- #1
pc2-brazil
- 205
- 3
Good afternoon,
First of all, this is not a homework question, but I'm not sure whether it should be posted on the Homework section.
My problem arose when I tried to think about what happens when a spring does work in two blocks.
First, consider two blocks (1 and 2) on a frictionless horizontal plane, and a spring. The two blocks are attached to each side of the spring, and the spring is initially on its equilibrium position.
Then, the two blocks are pulled away from each other by two external forces opposite to each other (F1 and F2), with the same magnitude, parallel to the spring, so that the spring is decompressed. So, the spring applies a restoring elastic force on each block. Let's call the restoring forces Fe1 (opposite to F1) and Fe2 (opposite to F2).
The displacement of blocks 1 and 2 are respectively x1 and x2. So, the spring is decompressed by a length of x = (x1 + x2) (which makes Fe1 = Fe2 = -k(x1 + x2)).
It means that the system stores a potential energy defined by:
[tex]\Delta U=\frac{1}{2}k(x_1+x_2)^2[/tex]
Right?
But, then, a problem arose in the reasoning:
Individually, each restoring force does negative work on each block. For block 1, the work done by the restoring force is:
[tex]W_{Fe1}=-\frac{1}{2}kx_1^2[/tex]
For block 2:
[tex]W_{Fe2}=-\frac{1}{2}kx_2^2[/tex]
But the sum of the works by the conservative forces equals the negative of the variation of potential energy of the system.
So, the sum of these works, [tex]W=W_{Fe1}+W_{Fe2}[/tex], should give:
[tex]W=W_{Fe1}+W_{Fe2}=-\Delta U[/tex]
But, in this case, the above is not true. What is wrong with this reasoning?
Thank you in advance.
First of all, this is not a homework question, but I'm not sure whether it should be posted on the Homework section.
My problem arose when I tried to think about what happens when a spring does work in two blocks.
First, consider two blocks (1 and 2) on a frictionless horizontal plane, and a spring. The two blocks are attached to each side of the spring, and the spring is initially on its equilibrium position.
Then, the two blocks are pulled away from each other by two external forces opposite to each other (F1 and F2), with the same magnitude, parallel to the spring, so that the spring is decompressed. So, the spring applies a restoring elastic force on each block. Let's call the restoring forces Fe1 (opposite to F1) and Fe2 (opposite to F2).
The displacement of blocks 1 and 2 are respectively x1 and x2. So, the spring is decompressed by a length of x = (x1 + x2) (which makes Fe1 = Fe2 = -k(x1 + x2)).
It means that the system stores a potential energy defined by:
[tex]\Delta U=\frac{1}{2}k(x_1+x_2)^2[/tex]
Right?
But, then, a problem arose in the reasoning:
Individually, each restoring force does negative work on each block. For block 1, the work done by the restoring force is:
[tex]W_{Fe1}=-\frac{1}{2}kx_1^2[/tex]
For block 2:
[tex]W_{Fe2}=-\frac{1}{2}kx_2^2[/tex]
But the sum of the works by the conservative forces equals the negative of the variation of potential energy of the system.
So, the sum of these works, [tex]W=W_{Fe1}+W_{Fe2}[/tex], should give:
[tex]W=W_{Fe1}+W_{Fe2}=-\Delta U[/tex]
But, in this case, the above is not true. What is wrong with this reasoning?
Thank you in advance.
Last edited: