# Equations of motion of damped oscillations due to kinetic friction

• phantomvommand
In summary, the block moves in a straight line under the influence of the spring and the friction force.
phantomvommand
Take rightwards as positive.
There are 2 equations of motion, depending on whether ##\frac {dx} {dt} ## is positive or not.
The 2 equations are:
##m\ddot x = -kx \pm \mu mg##

Is this SHM?

Possible method to solve for equation of motion:
- Solve the 2nd ODE, although “joining” the equations when ##\dot x ## changes from positive to negative is not easy.

We may write the equation of motion as
$$m\ddot{x}=-kx- sgn(\dot{x})\mu mg$$
where sgn(a)=1 for a>0, 0 for a=0, -1 for a<0.

phantomvommand
anuttarasammyak said:
We may write the equation of motion as
$$m\ddot{x}=-kx- sgn(\dot{x})\mu mg$$
where sgn(a)=1 for a>0, 0 for a=0, -1 for a<0.
Thanks for this; do you then go on to solve the 2nd ODE?
Also, is this SHM?

I observe the equation is a non linear one and do not expect to find general solution easily.

phantomvommand
anuttarasammyak said:
I observe the equation is a non linear one and do not expect to find general solution easily.
Is it just the sum of a particular function and complementary function?

Say k=0 and ##\dot{x}_0>0## we get familiar relation of
$$\dot{x}=-\mu g t + \dot{x}_0$$
$$x=-\frac{1}{2}\mu g t^2 + \dot{x}_0 t+x_0$$
for 0<t<##\frac{\dot{x}_0}{\mu g}##, x= ##\frac{\dot{x}_0^2}{2\mu g}+x_0## for t beyond.

For k##\neq##0 similarly you can solve the equation until when ##\dot{x}=0##. Then for time beyond it change sign of friction term until next time of ##\dot{x}=0## and so on.

Last edited:
phantomvommand said:
Summary:: A spring has 1 end fixed to the wall, and the other end is connected to a block. Find the equation of motion of the block, given that it experiences only the spring force and a friction force = ##\mu mg##.

Is this SHM?
No. SHM does not have anything like the friction force.

phantomvommand

## 1. What is an equation of motion for damped oscillations due to kinetic friction?

An equation of motion for damped oscillations due to kinetic friction is a mathematical representation that describes the motion of an object that is experiencing both damping (decrease in amplitude) and friction (resistance to motion) as it oscillates back and forth.

## 2. How is kinetic friction related to damped oscillations?

Kinetic friction is a type of friction that occurs when two surfaces are in contact and are sliding against each other. In the context of damped oscillations, kinetic friction acts as a damping force, reducing the amplitude of the oscillations over time.

## 3. What factors affect the equation of motion for damped oscillations due to kinetic friction?

The equation of motion for damped oscillations due to kinetic friction is affected by several factors, including the mass of the object, the spring constant of the oscillating system, the coefficient of kinetic friction between the surfaces, and the initial conditions (e.g. initial displacement and velocity).

## 4. Can the equation of motion for damped oscillations due to kinetic friction be solved analytically?

Yes, the equation of motion for damped oscillations due to kinetic friction can be solved analytically using techniques such as differential equations and calculus. However, in some cases, numerical methods may be needed to obtain a solution.

## 5. How does the presence of kinetic friction affect the period and frequency of damped oscillations?

The presence of kinetic friction increases the damping in the oscillating system, causing the amplitude to decrease over time. This results in a longer period and lower frequency of oscillation compared to a system without friction. As the amplitude continues to decrease, the period and frequency will also decrease until the oscillations eventually stop.

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