# Oscillating horizontal mass attached to a spring with Friction

• Miguel Orta
In summary: There are two forces acting on the block: the spring force and the friction force. The spring force is constant, but the friction force depends on the amount of contact between the block and the surface.
Miguel Orta

## Homework Statement

The question is similar to last week’s, except that we will consider how friction may damp the oscillation with time. A block with mass m shown in the drawing is acted on by a spring with spring constant k. The block is pulled distance [x[/0] from equilibrium position (x=0) and released. The block would oscillate back and forth around equilibrium position (x=0). From the last week’s work, we know that if the friction between
the block and the surface that the block is sitting on can be ignored, the block will oscillate forever with block’s position given as x(t)=X0cos(ωt) where ω=√(k/m) is the angular frequency of the oscillation.

Now consider that in reality there is actually friction between the block and the surface, and the friction coefficient is μ<<1. The block is again pulled distance X0 from equilibrium and released. It oscillates many times and eventually comes to rest. Show that the decrease of amplitude is the same for each cycle of oscillations. Find the number of cycles n the mass oscillates before coming to rest.
Hint:You can still write down x(t) = Acos(ωt) but the amplitude (A) decreases with time. Consider the work done by the friction and what it means to the elastic potential energy of the spring.

x(t) = Acos(ωt)

## The Attempt at a Solution

I've tried multiple ways today with not much in terms of progress. Below is an example

ΔPe= 1/2(KX0^2) - fAn f = force by friction, n = number of oscillations, A = amplitude
-1/2mv^2 = 1/2(KX0^2) - fAn I then set it equal to the kinetic energy times -1. I did this because the kinetic at it's maximum is equal to the potential energy at its maximum. My reasoning was that the friction would do work on the block. And then I would be able to find the amplitude. I could then find a ratio between the new amplitude and the original one.

That's as far as I got, I tried cancelling some things out and plugging in some more stuff but nothing has worked so far.

Miguel Orta said:
ΔPe= 1/2(KX0^2) - fAn f = force by friction, n = number of oscillations, A = amplitude
That is not going to work because the "amplitude" is constantly changing.
Start from first principles: consider the forces during a half cycle, i.e. from one extreme displacement to the other.

## 1. What is an oscillating horizontal mass attached to a spring with friction?

An oscillating horizontal mass attached to a spring with friction is a physical system in which a mass is attached to a horizontal spring and placed on a surface with friction. The mass is able to move back and forth due to the force of the spring, but the motion is hindered by the opposing force of friction.

## 2. How does adding friction affect the oscillation of the mass?

Friction acts as a damping force on the oscillating mass, which means it decreases the amplitude and velocity of the oscillations over time. This results in a shorter period and a quicker decay of the oscillations.

## 3. What is the role of the spring in this system?

The spring provides the restoring force that allows the mass to oscillate. As the mass moves away from its equilibrium position, the spring exerts a force in the opposite direction, pulling the mass back towards the center and allowing it to continue oscillating.

## 4. How is the motion of the mass affected by the properties of the spring?

The properties of the spring, such as its stiffness or spring constant, can affect the frequency and amplitude of the oscillations. A stiffer spring will result in a higher frequency and smaller amplitude, while a more flexible spring will result in a lower frequency and larger amplitude.

## 5. What factors can affect the friction in this system?

The friction in this system can be affected by the type of surface the mass is placed on, the mass of the object, and the velocity of the oscillations. Additionally, the coefficient of friction between the mass and the surface can also impact the amount of friction present in the system.

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