Find the work done by a spring using potential energy considerations

AI Thread Summary
To find the work done by a spring from -x to +x, the potential energy change must be calculated using the formula U = 1/2kx^2. The spring is compressed at -x and extended at +x, with the elastic potential energy (EPE) decreasing as it moves from -x to 0 and increasing from 0 to +x. The overall change in EPE is determined by subtracting the initial EPE from the final EPE, which reflects the work done by the spring. The work done by the spring is equal to the negative of this change in potential energy. Understanding these energy transitions is crucial for solving the problem accurately.
good jelly
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Homework Statement
Find the work done by spring from -x to x
Relevant Equations
W=FS U=1/2kx^2
At first I tried using U1 = 1/2kx^2 from -x to zero then U2 from zero to +x but I don't know shall I add U1 and U2 or subtract them?
 
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Does x=0 imply the spring Isn't compressed ir stretched?
 
good jelly said:
Homework Statement: Find the work done by spring from -x to x
Relevant Equations: W=FS U=1/2kx^2

At first I tried using U1 = 1/2kx^2 from -x to zero then U2 from zero to +x but I don't know shall I add U1 and U2 or subtract them?
Which is reasonable do you think ?
 
@good jelly, remember that x is the spring’s length minus it natural (unloaded) length.

Can you answer the following questions?

1. In the initial state (‘-x’), is the spring extended or compressed?

2. What is the value of the elastic potential energy (EPE) in the initial state?

3. Does the EPE increase or decrease going from the initial state to the unloaded (x=0) state? By how much?

4. In the final state (‘+x’), is the spring extended or compressed?

5. What is the value of the EPE in the final state?

6. Does the EPE increase or decrease going from the unloaded (x=0) state to the final state? By how much?

7. Considering your answers to Q3 and Q6, what is the overall change in EPE?

Minor edit.
 
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By definition, the change in potential energy from point A to point B is the negative of the work done by the conservative force on the system from point A to point B.

Can you find the potential energy change when the spring is stretched from -x to +x? Hint: Change is final value minus initial value.
 
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