At a point where \(E=30\,\mathrm{J}\) and \(U=18\,\mathrm{J}\), find \(K\).

What condition defines a turning point on an energy diagram?

If \(U(x)>E\), is the particle allowed at that position?

What does the vertical gap between \(E\) and \(U(x)\) represent?

A \(2\,\mathrm{kg}\) particle has \(E=40\,\mathrm{J}\) and \(U=24\,\mathrm{J}\). Find its speed.

For \(U(x)=x^2\,\mathrm{J}\) and \(E=9\,\mathrm{J}\), find the turning points.

For \(U(x)=x^2\,\mathrm{J}\) and \(E=9\,\mathrm{J}\), state the allowed region.

A particle is at a local minimum of \(U(x)\). What kind of equilibrium is this?

For \(U(x)=x^2-4\) and \(E=5\), find the turning points and allowed region.

For \(U(x)=x^4-5x^2\), identify the equilibrium positions.

For \(U(x)=x^4-5x^2\), classify the equilibria and describe the allowed regions for energies satisfying \(-25/4<E<0\).

For \(U(x)=\alpha x^4-\beta x^2\), with \(\alpha>0\) and \(\beta>0\), derive the energy regimes for motion: no motion, rest at a stable equilibrium, trapped in one well, just reaches the central barrier, and crosses between wells.