How Is Compression Distance Calculated in a Friction-Affected Spring System?

[rmc112]

A 6 kg mass slides to the right on a surface
having a coeffcient of friction 0.78 as shown
in the figure. The mass has a speed of 8 m/s
when contact is made with a spring that has
a spring constant 147 N/m. The mass comes
to rest after the spring has been compressed
a distance d. The mass is then forced toward
the left by the spring and continues to move
in that direction beyond the unstretched posi-
tion. Finally the mass comes to rest a distance
D to the left of the unstretched spring.
The acceleration of gravity is 9.8 m/s^2.
Find the compressed distance d. Answer in
units of m. Also find Vf and D.

im thinking i would use W=(1/2)kx^2 with k=147 N/m and x being the distance i need to find. i don't know what to set W equal to.

http://pics.bbzzdd.com/users/turtle/spring.JPG

 

[gunblaze]

A 6 kg mass slides to the right on a surface
having a coeffcient of friction 0.78 as shown
in the figure. The mass has a speed of 8 m/s
when contact is made with a spring that has
a spring constant 147 N/m. The mass comes
to rest after the spring has been compressed
a distance d. The mass is then forced toward
the left by the spring and continues to move
in that direction beyond the unstretched posi-
tion. Finally the mass comes to rest a distance
D to the left of the unstretched spring.
The acceleration of gravity is 9.8 m/s^2.
Find the compressed distance d. Answer in
units of m. Also find Vf and D.

im thinking i would use W=(1/2)kx^2 with k=147 N/m and x being the distance i need to find. i don't know what to set W equal to.

http://pics.bbzzdd.com/users/turtle/spring.JPG

clue: In this question, you need to apply your knowledge on energy conservation. Yes, you do need your elastic potential energy as your spring is compressed by the mass. But what else? When your mass is moving, what kind of energy does it possess? Will there be any energy loss due to friction? Equate the energy that causes the mass to move forward to the energy that causes the mass to slow down in its directed motion and you should be able to solve for ur d.

 

[rmc112]

would it be kinetic energy - energy lost to friction = 1/2kx^2 ?

(1/2mv^2) - (m_k*9.8*m) = 1/2kx^2

i solved for x and got 1.41 which was wrong.

where did i go wrong?

 

[OlderDan]

would it be kinetic energy - energy lost to friction = 1/2kx^2 ?

(1/2mv^2) - (m_k*9.8*m) = 1/2kx^2

i solved for x and got 1.41 which was wrong.

where did i go wrong?

You left something out. It would help you to check your equation for consistent dimensions. Two of your terms have dimensions of energy. One does not.

 

[civil_dude]

Ditto Dan. If you don't get the correct answer, the first thing you should check is your units. If they don't check, then you are missing a value, and if you are sharp, you can figure out what term, and then what unit to look for.

 

[gunblaze]

would it be kinetic energy - energy lost to friction = 1/2kx^2 ?

(1/2mv^2) - (m_k*9.8*m) = 1/2kx^2

i solved for x and got 1.41 which was wrong.

where did i go wrong?

Don't go careless now. Your steps are right. You are simply losing a value in finding ur energy due to friction.

 

[rmc112]

ok i finally got it and found that the spring compresses 1.334 meters. now to find the final velocity i thought i would use 1/2mv^2=1/2kx^2 solve for v and it was also wrong. I am not having too much luck on this problem...

 

[OlderDan]

ok i finally got it and found that the spring compresses 1.334 meters. now to find the final velocity i thought i would use 1/2mv^2=1/2kx^2 solve for v and it was also wrong. I am not having too much luck on this problem...

When the spring is fully compressed, all remaining energy is stored in the spring. When the mass comes to rest, all that energy has been dissipated by friction. In the end, there is no velocity, and since firction is doing work during the compression and restoration, the velocity of the mass is less than 8m/s when the spring is restored.

If you have incorporated all the friction work correctly, make sure you are finding the distance moved from the restored spring position, not the distance moved from the compressed position.

 

[rmc112]

im sorry, i still don't get it at all. i have tried this problem so many ways and looked in my textbook and the internet for over an hour. i either need it explained where an idiot can get it, or an equation.

 

[OlderDan]

im sorry, i still don't get it at all. i have tried this problem so many ways and looked in my textbook and the internet for over an hour. i either need it explained where an idiot can get it, or an equation.

You got the correct value for the compression of the spring by fixing your equation to read

(1/2mv^2) - (m_k*9.8*m)x = 1/2kx^2

That x that was added to the second term makes that term the work done by friction during the compression. As the spring is restored, it gives its stored potential energy to the mass, except for some energy lost to friction. The mass acquires kinetic energy which is eventually lost to friction. You do not need to know how much kinetic energy the mass acquired, because it is all lost to friction in the end. All you need to know is that the energy that had been stored in the spring is ultimately all converted to heat as friction does work on the mass. All you need to do is set the spring energy when compressed equal to the work done by friction and solve for the distance the mass has to move for this much work to be done. D is the distance the friction force acts minus the compression distance of the spring.