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Mechanics
Potential Energy in Mechanics
Potential energy is the mechanical energy stored within a system due to the spatial configuration or relative positions of its interacting bodies under conservative forces. It is defined mathematically through the negative line integral of a conservative force field.
UP TGT ScienceUP PGT PhysicsStudy Notes + Practice
Scope and prerequisites
TGT core
Definition of potential energy, work-energy relation for conservative forces, gravitational potential energy near the Earth's surface (U = mgh), elastic potential energy of a spring (U = &frac{1}{2}kx²), and conservation of mechanical energy.
PGT extension
General conservative force fields, derivation of force from potential (F = -∇U), central force potentials, gravitational potential due to mass distributions, stability analysis of equilibrium using potential energy curves (d²U/dx² > 0), and potential energy in non-inertial frames.
Prerequisites
Work, power, and kinetic energy theorem
Line integrals and vector calculus basics
Newton's laws of motion and conservative forces
Definition and Conservative Force Field
Potential energy U is defined only for conservative forces where work done is independent of the path taken.
The change in potential energy ΔU between two points is equal to the negative of the work done by the conservative force: Δ$U = -W = -\int _{A}^{B} F \cdot dr$.
The dimension of potential energy is [ML²T&sup-;²] and its SI unit is the joule (J).
Relation Between Force and Potential Energy
In one dimension, the conservative force is related to potential energy by F(x) = -&frac{dU}{dx}.
In three dimensions, force is the negative gradient of the potential energy scalar field: $F = -\nabla U = -(\frac{\partial U}{\partial x}i + \frac{\partial U}{\partial y}j + \frac{\partial U}{\partial z}k)$.
The negative sign indicates that the force always acts in the direction of decreasing potential energy.
Mechanical Energy Conservation
The total mechanical energy $E = K + U of a system remains constant if only conservative internal forces do work$.
Any increase in kinetic energy corresponds to an equal decrease in potential energy and vice versa.
Non-conservative forces like friction dissipate mechanical energy into thermal energy, violating simple mechanical energy conservation.
Equilibrium and Potential Energy Curves
Stable equilibrium occurs at local minima of potential energy, where dU/dx = 0 and d²U/dx² > 0.
Unstable equilibrium occurs at local maxima of potential energy, where dU/dx = 0 and d²U/dx² < 0.
Neutral equilibrium occurs where dU/dx = 0 and d²U/dx$^{2} = 0 over a finite displacement region$.
Derivations
Gravitational Potential Energy Near Earth's Surface
Conditions: Uniform gravitational field (g is constant), vertical displacement y.
Write the gravitational force acting on a mass m pointing downward: $F = -mg j$.
Express the change in potential energy using the line integral: Δ$U = -\int _{y1}^{y_{2}} F_{y} dy$.
U = potential energy function Dimensionless criteria
Conservative mechanical system
Conservation of Mechanical Energy
$$E = K + U = \text{constant}$$
E = total mechanical energy, K = kinetic energy, U = potential energy Joules (J)
Only conservative forces perform work
Graphs and experiments
Potential Energy Well and Phase Interpretation
Draw: A curve of U(x) versus x showing a smooth bowl-like minimum, with horizontal total energy line E intersecting the curve.
Interpret: The region where E > U represents allowed motion where kinetic energy is positive. The turning points occur where E = U. The minimum represents stable equilibrium.
Exam use: To determine turning points of motion, particle velocity at various positions, and stability analysis.
Force Determination from U-x Graph
Draw: An arbitrary non-linear potential energy curve U(x) with tangent lines drawn at various points.
Interpret: The force at any point is the negative slope (-dU/dx) of the tangent at that point. Steeper slopes indicate larger magnitudes of force.
Exam use: Direct qualitative and quantitative questions asking for the direction and relative magnitude of force from a given U-x graph.
Verification of Hooke's Law and Elastic PE
Draw: A graph of restoring force vs displacement (straight line passing through origin) and work done (area under the curve representing U).
Interpret: The area enclosed under the F-x graph up to extension x equals the potential energy stored in the spring (&frac{1}{2}kx²).
A body of mass 2 kg is lifted vertically upwards through a height of 5 m. Calculate the change in its gravitational potential energy. (Take $g = 9.8 m/s^{2})$
Identify given values: m = 2 kg, h = 5 m, $g = 9.8 m/s^{2}$.
Apply the formula for gravitational potential energy: ΔU = mgh.
The potential energy of a particle of mass 1 kg moving in one dimension is given by U(x) = 8x² - 4x³ Joules. Determine the position of stable equilibrium and the angular frequency of small oscillations about this equilibrium point.
Find the first derivative of U(x) with respect to x: dU/d$x = 16x - 12x^{2}$.
Set dU/dx = 0 for equilibrium: 4x(4 - 3x) = 0, giving x = 0 and $x = 4/3 m$.
Find the second derivative: d²U/dx$^{2} = 16 - 24x$.
Test stability at $x = 0: d^{2}U/dx^{2} = 16 \rangle 0 (Stable equilibrium)$.
Calculate angular frequency at stable equilibrium ($x = 0): \omega = \sqrt{k/m}$ where $k = d^{2}U/dx^{2} = 16 N/m$. Thus $\omega = \sqrt{16 / 1} = 4 rad/s$.
Answer: Stable equilibrium at $x = 0; Angular frequency \omega = 4 rad/s$
PGT extension
A conservative force field is given by $F = (2xy + z^{3}) i + x^{2} j + 3xz^{2} k$. Find the potential energy function U(x, y, z), assuming U = 0 at the origin (0, 0, 0).
Use the relation $F = -\nabla U$, which gives &partial;U/&partial;$x = -(2xy + z^{3})$, &partial;U/&partial;$y = -x^{2}$, and &partial;U/&partial;$z = -3xz^{2}$.
Integrate &partial;U/&partial;x with respect to x: U(x,y,z) = -∫ (2xy + z³) d$x = -x^{2}y - xz^{3} + f(y$, z).
Differentiate this intermediate U with respect to y and equate to -&partial;U/&partial;y (-x²): -x² + &partial;f/&partial;$y = -x^{2} \implies \partial f/\partial y = 0$, so f depends only on z: f(y,z) = g(z).
Differentiate $U = -x^{2}y - xz^{3} + g(z) with respect$ to z and equate to -3xz²: -3xz² + g'(z) = -3xz² &implies; g'(z) = 0 &implies; g(z) = C (constant).
Using the boundary condition U(0,0,0) = 0, find that C = 0.
Answer: U(x, y, z) = -x²y - xz³ J
Assumptions and limitations
Potential energy is strictly defined only for conservative forces; dissipative forces like friction cannot be represented by a potential energy function.
The absolute value of potential energy is arbitrary and depends on the choice of the reference zero level; only potential energy differences have physical significance.
Common misconceptions
Common error: Potential energy is an intrinsic property of a single isolated object.
Correct: Potential energy is a property of a system of two or more interacting bodies interacting via a conservative force.
Gravitational or elastic potential energy depends on the relative separation between the object and the Earth or spring respectively.
Common error: A negative potential energy means the particle is bound and cannot move.
Correct: Negative potential energy simply reflects the choice of setting U = 0 at infinity; particles with negative total mechanical energy are bound, but they can still move between turning points.
Zero potential energy is a matter of convention; kinetic energy can still be positive.
Quick revision
Potential energy is defined via change in energy as Δ$U = -W_{conservative}$.
Force is the negative spatial derivative of potential energy: $F = -dU/dx$.
Stable equilibrium corresponds to a local minimum of potential energy (dU/dx = 0, d²U/dx² > 0).
Must remember
Potential energy $U$ is defined exclusively for conservative forces, where the change in potential energy is $\Delta U = -W = -\int_{A}^{B} F \cdot dr$.
In one dimension, conservative force is related to potential energy by $F(x) = -\frac{dU}{dx}$.
In three dimensions, force is the negative gradient of the potential energy scalar field: $F = -\nabla U$.
Stable equilibrium occurs at local minima of potential energy where $\frac{dU}{dx} = 0$ and $\frac{d^2U}{dx^2} > 0$.
Unstable equilibrium occurs at local maxima of potential energy where $\frac{dU}{dx} = 0$ and $\frac{d^2U}{dx^2} < 0$.
Neutral equilibrium occurs where $\frac{dU}{dx} = 0$ and $\frac{d^2U}{dx^2} = 0$ over a finite region.
The total mechanical energy $E = K + U$ remains constant if only conservative internal forces perform work.
Gravitational potential energy near the Earth's surface is given by $U = mgh$, while the universal form is $U(r) = -\frac{GMm}{r}$.
Elastic potential energy stored in a stretched or compressed spring is $U = \frac{1}{2}kx^2$ within the elastic limit.
Potential energy is a property of an interacting system of bodies, not of a single isolated object.
Glossary
Conservative ForceA force for which work done in moving between two points is independent of the path taken and depends only on the endpoints.
Potential Energy ($U$)The energy possessed by a system due to the relative positions of its constituent parts within a conservative force field.
Gradient Operator ($\nabla$)A vector differential operator used in three dimensions to find the spatial rate of change and direction of maximum increase of a scalar field.
Stable EquilibriumA state of a system where a slight displacement creates a restoring force pointing back toward the initial equilibrium position.
Unstable EquilibriumA state of a system where a slight displacement creates a force that pushes the system further away from the equilibrium position.
Neutral EquilibriumA state of a system where a slight displacement results in no net force, and the system remains in equilibrium at its new position.
Mechanical Energy ($E$)The sum of the kinetic energy and potential energy of a mechanical system: $E = K + U$.
Spring Constant ($k$)A measure of the stiffness of a spring, representing the restoring force per unit displacement according to Hooke's law.
Types of Equilibrium Based on Potential Energy Curves
Equilibrium Type
First Derivative ($\frac{dU}{dx}$)
Second Derivative ($\frac{d^2U}{dx^2}$)
Stable Equilibrium
$0$
$> 0$ (Local Minimum)
Unstable Equilibrium
$0$
$< 0$ (Local Maximum)
Neutral Equilibrium
$0$
$= 0$ (Flat Region)
Gravitational Potential Energy Forms
Form
Equation
Validity Condition
Uniform Field
$U = mgh$
Near Earth's surface where $g$ is constant
Universal Field
$U(r) = -\frac{GMm}{r}$
Point masses or spherically symmetric bodies, reference $U=0$ at $r = \infty$
Practice Quiz
Choose an option to reveal the explanation.
1. Which of the following conditions must be satisfied for a force field to allow the definition of a potential energy function?
Explanation: Potential energy $U$ is defined exclusively for conservative forces where the work done is independent of the path taken between two points.
2. What is the relationship between a one-dimensional conservative force $F(x)$ and its potential energy function $U(x)$?
Explanation: In one dimension, the conservative force is related to potential energy by the negative spatial derivative: $F(x) = -\frac{dU}{dx}$.
3. In three-dimensional space, how is a conservative force vector $F$ expressed in terms of the potential energy scalar field $U$?
Explanation: In three dimensions, force is the negative gradient of the potential energy scalar field: $F = -\nabla U = -(\frac{\partial U}{\partial x}i + \frac{\partial U}{\partial y}j + \frac{\partial U}{\partial z}k)$.
4. What are the dimensional formula and SI unit of potential energy?
Explanation: Potential energy has the same dimension as work and energy, which is $[ML^2T^{-2}]$, and its SI unit is the joule (J).
5. A particle is in stable equilibrium at a position $x_0$ if the potential energy $U(x)$ satisfies which of the following?
Explanation: Stable equilibrium occurs at local minima of the potential energy curve, where the first derivative is zero and the second derivative is strictly positive ($\frac{d^2U}{dx^2} > 0$).
6. Under what condition does the total mechanical energy $E = K + U$ of a system remain conserved?
Explanation: The total mechanical energy remains constant if and only if only conservative internal forces do work within the system.
7. What is the elastic potential energy stored in a spring of stiffness constant $k$ compressed by a distance $x$ within its elastic limit?
Explanation: The elastic potential energy stored in an ideal spring obeying Hooke's law is given by $U = \frac{1}{2}kx^2$.
8. What does a negative sign in the relation $F = -\nabla U$ physically indicate?
Explanation: The negative sign signifies that the conservative force always acts in the direction pointing toward decreasing potential energy.
9. What is the universal gravitational potential energy between two masses $M$ and $m$ separated by distance $r$, taking $U=0$ at $r = \infty$?
Explanation: Universal gravitational potential energy is given by $U(r) = -\frac{GMm}{r}$, adopting the standard convention that $U=0$ at infinity.
10. Which of the following statements correctly distinguishes potential energy from kinetic energy?
Explanation: Potential energy is a property of a system of interacting bodies via conservative forces, whereas kinetic energy depends on an object's speed relative to a reference frame.
11. At a point of unstable equilibrium in a potential energy curve, what are the properties of the first and second derivatives?
Explanation: Unstable equilibrium occurs at local maxima of the potential energy function where $\frac{dU}{dx} = 0$ and $\frac{d^2U}{dx^2} < 0$.
12. If a potential energy function is given by $U(x) = ax^3 - bx^2$, what is the corresponding force $F(x)$?
Explanation: Using $F(x) = -\frac{dU}{dx}$, we get $F(x) = -\frac{d}{dx}(ax^3 - bx^2) = -(3ax^2 - 2bx) = -3ax^2 + 2bx$.
13. What happens to the total mechanical energy of a system when non-conservative forces like friction perform work?
Explanation: Non-conservative forces such as friction dissipate mechanical energy into thermal energy, meaning mechanical energy is not conserved.
14. What condition characterizes neutral equilibrium in a mechanical system?
Explanation: Neutral equilibrium occurs where both the first derivative and second derivative of potential energy are zero ($\frac{dU}{dx} = 0$ and $\frac{d^2U}{dx^2} = 0$) over a finite region of displacement.
15. How is the change in potential energy $\Delta U$ between points A and B mathematically related to the conservative work $W$?
Explanation: The change in potential energy is defined as the negative of the work done by the conservative force: $\Delta U = -W = -\int_{A}^{B} F \cdot dr$.
PYQ Patterns
These are pattern-based practice questions. Authentic year labels are only used when verified from a source.
1. The potential energy of a particle moving along the x-axis is given by $U(x) = \frac{a}{x^2} - \frac{b}{x}$, where $a$ and $خاذ$ are positive constants. What is the position of stable equilibrium?
Explanation: For equilibrium, $\frac{dU}{dx} = -\frac{2a}{x^3} + \frac{b}{x^2} = 0 \implies \frac{2a}{x^3} = \frac{b}{x^2} \implies x = \frac{2a}{b}$. Testing the second derivative yields $\frac{d^2U}{dx^2} > 0$ at this point, confirming stable equilibrium.
2. A force $F = (2xy + z^3)i + x^2 j + (3xz^2)k$ acts on a particle. What is the potential energy function $U(x, y, z)$ associated with this conservative force, assuming $U(0,0,0) = 0$?
Explanation: Using $F = -\nabla U$, we have $-\frac{\partial U}{\partial x} = 2xy + z^3 \implies U = -x^2 y - xz^3 + C$. Integrating with respect to $y$ and $z$ confirms consistency, giving $U = -x^2 y - xz^3$.
3. A particle of mass $m$ is released from rest at height $h$ in a uniform gravitational field. What is its speed just before hitting the ground using mechanical energy conservation?
Explanation: By conservation of mechanical energy, initial potential energy $mgh$ equals final kinetic energy $\frac{1}{2}mv^2$. Thus, $mgh = \frac{1}{2}mv^2 \implies v = \sqrt{2gh}$.
4. If the potential energy between two atoms in a diatomic molecule is given by $U(x) = \frac{A}{x^{12}} - \frac{B}{x^6}$, what is the equilibrium separation between the atoms?
5. A block of mass $m$ is attached to a vertical spring of constant $k$. When gently lowered until it hangs in equilibrium, the extension of the spring is $y_0$. What is the elastic potential energy stored in the spring at this equilibrium position?
Explanation: At equilibrium, $ky_0 = mg \implies y_0 = \frac{mg}{k}$. The elastic potential energy is $U = \frac{1}{2}ky_0^2 = \frac{1}{2}k\left(\frac{mg}{k}\right)^2 = \frac{m^2 g^2}{2k}$.
6. A conservative force acts on a particle in one dimension such that $F = -kx + bx^3$, where $k$ and $b$ are positive constants. What is the potential energy function $U(x)$ if $U(0) = 0$?
Explanation: Since $F = -\frac{dU}{dx}$, we have $U(x) = -\int (-kx + bx^3)dx = \frac{1}{2}kx^2 - \frac{1}{4}bx^4$ (taking integration constant as zero for $U(0)=0$).
Timed Topic Test
Attempt these questions under a ten-minute limit using a separate timer.
1. Which of the following force fields is non-conservative, prohibiting the definition of a scalar potential energy function?
Explanation: Friction is a non-conservative dissipative force whose work depends on the path taken; thus, no potential energy function can be defined for it.
2. The potential energy of a body of mass $m$ is given by $U = ax + by$, where $a$ and $b$ are constants. What is the magnitude of the force acting on the body?
Explanation: The force components are $F_x = -\frac{\partial U}{\partial x} = -a$ and $F_y = -\frac{\partial U}{\partial y} = -b$. The magnitude is $|F| = \sqrt{(-a)^2 + (-b)^2} = \sqrt{a^2 + b^2}$.
3. If a particle oscillates about a stable equilibrium point with small amplitude, what determines the angular frequency $\omega$ of oscillation?
Explanation: For small oscillations around a minimum of potential energy, the effective spring constant is $k = \left.\frac{d^2U}{dx^2}\right|_{x=x_0}$, giving angular frequency $\omega = \sqrt{\frac{k}{m}} = \sqrt{\frac{d^2U/dx^2}{m}}$.
4. What is the work done by a conservative force in moving a particle around any closed loop?
Explanation: By definition, the work done by a conservative force around any closed path is identically zero: $\oint F \cdot dr = 0$.
5. A particle moves in a potential energy field $U(x) = E_0 \left(\frac{x}{a} - \frac{a}{x}\right)$. At what position is the potential energy zero?
6. Which misconception about potential energy is corrected by proper physical formulation?
Explanation: Potential energy is fundamentally a property of an interacting system of two or more bodies (such as mass-Earth or charges), not of a single isolated object.
7. If the potential energy of a conservative system is given by $U(r) = \frac{k}{r^2}$ in spherical coordinates where $r$ is the radial distance, what is the radial force $F_r$?
Explanation: The radial force is $F_r = -\frac{dU}{dr} = -\frac{d}{dr}\left(\frac{k}{r^2}\right) = -\left(-\frac{2k}{r^3}\right) = \frac{2k}{r^3}$? Wait, derivative of $r^{-2}$ is $-2r^{-3}$, so $-\frac{d}{dr}(kr^{-2}) = -(-2kr^{-3}) = \frac{2k}{r^3}$. Let's re-verify: $F_r = -\frac{dU}{dr} = -(-2k/r^3) = 2k/r^3$. Ah, wait, option B is $-\frac{2k}{r^3}$ and option A is $\frac{2k}{r^3}$. Let's make sure correct index matches A.
8. What is the physical significance of negative potential energy values in gravitational or electrostatic systems?
Explanation: Negative potential energy simply arises from setting $U=0$ at infinite separation; particles with negative total mechanical energy are bound within turning points but can still possess positive kinetic energy.
9. A spring with spring constant $k$ is stretched by $x_1$ and then further stretched to $x_2$. What is the work done by the external agent against the spring force during this process?
Explanation: The work done by an external agent equals the change in elastic potential energy: $W_{\text{ext}} = U_2 - U_1 = \frac{1}{2}kx_2^2 - \frac{1}{2}kx_1^2 = \frac{1}{2}k(x_2^2 - x_1^2)$.
10. In a conservative force field, if a particle is released from rest at a point where potential energy is at a maximum (unstable equilibrium), what happens initially?
Explanation: At a local maximum (unstable equilibrium), any slight displacement results in a force directed away from the maximum (since $F = -\frac{dU}{dx}$ points toward decreasing $U$), causing the particle to accelerate away.