How many oscillations until a damped block comes to rest?

[VishalChauhan]

Homework Statement

A block of mass m is attached to a spring of spring constant k. It lies on a floor with coefficient of friction μ. The spring is stretched by a length a and released. Find how many oscillations it takes for the block to come to rest.

Homework Equations

d2x/dt2 + k/m x = +_ μg

Also the block will stop when the spring is unstretched and the velocity is 0

The Attempt at a Solution

I tried to write equations for each part of the motion (ie when the frition acts along and opposite to the spring force) but its too messy. Is there another than solving the differential equation for each part of the motion.

 

[Simon Bridge]

conservation of energy?
look up the equation?

 

[VishalChauhan]

conservation of energy?

Thought so. But how would it give me the NUMBER of oscillations.

 

[Chestermiller]

You can use conservation of energy to only look at the times when the block is in its extreme positions, so that you don't have to consider the change in kinetic energy. So, on any given swing, the change in stored elastic energy of the spring is equal to the work done against friction. This will tell you how much the amplitude decreases in any given swing.

Chet

 

[Nathanael]

Do you have the answer? If so can you please share it? I'd like to check my answer.

Try writing a sequence (if that's the correct term) for the amplitude after half-oscillations (if that makes sense).

 

[Simon Bridge]

The problem is usually given as an exercise in learning to solve problems in general.
That means that part of the learning process is going through the painful process of running into dead ends, trying other things out and so on.
There are several ways to approach the answer, you are supposed to discover one of them.

It is not as simple as getting a nice equation to plug numbers into - you do actually have to play around a bit to figure it out.

It is very common, with real problems, that you have to start out with no clear idea how the endpoint is to be reached, you should get used to it ... in this case it is not immediately obvious how to get the number of oscilations from the conservation of energy ... but as you start writing down the expressions and thinking about what you know about how energy works for this system, you'll start to see possible relationships ... like the amount of energy lost related to the distance traveled in half a swing.

BTW: you can actually use the differential equation approach you started with - I don't think there is a way through to the solution that you would consider non-messy.

Learning to play about, to explore the models you use, is the point of the exercise.

 

[VishalChauhan]

Do you have the answer? If so can you please share it? I'd like to check my answer.

Try writing a sequence (if that's the correct term) for the amplitude after half-oscillations (if that makes sense).

For m=0.5kg,friction coefficient=0.1,a=3cm and k=2.45N/cm the number of oscillations is 3.5