What is the total mechanical energy of a particle falling on an extended string?

[Jessica8956]

1. Find an expression for the total mechanical energy when the string is extended and find the maximum distance that the particle falls in terms of g
Unsure
3. My attempt at solutionHi Here is the question I am looking at http://imgur.com/a6T8zH7I take the datum as the fixed point so there is no kinetic energy to start with.

There is potential energy of mgx which is (4kg)(9.81m/s^2)(0.5) = 19.62 Joules

Kinetic energy is (1/2)mv^2, therefore 4v^2 as mass is 4kg

So total mechanical energy is (kinetic energy) + (potential energy) = 5.886J + 4v^2

Not sure if that is correct as I find this very confusing, and can't get an idea of how to answer the question that asks for the maximum distance that the particle falls

 

[Simon Bridge]

When the particle falls, does it gain or lose gravitational potential energy?

If "gain" - where does the energy come from?
If "lose" - where does the energy go?

Notice that energy may also be stored in the spring.
I think part (a) is just asking how much energy is stored in the spring for a given extension x, and they just want the equation.

 

[Jessica8956]

When the particle falls, does it gain or lose gravitational potential energy?

If "gain" - where does the energy come from?
If "lose" - where does the energy go?

Notice that energy may also be stored in the spring.
I think part (a) is just asking how much energy is stored in the spring for a given extension x, and they just want the equation.

Thanks for the reply :)

I'd say it loses gravitational potential energy and gain kinetic energy. So goes towards kinetic.

The question is using a string and not a spring if that makes any difference?

 

[BvU]

Initially I thought differently from Simon, but he might be right, since otherwise the answer for b) would already be featuring in a).

So yes, the first 0.5 m all that happens is the conversion of potential energy from gravity into kinetic energy. After that, ... (ever done a bungee jump? I didn't but I've seen videos)

I understand the 19.62 J value, but I don't understand the sign -- you start at your reference point with h = 0 and v =0, so I would say the total energy is 0 when letting go. It should still be zero after 0.5 m, so "total mechanical energy is ..." looks wrong to me.

I also have trouble believing 1/2 m is 4 kg in the kinetic energy expression just above...

 

[Simon Bridge]

BvU has it right.

The point of it being a string and not a spring is that there would be energy stored in a spring at the start (the spring would be compressed). This is a bungee jump problem - with an ideal bungee cord so don't use this sort of calculation on a real bungee cord.

Initially gravitational PE is being exchanged for KE - but when the unstretched length of the string is reached, what happens?

Look at it another way:
The mass will fall until the string has extended enough to stop it.
If the natural length of the string is L, and the particle falls from y=0 to y=y, then
How much gravitational PE did it lose?
Where has it all gone?
What is the final extension of the string?
What is the equation relating the energy stored in a spring to the extension of the spring?

 

[Jessica8956]

So total mechanical energy = gravitational potential + kinetic energy + elastic potential

The gravitational potential is mg x height

Which is 4(9.81)(0.5) = 19.62 JoulesThe elastic potential is (1/2)kx^2

Which is (1/2)100x^2 = 50x^2The kinetic energy is (1/2)mv^2

Which is (1/2)4v^2 = 2V^2So total potential energy = 19.62 + 50x^2 + 2v^2

I've read your comments but I still think I lack some of the knowledge of what equations are required to be used :/

 

[BvU]

Equations plenty. Now give them something to work on by choosing a coordinate system with a direction and by expressing these three energies in a coordinate.
What is x, when is it positive, when negative
What is v, when is it positive, when negative
If height has to do with x, what is their relationship ? Drop one of them.

 

[Simon Bridge]

So total mechanical energy = gravitational potential + kinetic energy + elastic potential

Where does the kinetic energy come from?
Where does it go?

Where does the elastic potential come from?

Certainly at any position below the natural length of the string, there will be all three.

Each is related to the others in some way, so a simplistic treatment is not going to work.
The way you were setting things up in post #1, at t=0, the total mechanical energy is zero.
http://en.wikipedia.org/wiki/Mechanical_energy