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Potential Energy and the Conservation of Mechanical Energy

Store energy in height and in stretched springs, meet conservative forces, and use conservation of mechanical energy to solve motion without touching a single force equation.

Stored energy

Potential energy is energy an object has because of where it is or how it is arranged, waiting to be released as motion. Lift a book and you store gravitational potential energy; compress a spring and you store elastic potential energy. Let go, and either one turns into kinetic energy.

U_g = m g h

Gravitational potential energy near Earth's surface: mass times g times height.

U_s = \tfrac{1}{2} k x^2

Elastic potential energy stored in a spring stretched or compressed by x (from Hooke's law).

Conservative vs nonconservative forces

Only certain forces can be stored as potential energy. A conservative force does work that depends only on the start and end points, not on the path taken. Gravity and ideal springs are conservative: carry a mass up a straight ladder or a winding staircase to the same height and gravity does the same (negative) work. A round trip does zero net work, so the energy is fully recoverable.

A nonconservative force such as kinetic friction fails this test: drag a box the long way round and friction steals more energy than by the short way. Path-dependent forces cannot be bottled up as a potential energy — they dissipate energy instead, usually as heat.

F_x = -\frac{dU}{dx}

A conservative force is minus the slope of its potential-energy curve — the object is pushed 'downhill' on the energy graph.

Conservation of mechanical energy

Add kinetic and potential energy together and you get the mechanical energy, E = KE + U. Here is the payoff: when only conservative forces do work, mechanical energy is constant. This is the conservation of mechanical energy.

E = KE + U = \text{constant}

With only conservative forces, the total mechanical energy never changes.

\tfrac{1}{2} m v_i^2 + m g h_i = \tfrac{1}{2} m v_f^2 + m g h_f

Energy at the start equals energy at the end — the workhorse equation for frictionless problems.

Set a starting height and release the ball on the frictionless track: watch kinetic and potential energy swap continuously while their sum — the total bar — stays pinned. Then add friction and watch the total start to fall.

Energy diagrams

A potential-energy diagram plots U against position and turns a motion problem into a picture of a ball rolling on a landscape. Draw a horizontal line at the fixed total energy E; the object can only go where E \ge U. Where the line meets the curve, KE = 0 and the object turns around — those are the turning points.

Worked example: speed at the bottom of a drop

A cart starts from rest at height h on a frictionless track (a roller-coaster drop, say) and rolls to the bottom. How fast is it going there? Strategy: put the reference level U = 0 at the bottom. All the starting potential energy becomes kinetic energy — no forces needed.

m g h = \tfrac{1}{2} m v^2 \;\Rightarrow\; v = \sqrt{2 g h}

The mass cancels: everything reaches the same speed from the same height on a frictionless track.

For h = 5.0 m, v = \sqrt{2 \times 9.8 \times 5.0} \approx 9.9 m/s. The striking result is that mass cancelled — a marble and a boulder released from the same height arrive at the same speed. Note that this is the same v = \sqrt{2gh} you would get for an object in free fall, which makes sense: the track just redirects the motion without doing work.