# Harmonic Motion of Oscillating Particle

• MD_Programmer
In summary, a particle moves along the x axis with an initial position of 0.280 m, velocity of 0.200 m/s, and acceleration of -0.450 m/s^2. After 4.10 s of constant acceleration, its position is -2.68 m and its velocity is -1.65 m/s. When the particle oscillates in simple harmonic motion for 4.10 s with the same initial conditions, the angular frequency can be found using the equation a = -w^2x, and the amplitude can be found using the conservation of energy equation. The phase constant can be found by taking the inverse of a trig function, and the position and velocity after 4.10 s can
MD_Programmer

## Homework Statement

A particle moves along the x axis. It is moving initially at the position 0.280 m, moving with velocity 0.200 m/s and acceleration -0.450 m/s^2. Suppose it moves with constant acceleration for 4.10 s.

(a) Find the position of the particle after this time.

(b) Find its velocity at the end of this time interval.

We take the same particle and give it the same initial conditions as before. Instead of having a constant acceleration, it oscillates in simple harmonic motion for 4.10 s around the equilibrium position x = 0.
(c) Find the angular frequency of the oscillation. Hint: in SHM, a is proportional to x.

(d) Find the amplitude of the oscillation. Hint: use conservation of energy.

(e) Find its phase constant 0 if cosine is used for the equation of motion. Hint: when taking the inverse of a trig function, there are always two angles but your calculator will tell you only one and you must decide which of the two angles you need.

(f) Find its position after it oscillates for 4.10 s.

(g) Find its velocity at the end of this 4.10 s time interval.

## Homework Equations

x(t) = A cos(wt + phi)
w = sqrt(k/m)
v = dx/dt = -w Asin(wt + phi)
Vmax = w A = sqrt(k/m) A
a = d^2x/dt^2 = -w^2 Acos(wt + phi)
T = 2pi sqrt(m/k)
f = 1/T
w = 2pi f

## The Attempt at a Solution

found a to be -2.68 m
found b to be -1.65 m/s

found a to be -2.68 m
found b to be -1.65 m/s
... please show your working and your reasoning behind each of these answers, and you best attempt so far for the last ones: how you think about these problems will help me guide you.

Guessing: for c. the acceleration is no longer constant.
The given number is the instantaneous acceleration at the given position.
(that help)

Also you are missing an equation - very important one from the definition of SHM.
Usually about the way the restoring force varies with position.

Last edited:

## 1. What is harmonic motion?

Harmonic motion refers to the repetitive movement of an oscillating particle around an equilibrium point. This type of motion follows a sinusoidal pattern, with the particle moving back and forth between two points.

## 2. What factors affect the harmonic motion of an oscillating particle?

The harmonic motion of an oscillating particle is affected by three main factors: the mass of the particle, the spring constant of the system, and the amplitude of the oscillation. A larger mass will result in a slower oscillation, a higher spring constant will result in a faster oscillation, and a larger amplitude will result in a wider range of motion.

## 3. What is the relationship between harmonic motion and simple harmonic motion?

Harmonic motion is a more general term that encompasses all types of oscillatory motion, including simple harmonic motion. Simple harmonic motion refers specifically to the motion of an object that is subject to a restoring force that is directly proportional to the displacement from equilibrium. In other words, simple harmonic motion follows a specific mathematical equation, while harmonic motion can encompass a wider variety of equations.

## 4. How is the period of a harmonic motion calculated?

The period of a harmonic motion can be calculated using the equation T = 2π√(m/k), where T represents the period, m represents the mass of the particle, and k represents the spring constant of the system. This equation is derived from the fact that the period of a simple harmonic motion is directly proportional to the square root of the mass and inversely proportional to the square root of the spring constant.

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

Some common examples of harmonic motion in real life include the swinging of a pendulum, the vibrations of a guitar string, and the motion of a mass attached to a spring. These examples all follow a sinusoidal pattern and are subject to a restoring force that brings the particle back to its equilibrium position.

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