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3. A particle moves in x - y plane according to the rule, x = asinot and y = a * cos(wt pi/2)) The particle?
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3. A particle moves in x - y plane according to the rule, x = asinot a...
The given equation represents the motion of a particle in the x-y plane. Let's break down the equation and understand it in detail.
- Equation: x = a * sin(ωt) and y = a * cos(ωt * π/2)
Explanation:
The equation consists of two parts, one for the x-coordinate (x) and the other for the y-coordinate (y). Both coordinates are represented as functions of time (t).
x-coordinate:
The x-coordinate is given by the equation x = a * sin(ωt), where a represents the amplitude of the motion and ω represents the angular frequency.
- Amplitude (a): The amplitude determines the maximum displacement of the particle in the x-direction from its equilibrium position. It represents the maximum value of the sine function.
- Angular Frequency (ω): The angular frequency determines the speed at which the particle oscillates in the x-direction. It represents the number of complete oscillations per unit time.
y-coordinate:
The y-coordinate is given by the equation y = a * cos(ωt * π/2), where a represents the amplitude of the motion and ω represents the angular frequency.
- Amplitude (a): The amplitude determines the maximum displacement of the particle in the y-direction from its equilibrium position. It represents the maximum value of the cosine function.
- Angular Frequency (ω): The angular frequency determines the speed at which the particle oscillates in the y-direction. It represents the number of complete oscillations per unit time.
Motion of the Particle:
Based on the given equations, we can observe the following characteristics of the particle's motion:
- The particle moves in a circular path centered at the origin (0,0) of the x-y plane.
- The amplitude (a) determines the radius of the circular path.
- The angular frequency (ω) determines the speed at which the particle moves along the circular path.
- The x-coordinate and y-coordinate of the particle vary sinusoidally with time.
- The particle completes one full revolution in the x-y plane for each complete cycle of the sine and cosine functions.
Visualization:
To visualize the motion of the particle, we can plot the x-coordinate and y-coordinate as functions of time. This will help us understand how the particle moves in the x-y plane.
- The x-coordinate (x = a * sin(ωt)) can be plotted on the x-axis.
- The y-coordinate (y = a * cos(ωt * π/2)) can be plotted on the y-axis.
By plotting these coordinates for different values of time, we can observe a circular motion pattern. The particle will move along the circular path with a certain speed determined by the angular frequency (ω) and a certain radius determined by the amplitude (a).
In conclusion, the given equation represents the motion of a particle in the x-y plane. The particle moves in a circular path centered at the origin, and its motion is characterized by the amplitude and angular frequency values. By visualizing the x-coordinate and y-coordinate as functions of time, we can understand the circular motion pattern of the particle.
- Equation: x = a * sin(ωt) and y = a * cos(ωt * π/2)
Explanation:
The equation consists of two parts, one for the x-coordinate (x) and the other for the y-coordinate (y). Both coordinates are represented as functions of time (t).
x-coordinate:
The x-coordinate is given by the equation x = a * sin(ωt), where a represents the amplitude of the motion and ω represents the angular frequency.
- Amplitude (a): The amplitude determines the maximum displacement of the particle in the x-direction from its equilibrium position. It represents the maximum value of the sine function.
- Angular Frequency (ω): The angular frequency determines the speed at which the particle oscillates in the x-direction. It represents the number of complete oscillations per unit time.
y-coordinate:
The y-coordinate is given by the equation y = a * cos(ωt * π/2), where a represents the amplitude of the motion and ω represents the angular frequency.
- Amplitude (a): The amplitude determines the maximum displacement of the particle in the y-direction from its equilibrium position. It represents the maximum value of the cosine function.
- Angular Frequency (ω): The angular frequency determines the speed at which the particle oscillates in the y-direction. It represents the number of complete oscillations per unit time.
Motion of the Particle:
Based on the given equations, we can observe the following characteristics of the particle's motion:
- The particle moves in a circular path centered at the origin (0,0) of the x-y plane.
- The amplitude (a) determines the radius of the circular path.
- The angular frequency (ω) determines the speed at which the particle moves along the circular path.
- The x-coordinate and y-coordinate of the particle vary sinusoidally with time.
- The particle completes one full revolution in the x-y plane for each complete cycle of the sine and cosine functions.
Visualization:
To visualize the motion of the particle, we can plot the x-coordinate and y-coordinate as functions of time. This will help us understand how the particle moves in the x-y plane.
- The x-coordinate (x = a * sin(ωt)) can be plotted on the x-axis.
- The y-coordinate (y = a * cos(ωt * π/2)) can be plotted on the y-axis.
By plotting these coordinates for different values of time, we can observe a circular motion pattern. The particle will move along the circular path with a certain speed determined by the angular frequency (ω) and a certain radius determined by the amplitude (a).
In conclusion, the given equation represents the motion of a particle in the x-y plane. The particle moves in a circular path centered at the origin, and its motion is characterized by the amplitude and angular frequency values. By visualizing the x-coordinate and y-coordinate as functions of time, we can understand the circular motion pattern of the particle.
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3. A particle moves in x - y plane according to the rule, x = asinot and y = a * cos(wt pi/2)) The particle?
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3. A particle moves in x - y plane according to the rule, x = asinot and y = a * cos(wt pi/2)) The particle? for NEET 2024 is part of NEET preparation. The Question and answers have been prepared according to the NEET exam syllabus. Information about 3. A particle moves in x - y plane according to the rule, x = asinot and y = a * cos(wt pi/2)) The particle? covers all topics & solutions for NEET 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for 3. A particle moves in x - y plane according to the rule, x = asinot and y = a * cos(wt pi/2)) The particle?.
3. A particle moves in x - y plane according to the rule, x = asinot and y = a * cos(wt pi/2)) The particle? for NEET 2024 is part of NEET preparation. The Question and answers have been prepared according to the NEET exam syllabus. Information about 3. A particle moves in x - y plane according to the rule, x = asinot and y = a * cos(wt pi/2)) The particle? covers all topics & solutions for NEET 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for 3. A particle moves in x - y plane according to the rule, x = asinot and y = a * cos(wt pi/2)) The particle?.
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