Solving a Harmonic Motion Problem: Finding Position, Velocity, and Acceleration

In summary, the conversation discusses an engine piston that undergoes simple harmonic motion with a given position and time equation. The participants also solve for the piston's position, velocity, acceleration, and period and amplitude of motion.
  • #1
chocolatelover
239
0
Hi everyone,

Could someone please help me with this problem?

Homework Statement


In an engine, a piston oscilates with simple harmonic motion so that its position varies according to the following expression, where x is in centimeters and t is in seconds.
x=5.00cmcos(3t+pi/5)

a. At t=0, find the position of the piston
b. What is its velocity?
c. What is its acceleration?
d. Find the period and amplitude of the motion


Homework Equations





The Attempt at a Solution



a. 5cmcos(3(0)+pi/5)=5cos(pi/5)=4.05cm

b. -5sin(pi/5)=-2.9

c. -5cos(pi/5)=-4.05

d. 5cm=amplitude

Thank you very much
 
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  • #2
[itex]x=5cos(3t+\frac{\pi}{5})[/itex]

[tex]v=\frac{dx}{dt}=-5sin(3t+\frac{\pi}{5}) \times 3=-15sin(3t+\frac{\pi}{5}) [/tex]
 
  • #3
Thank you very much

Regards
 

1. What is harmonic motion?

Harmonic motion refers to the repetitive back-and-forth movement of an object around a central equilibrium point, such as a pendulum swinging or a mass on a spring oscillating.

2. How is harmonic motion different from other types of motion?

Harmonic motion is characterized by a restoring force that is directly proportional to the displacement of the object from its equilibrium position. This means that the further the object is from its equilibrium point, the stronger the force pulling it back towards that point. Other types of motion, such as linear or circular motion, do not have this restoring force.

3. What is a simple harmonic oscillator?

A simple harmonic oscillator is a system that exhibits harmonic motion, meaning it has a restoring force that is directly proportional to the displacement of the object from its equilibrium position. This can be represented mathematically using Hooke's Law, which states that the force exerted by a spring is equal to the spring's spring constant multiplied by the displacement of the spring from its equilibrium position.

4. How is a simple harmonic oscillator described mathematically?

A simple harmonic oscillator can be described using the equation x(t) = A*cos(ωt + φ), where x(t) represents the displacement of the object from its equilibrium position at time t, A is the amplitude of the oscillation, ω is the angular frequency, and φ is the phase angle. This equation is derived from the solution to the differential equation for harmonic motion.

5. What are some real-life examples of harmonic motion?

Some common examples of harmonic motion include the swinging of a pendulum, the motion of a mass attached to a spring, and the vibrations of a guitar string. Other examples include the motion of a mass attached to a rubber band, the motion of a diving board, and the motion of a child on a swing.

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