Chapter 8 Lectures: Potential Energy & Energy Conservation (Slides 1-10)

Fundamentals of Work, Energy, and Closed Paths

• Potential energy and the conservation of energy expand directly upon the concepts of mechanical work and kinetic energy.

• Gravitational potential energy near the surface of the Earth is commonly expressed as $U = mgh$, a formula that originates directly from the fundamental definition of work done by a conservative force against gravity.

• A closed path (or closed loop) is defined as any trajectory through space where the initial position vector and the final position vector of an object are identical:   $\mathbf{r}_i = \mathbf{r}_f$

• In a closed path, an object departs from a starting position, traverses an arbitrary continuous route in space, and ultimately returns to terminate at that exact initial starting coordinate.

• The net work ($W_{\text{net}}$) performed on a system or an object is a scalar quantity calculated as the algebraic sum of the individual works performed by all acting forces over their respective displacements:   $W_{\text{net}} = \sum_{i} W_i$

• For a constant force, work is evaluated via the vector dot product:   $W = \mathbf{F} \cdot \Delta\mathbf{r}$

• For a variable force whose magnitude or direction alters along the path, calculus is required, evaluating work as a line integral along the trajectory from the initial position to the final position:   $W = \int_{\mathbf{r}_i}^{\mathbf{r}_f} \mathbf{F} \cdot d\mathbf{r}$

• The classification of a force as either conservative or non-conservative is determined by analyzing the net work executed by that specific force over an arbitrary closed path.

Case Study 1: Spring Force Along a Closed Path

• A block of mass $m$ is attached to a horizontal spring mounted to a rigid wall, resting on a frictionless table.

• The spring possesses a characteristic stiffness designated by the spring constant $k$.

• A one-dimensional coordinate system along the horizontal $x$-axis is established where the equilibrium position (the natural, relaxed state where no forces act on the spring) is positioned at:   $x = 0$

• When the spring is displaced from $x = 0$, it exerts a restoring force governed by Hooke's Law in one dimension:   $\mathbf{F}_s = -k x\,\hat{\mathbf{i}}$

• The negative sign indicates that the spring force always opposes the displacement vector from equilibrium:

  • If the spring is compressed ($x < 0$), the force points in the positive direction ($\mathbf{F}_s > 0$).

  • If the spring is stretched ($x > 0$), the force points in the negative direction ($\mathbf{F}_s < 0$).

• The infinitesimal displacement vector in one dimension along the horizontal axis is given by:   $d\mathbf{r} = dx\,\hat{\mathbf{i}}$

• In a multi-dimensional coordinate space, the general displacement vector is:   $d\mathbf{r} = dx\,\hat{\mathbf{i}} + dy\,\hat{\mathbf{j}} + dz\,\hat{\mathbf{k}}$

• Since motion and force are strictly confined along the $x$-axis, the dot product with the transverse unit vectors evaluates to zero because they are orthogonal ($90^\circ$):   $\hat{\mathbf{i}} \cdot \hat{\mathbf{j}} = 0$   $\hat{\mathbf{i}} \cdot \hat{\mathbf{k}} = 0$   $\hat{\mathbf{i}} \cdot \hat{\mathbf{i}} = 1$

• The block is initially compressed by a distance $d$ to $x = -d$, storing elastic energy in the system, and then released from rest.

• In the absence of friction, momentum carries the block past the equilibrium point out to $x = +d$, before returning back to the starting point $x = -d$, completing one full closed-loop cycle.

• The complete closed-path cycle is divided into four discrete sequential segments:

  • Segment 1: Motion from $x_i = -d$ to $x_f = 0$

  • Segment 2: Motion from $x_i = 0$ to $x_f = +d$

  • Segment 3: Motion from $x_i = +d$ to $x_f = 0$

  • Segment 4: Motion from $x_i = 0$ to $x_f = -d$

• The work done along Segment 1 ($x_i = -d$ to $x_f = 0$):   $W_1 = \int_{-d}^{0} (-kx\,\hat{\mathbf{i}}) \cdot (dx\,\hat{\mathbf{i}}) = -k \int_{-d}^{0} x\,dx = -\frac{k}{2} \left[ x^2 \right]_{-d}^0 = -\frac{k}{2} \left( 0^2 - (-d)^2 \right) = -\frac{k}{2} (-d^2) = +\frac{1}{2}kd^2$

  • Because the restoring force points in the direction of motion (both point toward the positive direction), the force adds energy to the block, resulting in positive work.

• The work done along Segment 2 ($x_i = 0$ to $x_f = +d$):   $W_2 = \int_{0}^{d} (-kx\,\hat{\mathbf{i}}) \cdot (dx\,\hat{\mathbf{i}}) = -k \int_{0}^{d} x\,dx = -\frac{k}{2} \left[ x^2 \right]_{0}^d = -\frac{k}{2} \left( d^2 - 0^2 \right) = -\frac{1}{2}kd^2$

  • As the spring stretches, the restoring force pulls back toward equilibrium, opposing the direction of displacement ($\theta = 180^\circ$), removing energy from the block and performing negative work.

• The work done along Segment 3 ($x_i = +d$ to $x_f = 0$):   $W_3 = \int_{d}^{0} (-kx\,\hat{\mathbf{i}}) \cdot (dx\,\hat{\mathbf{i}}) = -k \int_{d}^{0} x\,dx = -\frac{k}{2} \left[ x^2 \right]_{d}^0 = -\frac{k}{2} \left( 0^2 - d^2 \right) = -\frac{k}{2} (-d^2) = +\frac{1}{2}kd^2$

  • The restoring force and the displacement vector both point inward toward $x = 0$, speeding up the block and doing positive work.

• The work done along Segment 4 ($x_i = 0$ to $x_f = -d$):   $W_4 = \int_{0}^{-d} (-kx\,\hat{\mathbf{i}}) \cdot (dx\,\hat{\mathbf{i}}) = -k \int_{0}^{-d} x\,dx = -\frac{k}{2} \left[ x^2 \right]_{0}^{-d} = -\frac{k}{2} \left( (-d)^2 - 0^2 \right) = -\frac{k}{2} (d^2) = -\frac{1}{2}kd^2$

  • The restoring force points toward the positive direction while displacement is negative, opposing motion and performing negative work.

• The total net work performed by the spring force over the entire closed loop is the sum of the four segments:   $W_{\text{net}} = W_1 + W_2 + W_3 + W_4 = \left(+\frac{1}{2}kd^2\right) + \left(-\frac{1}{2}kd^2\right) + \left(+\frac{1}{2}kd^2\right) + \left(-\frac{1}{2}kd^2\right) = 0$

• The net work done by the spring force over the complete closed cycle is identically zero.

Case Study 2: Gravitational Force Near Earth's Surface Along a Closed Path

• Near the surface of the Earth, the acceleration due to gravity $g$ is approximately constant, making the gravitational force on an object of mass $m$ a constant downward vector:   $\mathbf{F}_g = -mg\,\hat{\mathbf{j}}$

• Consider a ball launched vertically upward from ground level ($y_i = 0$) to a peak vertical height $h$, which then falls directly back down to ground level ($y_f = 0$).

• The infinitesimal displacement vector in a vertical planar coordinate frame is:   $d\mathbf{r} = dx\,\hat{\mathbf{i}} + dy\,\hat{\mathbf{j}}$

• Taking the dot product of the gravitational force vector with the displacement vector:   $\mathbf{F}_g \cdot d\mathbf{r} = (-mg\,\hat{\mathbf{j}}) \cdot (dx\,\hat{\mathbf{i}} + dy\,\hat{\mathbf{j}}) = -mg\,dx(\hat{\mathbf{j}} \cdot \hat{\mathbf{i}}) - mg\,dy(\hat{\mathbf{j}} \cdot \hat{\mathbf{j}}) = -mg\,dy$

• The closed path consists of two discrete stages:

  • Segment 1: Ascent from $y_i = 0$ to $y_f = h$

  • Segment 2: Descent from $y_i = h$ to $y_f = 0$

• Evaluating the work done during Segment 1 (ascent):   $W_1 = \int_{0}^{h} (-mg)\,dy = -mg \left[ y \right]_0^h = -mg(h - 0) = -mgh$

  • Because the gravitational force vector points downward while displacement points upward ($\theta = 180^\circ$), gravity performs negative work and slows the object down.

• Evaluating the work done during Segment 2 (descent):   $W_2 = \int_{h}^{0} (-mg)\,dy = -mg \left[ y \right]_h^0 = -mg(0 - h) = +mgh$

  • Because the gravitational force vector points downward and displacement points downward ($\theta = 0^\circ$), gravity accelerates the object and performs positive work.

• The total net work executed by gravity over the round-trip closed trajectory is:   $W_{\text{net}} = W_1 + W_2 = -mgh + (+mgh) = 0$

• The net work done by the gravitational force over any closed vertical loop near Earth's surface is identically zero.

Case Study 3: Kinetic Friction Along a Closed Circular Path

• A puck of mass $m$ is restrained by a string of fixed radius $R$ and moves in a horizontal circular path on a flat table possessing surface friction.

• The coefficient of kinetic friction between the puck and the tabletop is $\mu_k$.

• The normal force exerted by the horizontal table on the puck balances gravity:   $N = mg$

• The magnitude of the kinetic friction force is constant throughout the motion:   $f_k = \mu_k N$

• In circular motion, arc length $s$ is related to the subtended angle $\theta$ by:   $s = R\theta$

  • This relationship is valid only when $\theta$ is expressed in dimensionless units of radians.

• Taking the differential of arc length when radius $R$ is held constant gives the infinitesimal displacement element:   $ds = R\,d\theta$

• Kinetic friction is a reactionary contact force that acts tangentially to the trajectory, always directly opposing the instantaneous displacement vector ($d\mathbf{s}$).

• The angle between the kinetic friction vector $\mathbf{f}_k$ and the differential displacement vector $d\mathbf{s}$ is continuously and invariably:   $\theta = 180^\circ$

• Applying the geometric definition of the dot product:   $\mathbf{f}_k \cdot d\mathbf{s} = |\mathbf{f}_k| |d\mathbf{s}| \cos(180^\circ) = (\mu_k N)(ds)(-1) = -\mu_k N R\,d\theta$

• The circular closed loop is divided into two semicircular halves:

  • Path from $A$ to $B$: Sweeping from $\theta = 0$ to $\theta = \pi$

  • Path from $B$ to $A$: Sweeping from $\theta = \pi$ to $\theta = 2\pi$

• Evaluating the work done by friction along the first semicircle ($A \rightarrow B$):   $W_{A \rightarrow B} = \int_{0}^{\pi} (-\mu_k N R)\,d\theta = -\mu_k N R [\theta]_0^\pi = -\mu_k \pi R N$

• Evaluating the work done by friction along the completing semicircle ($B \rightarrow A$):   $W_{B \rightarrow A} = \int_{\pi}^{2\pi} (-\mu_k N R)\,d\theta = -\mu_k N R [\theta]_\pi^{2\pi} = -\mu_k N R (2\pi - \pi) = -\mu_k \pi R N$

• The net work done by kinetic friction over one complete circular closed loop is:   $W_{\text{net}} = W_{A \rightarrow B} + W_{B \rightarrow A} = (-\mu_k \pi R N) + (-\mu_k \pi R N) = -2\mu_k \pi R N \neq 0$

• The net work performed by friction over a closed path is strictly non-zero and negative, dissipating mechanical energy permanently into thermal energy.

Mathematical Definition and Properties of Conservative Forces

• A force is formally defined as a conservative force if and only if the total net work it performs on a particle traversing any closed loop is zero:   $\oint \mathbf{F} \cdot d\mathbf{r} = 0$

• Examples of conservative forces include:

  • The spring force ($\mathbf{F}_s = -kx\,\hat{\mathbf{i}}$).

  • The gravitational force near the surface of Earth ($\mathbf{F}_g = -mg\,\hat{\mathbf{j}}$).

• A force is defined as a non-conservative force if the work performed along a closed path is non-zero:   $\oint \mathbf{F} \cdot d\mathbf{r} \neq 0$

• Examples of non-conservative forces include kinetic friction and aerodynamic drag (air resistance), both of which continuously dissipate energy out of the mechanical system.

• A direct mathematical corollary to the closed-loop definition is that the work done by a conservative force between two spatial coordinates is path-independent.

• Path independence asserts that the total work done moving from an arbitrary point $A$ to a destination point $B$ depends solely on the positions of endpoints $A$ and $B$, and is entirely independent of the geometric route traversed between them:   $W_{\text{Path 1}} = W_{\text{Path 2}}$

• Proof of the equivalence between path independence and zero closed-loop work:

  • Construct a closed loop consisting of forward movement from $A$ to $B$ via Path 1, followed by return movement from $B$ to $A$ via Path 1:     $W_{1, A \rightarrow B} + W_{1, B \rightarrow A} = 0$     $W_{1, A \rightarrow B} = -W_{1, B \rightarrow A}$

  • Construct an alternate closed loop traversing forward from $A$ to $B$ along Path 1, and returning from $B$ to $A$ along an arbitrary Path 2:     $W_{1, A \rightarrow B} + W_{2, B \rightarrow A} = 0$     $W_{1, A \rightarrow B} = -W_{2, B \rightarrow A}$

  • Equating both expressions for $W_{1, A \rightarrow B}$:     $-W_{1, B \rightarrow A} = -W_{2, B \rightarrow A}$     $W_{1, B \rightarrow A} = W_{2, B \rightarrow A}$

  • Reversing the operational limits of integration inverts the sign of the line integral:     $W_{1, A \rightarrow B} = W_{2, A \rightarrow B}$

  • Because Path 1 and Path 2 are arbitrary trajectories connecting endpoints $A$ and $B$, the work executed by any conservative force is rigorously path-independent.

Questions & Discussion

• Question: A box is pushed across a rough floor with friction at a constant speed from Point 1 to Point 2. Two distinct trajectories exist between these points: Path A follows a semicircular arc of radius $R$, while Path B follows a straight line directly through the center connecting Point 1 to Point 2. Which path requires more work from the person pushing? Can the pushing force $F_P$ exerted on the box be considered a conservative force? Why or why not?

• Analysis of the physical system:

  • Point 1 is located at position $-R$ and Point 2 is located at position $+R$.

  • The box travels at constant speed, meaning acceleration is zero ($a = 0$).

  • By Newton's second law, the applied pushing force vector must exactly balance the instantaneous kinetic friction vector:     $\mathbf{F}_P + \mathbf{f}_k = 0 \implies \mathbf{F}_P = -\mathbf{f}_k$

  • The work performed by the pushing force is equal in magnitude and opposite in sign to the work performed by friction:     $W_P = -W_f$

• Explicit calculation for Path A (semicircular path of radius $R$):

  • Infinitesimal displacement is expressed in polar arc length coordinates as $ds = R\,d\theta$.

  • The angle sweeps from $\theta = 0$ to $\theta = \pi$.

  • Friction continuously opposes the displacement at $\theta = 180^\circ$:     $W_{f, A} = \int_{0}^{\pi} |\mathbf{f}_k| |d\mathbf{s}| \cos(180^\circ) = \int_{0}^{\pi} (\mu_k N)(R\,d\theta)(-1) = -\mu_k \pi R N$

  • The work done by the applied pushing force along Path A is therefore:     $W_{P, A} = -W_{f, A} = +\mu_k \pi R N$

• Explicit calculation for Path B (direct linear path across the diameter):

  • The motion is strictly along the horizontal Cartesian axis from $x_i = -R$ to $x_f = +R$.

  • The frictional force vector is oriented in the negative direction:     $\mathbf{f}_k = -\mu_k N\,\hat{\mathbf{i}}$

  • The linear differential displacement vector is:     $d\mathbf{r} = dx\,\hat{\mathbf{i}} + dy\,\hat{\mathbf{j}}$

  • Evaluating the frictional work along the straight line:     $W_{f, B} = \int_{-R}^{+R} (-\mu_k N\,\hat{\mathbf{i}}) \cdot (dx\,\hat{\mathbf{i}} + dy\,\hat{\mathbf{j}}) = \int_{-R}^{+R} -\mu_k N\,dx = -\mu_k N [x]_{-R}^{+R}$     $W_{f, B} = -\mu_k N (R - (-R)) = -2\mu_k R N$

  • The work done by the applied pushing force along Path B is therefore:     $W_{P, B} = -W_{f, B} = +2\mu_k R N$

• Comparison of work magnitudes:

  • Evaluating the numerical prefactors:     $\pi \approx 3.14159$     $2 < \pi$

  • Consequently:     $W_{P, A} > W_{P, B}$     $\mu_k \pi R N > 2\mu_k R N$

  • Path A requires substantially more work because the path length traversed against continuous friction is longer ($\pi R > 2R$).

• Classification of the pushing force:

  • The applied force $\mathbf{F}_P$ cannot be considered a conservative force.

  • The work required to move the box between identical spatial endpoints (Point 1 and Point 2) depends directly on the path taken ($W_{P, A} \neq W_{P, B}$.

• A conservative force must exhibit complete path independence, which is violated here due to direct coupling with non-conservative friction.

Theoretical Foundation for Potential Energy Functions

• A potential energy function, denoted as $U(\mathbf{r})$, can only be mathematically defined for conservative forces.

• The assignment of a definite scalar potential energy value to a specific spatial configuration relies fundamentally on the work being completely path-independent.

• If a force is conservative, the work performed against it to transition a system from a reference state to a given position depends exclusively on the endpoints:   $\Delta U = -W_{\text{conservative}} = -\int_{\mathbf{r}_i}^{\mathbf{r}_f} \mathbf{F}_{\text{cons}} \cdot d\mathbf{r}$

• If a force is non-conservative (such as friction), the mechanical energy expended varies with every unique geometric path, making it impossible to assign a unique, single-valued potential energy function to a spatial coordinate.