Lesson 6 · 30 min

Conservative Forces and Potential Energy

The weight and a spring have something friction lacks: their work depends only on where the particle starts and ends. That lets us book their work in advance, as energy stored in the particle's position. Lift a crate and the work you did is waiting there, ready to come back as speed when it falls.

Learning objectives

Conservative forces

Lesson 3 found two: the weight, \(U_W = -W\,\Delta y\), and a linear spring, \(U_s = -\left(\tfrac12 ks_2^2 - \tfrac12 ks_1^2\right)\). Gravity at any distance from the Earth is conservative too, as are electric forces.

Kinetic friction is not. It always opposes the motion, so going around a loop it takes work away on every part of it: the closed-path test fails. Air drag, a push from your hand and the force of a motor are nonconservative too. Their work depends on what happens along the way.

Example 6.1 — Around a closed path

A \(40\ \text{kg}\) crate is dragged slowly around a closed loop \(20\ \text{m}\) long on a level floor, back to where it started. The coefficient of kinetic friction is \(0.3\). Find the work of friction and the work of the weight over the trip.

Show solution
\[ \colF{U_f} = -\mu_k mg\,d = -(0.3)(40)(9.81)(20) = -2354\ \text{J}, \qquad \colVg{U_W} = -W\,\Delta y = 0 \]

Back at the start, the weight's work is zero, as it must be for a conservative force. Friction's is not: \(2.35\ \text{kJ}\) went into heating the floor and the crate.

Potential energy

For a conservative force, define a function of position, the potential energy \(V\), so that the force's work is the drop in \(V\):

Work of a conservative force

\[ U_{1\to2} = V_1 - V_2 = -\Delta V \]

When the force does positive work, its potential energy goes down, and the other way round.

Gravitational potential energy

Comparing with \(U_W = -W(y_2 - y_1)\) gives

Gravitational potential energy (near the Earth)

\[ \colVg{V_g = W y = m g y} \]

\(y\) is measured upward from a horizontal datum that you choose. Above the datum \(V_g \gt 0\); below it, \(V_g \lt 0\).

Only changes in \(V_g\) ever enter a calculation, so the datum is yours to choose. Put it where it saves arithmetic: at the lowest point of the motion, or at one of the two positions.

Figure 6.1 A \(2\ \text{kg}\) ball moves from a balcony at \(A\) down to a table at \(B\). Drag the datum (the blue dashed line) anywhere. Both potential energies change, but their difference does not, and neither does the weight's work \(U_W = V_A - V_B\).

Example 6.2 — Two datums, one answer

The ball of Figure 6.1 moves from the balcony, \(4\ \text{m}\) above the floor, to the table, \(1\ \text{m}\) above the floor. Find its potential energies at \(A\) and \(B\) with the datum (a) at the floor and (b) at the balcony, and the work of its weight.

Show solution

\(W = (2)(9.81) = 19.62\ \text{N}\).

(a) Datum at the floor: \(V_A = (19.62)(4) = 78.48\ \text{J}\), \(V_B = (19.62)(1) = 19.62\ \text{J}\).

(b) Datum at the balcony: \(V_A = 0\), \(V_B = (19.62)(-3) = -58.86\ \text{J}\).

Either way, \(U_W = V_A - V_B = 58.86\ \text{J}\), the same as \(-W\,\Delta y\).

Elastic potential energy

Comparing with \(U_s = -\left(\tfrac12 ks_2^2 - \tfrac12 ks_1^2\right)\) gives

Elastic potential energy of a linear spring

\[ \colVe{V_e = \tfrac12 k s^2} \]

\(s\) is the deformation from the unstretched length, so \(V_e \ge 0\), the same stretched or compressed by the same amount. There is no datum to choose: \(V_e = 0\) when the spring is unstretched.

Example 6.3 — From compressed to stretched

A spring with \(k = 1200\ \text{N/m}\) goes from \(0.05\ \text{m}\) of compression to \(0.10\ \text{m}\) of stretch. Find its elastic potential energy at each end and the work it does on the attached particle.

Show solution
\[ V_1 = \tfrac12(1200)(0.05)^2 = 1.5\ \text{J}, \qquad V_2 = \tfrac12(1200)(0.10)^2 = 6.0\ \text{J}, \qquad \colVe{U_s} = V_1 - V_2 = -4.5\ \text{J} \]

The sign of the deformation never matters: on the way the spring first gives back its \(1.5\ \text{J}\) as it passes through its unstretched length, then takes \(6.0\ \text{J}\) as it stretches.

Force and potential energy diagrams

Over a small move \(ds\) along the path, the work of a conservative force is \(F_t\,ds = -dV\). So the force's component along the path is minus the slope of its potential energy:

\[ F_t = -\frac{dV}{ds} \qquad \left(\text{in three dimensions, } \Fvec = -\nabla V\right) \]

Where \(V\) falls, the force pushes forward; where it climbs, it pushes back; where \(V\) is flat, there is no force along the path: an equilibrium position. Plotting \(V\) against position, together with a horizontal line at the total energy, shows the whole motion at a glance. Lesson 7 shows why: with only conservative forces, \(T = E - V\), so the gap between the line and the curve is the kinetic energy.

Figure 6.2 The potential energy of a \(1\ \text{kg}\) cart on a smooth track with two hills and a bumper spring at the right end: \(\colVg{V_g}\) follows the track's height, and \(\colVe{V_e}\) rises steeply in the spring. With total energy \(E\) (dashed), the cart can be only where \(V \le E\): the green gap is its kinetic energy, and it turns back where the line meets the curve. Pick a point to see the force \(-dV/ds\).

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Key takeaways