Respuesta :
Answer:
The maximum velocity and maximum acceleration of the particle are approximately 12.566 inches per second and 789.568 inches per square second.
Explanation:
The equation of motion for the position of a particle experimentating a simple harmonic motion ([tex]x(t)[/tex]), measured in inches, is described by the following expression:
[tex]x(t) = A\cdot \cos \left(\frac{2\pi\cdot t}{T} +\phi\right)[/tex] (1)
Where:
[tex]A[/tex] - Amplitude, measured in inches.
[tex]t[/tex] - Time, measured in seconds.
[tex]T[/tex] - Period, measured in seconds.
[tex]\phi[/tex] - Phase, measured in radians.
Then, we obtain the formulas for the velocity and acceleration of the particle by differentiating (1):
[tex]v(t) = -\frac{2\pi\cdot A}{T}\cdot \sin \left(\frac{2\pi\cdot t}{T}+\phi \right)[/tex] (2)
[tex]a(t) = -\left(\frac{2\pi}{T} \right)^{2}\cdot A\cdot \cos \left(\frac{2\pi\cdot t}{T}+\phi \right)[/tex] (3)
From (2) and (3) we find that maximum velocity ([tex]v_{max}[/tex]), measured in inches per second, and maximum acceleration ([tex]a_{max}[/tex]), measured in inches per square second, are defined by the following formulas:
[tex]v_{max} = \frac{2\pi\cdot A}{T}[/tex] (4)
[tex]a_{max} = \left(\frac{2\pi}{T} \right)^{2}\cdot A[/tex] (5)
If we know that [tex]A = 0.2\,in[/tex] and [tex]T = 0.1\,s[/tex], then the maximum velocity and maximum acceleration of the particle are, respectively:
[tex]v_{max} = \frac{2\pi\cdot (0.2\,in)}{0.1\,s}[/tex]
[tex]v_{max} \approx 12.566\,\frac{in}{s}[/tex]
[tex]a_{max} = \left(\frac{2\pi}{0.1\,s} \right)^{2}\cdot (0.2\,in)[/tex]
[tex]a_{max} \approx 789.568\,\frac{in}{s^{2}}[/tex]
The maximum velocity and maximum acceleration of the particle are approximately 12.566 inches per second and 789.568 inches per square second.
When an object executes a to and fro motion is said to be a simple harmonic motion. The maximum velocity and maximum acceleration of the particle will be 12.566 in/s and 789.568 in/s².
What is simple harmonic motion?
When an object executes a to and fro motion in the definite plane when it is tied with the string. The type of motion will be the simple harmonic motion.
Simple harmonic motion is a form of periodic motion in mechanics and physics in which the restoring force on the moving item is directly proportional to the size of the object.
The value of maximum velocity is given by the formula in the SHM motion is
[tex]\rm V_{max = \frac{2\pi A}{T}[/tex]
[tex]\rm V_{max} = \frac{2\times 3.14 \times 0.2 }{0.1}[/tex]
[tex]\rm V_{max = 12.566 \;in/sec[/tex]
The value of maximum acceleration is given by the formula in the SHM motion is
[tex]\rm a_{max}=(\frac{2\pi}{T} )^2.A\\\\ \rm a_{max}=(\frac{2\times 3.14 }{0.1} )^2 \times 0.2 \\\\ \rm a_{max}=789.568\; in/sec^2[/tex]
Hence the maximum velocity and maximum acceleration of the particle will be 12.566 in/s and 789.568 in/s²
To learn more about the simple harmonic motion refer to thr link;
https://brainly.com/question/17315536
