# Harmonic oscillation in classical mechanics

• sya deela
In summary, an object of mass 300g attached to a spring with constant k = 3.0Nm-1 and a resistive force linearly proportional to velocity is displaced 10mm to the right of equilibrium position and released. The equations of motion and the solution to the motion are derived, and the graph of position vs time is requested.
sya deela

## Homework Statement

An object of mass m = 300g is attached to a spring with a constant k = 3.0Nm-1 and is at rest on a smooth horizontal floor in a fluid where the resistive force is assumed to be linearly proportional to the velocity v. the object is then displaced 10mm to the right of the equilibrium position and released. Given the constant of proportionality c1 = 0.2kgs-1;

## Homework Equations

i) sketch the free body diagram (FBD) and write the equation of motion for the object immediately after it is set in motion
ii) write the solution to equation of motion in i) and describe the motion, and
iii) sketch the graph of the position versus time for the object.

## The Attempt at a Solution

i) F(x) = -kx
Fv-F(x)= m dv/dt
-c1v-kx= m dv/dt

ii) -c1v-kx= m dv/dt after inegrate both sides v0 is zero
then v=-kx/c1(1-e(-c1t/m))

i can't get the equation for position versus time...

Use ##v=dx/dt##.

eys_physics said:
Use ##v=dx/dt##.
i already try it..but as i know that for harmonic oscillation the graph should be cos / sin .So i don't know how to relate it.

sya deela said:

## Homework Statement

An object of mass m = 300g is attached to a spring with a constant k = 3.0Nm-1 and is at rest on a smooth horizontal floor in a fluid where the resistive force is assumed to be linearly proportional to the velocity v. the object is then displaced 10mm to the right of the equilibrium position and released. Given the constant of proportionality c1 = 0.2kgs-1;

## Homework Equations

i) sketch the free body diagram (FBD) and write the equation of motion for the object immediately after it is set in motion
ii) write the solution to equation of motion in i) and describe the motion, and
iii) sketch the graph of the position versus time for the object.

## The Attempt at a Solution

i) F(x) = -kx
Fv-F(x)= m dv/dt
-c1v-kx= m dv/dt

ii) -c1v-kx= m dv/dt after inegrate both sides v0 is zero
then v=-kx/c1(1-e(-c1t/m))

i can't get the equation for position versus time...

I don't understand what you are doing at ii). Here both ##x## and ##t## depends on ##t##. So, your equation is
$$-c1 v(t)-kx(t)=mdv/dt$$
By using ##v(t)=dx/dt## in this equation, you can derive a second-order differential equation for ##x(t)##.

## 1. What is harmonic oscillation in classical mechanics?

Harmonic oscillation in classical mechanics refers to a type of periodic motion where a system experiences a restoring force that is directly proportional to its displacement from equilibrium.

## 2. What are some real-life examples of harmonic oscillation?

Examples of harmonic oscillation in everyday life include a pendulum swinging back and forth, a mass on a spring bouncing up and down, and the motion of a child on a swing.

## 3. How is harmonic oscillation mathematically described?

Harmonic oscillation is described using the equation x = Acos(ωt+φ), where x is the displacement from equilibrium, A is the amplitude, ω is the angular frequency, and φ is the phase angle.

## 4. What is the relationship between mass, spring constant, and frequency in harmonic oscillation?

In harmonic oscillation, the frequency of the motion is directly proportional to the square root of the spring constant divided by the mass. This means that a higher mass or a stiffer spring will result in a lower frequency of oscillation.

## 5. How does energy behave in harmonic oscillation?

Energy in harmonic oscillation is conserved, meaning it remains constant throughout the motion. The energy is constantly being transferred between kinetic energy (when the object is in motion) and potential energy (when the object is at its maximum displacement).

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