What is the relationship between conservative work and potential energy change?

A conservative force does \(12\,\mathrm{J}\) of work. Find \(\Delta U\).

A particle returns to its starting point under a conservative force. What is the net work by that force?

Name two common conservative forces in introductory mechanics.

A conservative force changes potential energy from \(20\,\mathrm{J}\) to \(5\,\mathrm{J}\). Find the work done by the force.

A particle has \(K_i=8\,\mathrm{J}\), \(U_i=15\,\mathrm{J}\), and later \(U_f=6\,\mathrm{J}\). Find \(K_f\) if only conservative forces act.

Why can a potential energy function be assigned to a conservative force?

A block slides along two different frictionless paths between the same two heights. Compare work done by gravity.

A particle moves under a conservative force from A to B. Along one path the force does \(18\,\mathrm{J}\) of work. Along a second path from B back to A, find the work.

A particle moves from \(x=1\,\mathrm{m}\) to \(x=4\,\mathrm{m}\) under a conservative force with \(U(x)=3x^2\,\mathrm{J}\). Find the work done by the force.

A force field is \(\vec{F}=2axy\,\hat{\imath}+ax^2\,\hat{\jmath}\), where \(a\) is constant. Decide whether the force is conservative, find a potential \(U(x,y)\) with \(U(0,0)=0\), and find the work done by the force from \((0,0)\) to \((X,Y)\).

A particle of mass \(m\) moves in one dimension under the conservative force \(F(x)=-ax+bx^3\), with \(a>0\) and \(b>0\). It starts at \(x=0\) with speed \(v_0\). Construct \(U(x)\) using \(U(0)=0\), then classify the possible motion regimes as \(v_0\) varies.