Potential energy of an object-spring system
Potential energy of an object-spring system
Assuming a reference level in the unstretched position of the spring, the potential energy of an object-spring system is\[E_{\mathrm{P,\,spring}} = \frac12 k x^2,\]
where \(x\) is the stretch or compression of the spring.
|
|
The expression above follows from the formula for the work done by a spring, which we have already seen. Without loss of generality, we can place the origin of our coordinate system to the unstretched position of the spring: \(x_0 = 0.\) The potential energy stored in the spring in position (1): \[E_{\mathrm{P}1} = W_{1 \rightarrow 0} = \frac12 k x_1^2 - \frac12 k 0^2 = \frac12 k x_1^2.\] The potential energy stored in the spring in the reference position: \[E_{\mathrm{P}0} = W_{0 \rightarrow 0} = 0.\] The potential energy stored in the spring in position (2): \[E_{\mathrm{P}2} = W_{2 \rightarrow 0} = \frac12 k x_2^2 - \frac12 k 0^2 = \frac12 k x_2^2.\] |
