Energy Conservation and Potential-Energy Analysis in Simple Harmonic Motion

All flashcards

How is elastic potential energy stored in an ideal spring calculated?

With zero potential energy at equilibrium, $U=\frac{1}{2}kx^2$, where $k$ is stiffness and $x$ is displacement.

How is work by a conservative spring force related to potential energy?

The work done by the spring force equals the negative change in potential energy: $W_{\text{spring}}=-\Delta U$.

What does the slope of a one-dimensional potential-energy curve determine?

The force is the negative slope: $F_x=-\frac{dU}{dx}$.

What characterizes stable equilibrium?

A small displacement produces a restoring force that tends to return the system to equilibrium.

How does an ideal spring oscillator’s speed depend on position?

Its speed is $|v|=\sqrt{\frac{k}{m}(A^2-x^2)}$, where $A$ is amplitude.

Quiz questions

1. What is the total mechanical energy of an undamped mass–spring oscillator with amplitude $A$?Answer: $E=\frac{1}{2}kA^2$
2. At a turning point of an ideal spring oscillator, how is its mechanical energy distributed?Answer: It is entirely spring potential energy.
3. For an ideal spring oscillator, which expression gives speed at displacement $x$?Answer: $|v|=\sqrt{\frac{k}{m}(A^2-x^2)}$
4. On a one-dimensional potential-energy graph, what type of equilibrium is indicated by a local maximum?Answer: Unstable equilibrium
5. Where is the magnitude of the ideal spring force greatest during an oscillation of amplitude $A$?Answer: At the turning points, where $|x|=A$.