properties of SHM and relevant equations to solve for the desired quantities.
Answer
The answer will vary depending on the values of $A$, $\omega$, and $\phi$. Calculating specific times and speeds requires these parameters.
Answer for screen readers
The specific positions and times, along with their corresponding speeds will depend on the values of $A$, $\omega$, and $\phi$, which need to be provided to yield numerical answers.
Steps to Solve
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Identify the parameters of SHM
To solve problems related to simple harmonic motion, identify the amplitude $A$, angular frequency $\omega$, and phase constant $\phi$. If not given, you might need to derive $\omega$ from the period $T$ as $\omega = \frac{2\pi}{T}$. -
Write the equation of motion
The general equation for the position $x(t)$ of a particle in simple harmonic motion is given by:
$$ x(t) = A \cos(\omega t + \phi) $$
If $\phi = 0$, it simplifies to $x(t) = A \cos(\omega t)$. -
Determine the specific positions and times
Identify the specific positions that the question asks for and set $x(t)$ equal to those values. Solve for $t$:
$$ A \cos(\omega t + \phi) = x $$
Rearranging gives:
$$ \cos(\omega t + \phi) = \frac{x}{A} $$
Then solve for $\omega t + \phi$ using the inverse cosine function. -
Calculate the time values
Use the periodic nature of the cosine function to find multiple solutions for $t$:
$$ \omega t + \phi = \cos^{-1}\left(\frac{x}{A}\right) + 2\pi n $$ $$ \omega t + \phi = -\cos^{-1}\left(\frac{x}{A}\right) + 2\pi n \quad (n \in \mathbb{Z}) $$
From these equations, isolate $t$ to find specific time values. -
Find the speed at those positions
The speed $v(t)$ of the particle in SHM can be found by taking the derivative of the position function:
$$ v(t) = -A\omega \sin(\omega t + \phi) $$
Plug in the values of $t$ obtained in the previous steps to find the speed at those specific times.
The specific positions and times, along with their corresponding speeds will depend on the values of $A$, $\omega$, and $\phi$, which need to be provided to yield numerical answers.
More Information
Simple harmonic motion is a fundamental concept in physics, applicable to many real-world systems like pendulums and springs. In SHM, the motion is sinusoidal, and specific properties, such as maximum speed and amplitude, are straightforward to calculate using the defined equations.
Tips
- Forgetting to account for the periodic nature of the cosine function leading to multiple solutions for time.
- Not checking if the position values are within the amplitude limits (if $|x| > A$, then that position is unreachable).
- Confusing the phase shift $\phi$ with $t=0$ conditions.
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