Solving for an Unknown Mass Using Hooke's Law and Simple Materials

[daodude1987]

Hey guys, I'm a beginner physics student and I don't really know how to answer this question:

I have a spring, ruler,3 known masses, and 1 unknown mass.How would I find the unknown mass using these materials? Is it possible to solve using Hooke's Law? It would be very helpful if you guys can provide some equations or include any diagrams. Also how would I derive the needed equations from a graph? Thanks a lot!

 

[petmar]

... and, pencils down.

well, let's see what we can do here.
first of all, remember hooke's law for a vertical opposition system (a spring with a mass hanging from it):
F=mg=-ky
so, from this we know that the spring has a linear stretching dependent upon the mass placed on it. hang the spring from a hook, and then put the ruler next to it. now, put the first known mass on it. measure the displacement now (y).from this, you have one value of k, by doing -mg/y=k. (g=9.81m/sec^2, and the - denotes the downwards direction.) now, do the other two masses the same way, and average the three values. k=(k1+k2+k3)/3=((m1g/y1)+(m2g/y2)+(m3g/y3))/3
where k is the average k.
now we get to the unknown mass.
measure the mass by putting it on the spring the same way as the first three. now measure its displacement (y), then you get (ky)/g=m.
voila! you have the mass!

remember: masses are in kilograms (kg), forces in Newtons (N), and displacements in meters (m). hope this helps!

 

[HallsofIvy]

"I have a spring, ruler,3 known masses, and 1 unknown mass.How would I find the unknown mass using these materials? Is it possible to solve using Hooke's Law? It would be very helpful if you guys can provide some equations or include any diagrams. Also how would I derive the needed equations from a graph?"

Hooke's law should work nicely. Hooke's law says that (for small extensions or compressions) the force required to stretch (or compress) a spring is proportional to the amount of stretch (or compression): F= -kx. If you attach one of the masses to the spring and hold it up so the mass is hanging down, the force is the gravitational force on the object (its weight), -Mg so that -Mg= -kX.
Using the known masses you can solve for k: k= Mg/X (X, is the stretch caused by known mass M).

Once you know k, hook on the unknown mass, m, and use mg= kx=
(Mg/X)x so m= M(x/X).

Strictly speaking you should only need one "known mass" to do that. Since F= -kx, the graph should be a straight line passing through (0,0). It takes two points to determine a straight line so (theoretically) only one mass should be needed. In actual practice, because of "experimental" error (and the fact that Hooke's law is only approximate) the three masses will give three slightly different answers.

What you can do is this: Set up a coordinate system with "mass" on the x-axis and "amount of stretch" on the y axis. Mark the points for the three known masses and the corresponding stretch of the spring. Those three should lie approximately on a straight line- and (0,0) should also be on that line.

Put your unknown mass on the spring and measure the stretch. Now you don't need to worry about getting a "formula" or "equation" from the graph. Mark the measured stretch on the y-axis of your graph, go over horizontally to the graph and then down to the x-axis to read off the mass.

 

[daodude1987]

Thanks a lot you guys! Everything is much clearer!