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Mechanics
Central Forces and Central Field Mechanics
A central force is a force whose line of action always passes through a fixed center point, and its magnitude depends solely on the distance from that center. Central forces are conservative in nature and possess spherical symmetry.
UP TGT ScienceUP PGT PhysicsStudy Notes + Practice
Scope and prerequisites
TGT core
Definition of central forces, proof of conservation of angular momentum under central forces, areal velocity constancy (Kepler's second law relation), and inverse-square law basics (gravitational and electrostatic forces).
PGT extension
Binet's differential equation of orbits, reduced mass transformation in two-body central force problems, effective potential energy curves, apsidal angles, and planetary motion stability.
Prerequisites
Newton's laws of motion
Conservative force fields and potential energy
Conservation of linear and angular momentum
Definition and Vector Representation
A force F(r) is termed a central force if it can be expressed mathematically as F(r) = f(r) r^, where r^ is the unit position vector and f(r) is a scalar function of the distance r from the center of force.
The line of action of the force vector always passes through the origin or the fixed center of attraction/repulsion.
Central forces are inherently spherically symmetric, meaning the force magnitude depends only on r = |r|, not on the angular coordinates (theta, phi).
Conservation of Angular Momentum
Since the torque acting on a particle moving under a central force is zero (ta$u = r x F = r x f(r)r^ = 0)$, the angular momentum L of the particle remains strictly conserved.
Conservation of angular momentum dictates that the motion of the particle must be strictly confined to a fixed plane perpendicular to the constant vector L.
This planar restriction reduces the degrees of freedom from three (Cartesian coordinates x, y, z) to two (polar coordinates r, theta).
Constancy of Areal Velocity
The areal velocity, defined as the rate at which the position vector sweeps out area (dA/dt), is directly proportional to the specific angular momentum L/m.
Areal velocity is given by dA/d$t = L / (2m) = constant$, which is a direct mathematical consequence of the conservation of angular momentum in a central field.
This principle forms the basis of Kepler's Second Law of planetary motion.
Conservative Nature and Potential Energy
All central forces are conservative forces because the curl of F vanishes (nabla x F = 0).
A potential energy function V(r) can always be defined such that F(r) = - dV/dr r^.
The total mechanical energy $E = T + V(r) remains constant throughout the motion of the particle in a central field$.
Derivations
Conservation of Angular Momentum and Planar Motion
Conditions: Particle of mass m moving under any arbitrary central force F(r)r^
Write the torque acting on the particle: ta$u = r x F = r x [f(r) r^]$.
Substitute r^ = r / r into the cross product: ta$u = r x [f(r) (r / r)] = [f(r) / r] (r x r)$.
Since the cross product of any vector with itself is zero (r x r = 0), the torque tau = 0.
By Newton's second law for rotation, ta$u = dL/dt = 0$, which implies angular momentum L = r x p = constant.
Result: L is a constant vector, proving that motion is confined to a fixed plane.
Binet's Equation for Orbit in Central Field
Conditions: Particle under central force F(r), using transformation $u = 1/r$
Express angular momentum in polar coordinates: $L = m r^2 (d\theta/dt)$, giving d\theta/d$t = L / (mr^2) = (L u^2)/m$.
Substitute $r = 1/u into radial equation of motion: m [d^2r/dt^2 - r (d\theta/dt)^2] = F(r)$.
Evaluate derivatives with respect to theta using chain rule: dr/d$t = (dr/du)(du/d\theta)(d\theta/dt) = - (L/m) (du/d\theta)$.
Substitute derivatives back and simplify to obtain the differential equation for the orbit u(\theta).
A = area swept, L = angular momentum magnitude, m = mass m^2 s^-1
Constant in any central force field
Relation between Force and Potential Energy
$$F(r) = - \frac{dV}{dr}$$
V(r) = potential energy function Newtons (N)
Valid since central forces are conservative
Effective Potential Energy
$$V_{eff}(r) = V(r) + \frac{L^2}{2 m r^2}$$
V(r) = true potential, L = angular momentum, m = mass Joules (J)
Used in radial 1D equivalent motion analysis
Radial Equation of Motion
$$m \left( \ddot{r} - r \dot{\theta}^2 \right) = F(r)$$
r = radial distance, \dot{\theta} = angular velocity Newtons (N)
Polar coordinate formulation
Binet's Orbital Equation
$$\frac{d^2u}{d\theta^2} + u = - \frac{m}{L^2 u^2} F\left(\frac{1}{u}\right)$$
$u = 1/r$, L = angular momentum, F = force law m^-1 (dimensionless in theta)
Used to determine orbit shape from force law
Total Energy in Central Field
$$E = \frac{1}{2} m \dot{r}^2 + \frac{L^2}{2 m r^2} + V(r)$$
E = total energy, \dot{r} = radial velocity Joules (J)
Conserved quantity
Graphs and experiments
Effective Potential Energy vs Radial Distance (V_eff vs r)
Draw: A curve showing a potential well with a minimum point, having a steep centrifugal barrier term (1/r^2) dominating at small r and the attractive potential dominating at large r.
Interpret: The minimum of the effective potential curve corresponds to stable circular orbit radius. Horizontal lines representing total energy E indicate turning points (periapsis and apoapsis).
Exam use: Tested in PGT to analyze stability of circular orbits and determine turning points of planetary motion.
Kepler's Second Law Areal Sweep
Draw: An elliptical orbit with the sun at one focus, showing two sectors swept out by the radius vector in equal time intervals with equal areas.
Interpret: Demonstrates that equal areas are swept in equal times (dA/dt = constant), illustrating conservation of angular momentum.
Exam use: Frequently tested conceptually in TGT and PGT examinations regarding orbital speed variations.
Planetary Orbit Geometry in Polar Coordinates
Draw: An ellipse showing polar coordinates (r, theta) from the center of force (focus) to a point on the orbit, indicating perihelion r_min and aphelion r_max.
Interpret: Visualizes the inverse relationship between distance and orbital velocity due to angular momentum conservation.
Exam use: Used in numerical problem setup for calculating maximum and minimum velocities.
Solved numericals
TGT core
A particle moves under the influence of a central force given by F(r) = - k / r^3. If the particle's mass is m and its angular momentum is L, find the differential equation of its orbit using substitution $u = 1/r$.
A particle of mass m moves in a circular orbit of radius r_0 under an attractive central force F(r) = - k / r^4. Find the condition for the stability of this circular orbit.
Write the effective potential for the force F(r) = - k / r^4. Since F(r) = - dV/dr, we have V(r) = - k / (3r^3).
Construct the effective potential: $V_eff(r) = - \frac{k}{3r^3} + \frac{L^2}{2mr^2}$.
For a circular orbit, the first derivative of V_eff with respect to r must vanish at $r = r_0: dV_eff/dr|_{r=r_0} = \frac{k}{r_0^4} - \frac{L^2}{m r_0^3} = 0$, which gives $L^2 = \frac{mk}{r_0}$.
For stability, the second derivative of V_eff must be greater than zero at $r = r_0: d^2V_eff/dr^2|_{r=r_0} = -\frac{4k}{r_0^5} + \frac{3L^2}{m r_0^4} > 0$.
Substitute $L^2 = mk/r_0 into the second derivative condition: -\frac{4k}{r_0^5} + \frac{3(mk/r_0)}{m r_0^4} = -\frac{k}{r_0^5} > 0$, which requires k < 0 (repulsive) or if attractive (k > 0), examine the exact inequality constraint: -4k + 3k < $0 => -k < 0$, hence stable for specific power laws.
Answer: The circular orbit is stable if the effective potential curvature condition d^2V_{eff}/dr^2 > 0 is satisfied at r_0.
PGT extension
If a particle describes a circular orbit of radius R under a central force F(r) = - \frac{k}{r^2}, find the total energy of the particle in terms of k and R.
For an inverse-square attractive force, potential energy is V(r) = - k / r.
From Newton's second law for circular motion: \frac{mv^2}{R} = \frac{k}{R^2}, which gives kinetic energy $T = \frac{1}{2} mv^2 = \frac{k}{2R}$.
Potential energy at r = R is V(R) = - \frac{k}{R}.
Total energy $E = T + V = \frac{k}{2R} - \frac{k}{R} = - \frac{k}{2R}$.
Answer: $E = - \frac{k}{2R}$
Assumptions and limitations
The center of force is assumed to be stationary or located at the inertial frame origin (reduced mass is used if the center has finite mass).
Relativistic effects are ignored; classical mechanics formulations apply throughout.
Common misconceptions
Common error: All central forces are inverse-square forces.
Correct: Inverse-square forces (like gravity and electrostatics) are only a specific subclass of central forces.
A central force can follow any functional dependence on distance, such as F(r) = -kr (Hooke's law) or F(r) = -k/r^3.
Common error: Central force motion can take place in three dimensions arbitrarily.
Correct: Motion under a central force is strictly planar.
Conservation of angular momentum ensures that the position vector and velocity vector always lie in a fixed plane perpendicular to the constant angular momentum vector.
Quick revision
Central force magnitude depends only on distance r and points along the position vector.
Angular momentum L and areal velocity dA/dt are always conserved under central forces.
All central forces are conservative, allowing the definition of a potential energy function V(r).
Motion under central forces is always restricted to a single plane.
Must remember
A central force is directed radially towards or away from a fixed center and can be expressed as $F(r) = f(r) \hat{r}$.
The torque due to a central force is zero ($\tau = 0$), which leads directly to the conservation of angular momentum ($L = \text{constant}$).
Conservation of angular momentum restricts the motion of a particle under a central force to a fixed plane, reducing degrees of freedom to two.
The areal velocity of a particle in a central field is constant and given by $\frac{dA}{dt} = \frac{L}{2m}$.
All central forces are conservative because the curl of $\mathbf{F}$ vanishes ($\nabla \times \mathbf{F} = 0$), allowing the definition of a potential energy function $V(r)$ such that $F(r) = -\frac{dV}{dr}$.
Binet's orbital equation $\frac{d^2u}{d\theta^2} + u = - \frac{m}{L^2 u^2} F\left(\frac{1}{u}\right)$ (where $u = 1/r$) is used to determine the trajectory shape from a given force law.
Glossary
Central ForceA force whose line of action always passes through a fixed center and depends only on the distance $r$.
Areal VelocityThe rate at which the position vector sweeps out area, equal to $\frac{dA}{dt} = \frac{L}{2m}$.
Effective Potential EnergyThe sum of true potential energy and centrifugal potential energy: $V_{eff}(r) = V(r) + \frac{L^2}{2 m r^2}$.
Binet's EquationDifferential equation governing the shape of the orbit of a particle in a central field using variable $u = 1/r$.
Conservative ForceA force field for which work done is independent of path and curl of the force is zero.
1. Which of the following conditions is both necessary and sufficient for a force $\mathbf{F}$ to be classified as a central force?
Explanation: A central force is defined by $F(r) = f(r)\hat{r}$, meaning its line of action passes through a fixed center and its magnitude depends solely on the distance $r$ from the origin.
2. Why is angular momentum $L$ strictly conserved for a particle moving under any central force?
Explanation: The torque $\mathbf{\tau} = \mathbf{r} \times \mathbf{F} = \mathbf{r} \times f(r)hat{r} = 0$ because $\mathbf{r}$ and $hat{r}$ are collinear. Since torque is the rate of change of angular momentum, $\mathbf{L}$ remains constant.
3. The motion of a particle under a central force is strictly confined to:
Explanation: Since angular momentum vector $\mathbf{L} = \mathbf{r} \times \mathbf{p}$ is constant in direction and magnitude, the position vector $\mathbf{r}$ and velocity vector $\mathbf{v}$ always remain perpendicular to $\mathbf{L}$, confining motion to a single plane.
4. What is the areal velocity $\frac{dA}{dt}$ of a particle of mass $m$ having angular momentum $L$ in a central field?
Explanation: The areal velocity is given by $\frac{dA}{dt} = \frac{1}{2} r^2 dot{\theta} = \frac{L}{2m}$, which remains constant according to the conservation of angular momentum.
5. All central forces are conservative. Which mathematical property confirms this statement?
Explanation: A force is conservative if and only if its curl vanishes. For any central force $F(r)hat{r}$, $\nabla \times [f(r)hat{r}] = 0$ identically.
6. The relationship between a conservative central force $F(r)$ and its potential energy function $V(r)$ is expressed as:
Explanation: In a one-dimensional radial equivalent or conservative field, the force is the negative gradient of the potential energy: $F(r) = -\frac{dV}{dr}$.
7. In the radial equation of motion for a central force field, $m(\ddot{r} - rdot{\theta}^2) = F(r)$, what does the term $-mrdot{\theta}^2$ represent?
Explanation: In polar coordinates, the radial acceleration includes $-rdot{\theta}^2$, which corresponds to the centrifugal acceleration term when analyzing motion from a rotating or polar perspective.
8. What is the expression for the effective potential energy $V_{eff}(r)$ of a particle of mass $m$ and angular momentum $L$ moving in a central potential $V(r)$?
Explanation: The effective potential energy combines the true potential energy and the centrifugal barrier term: $V_{eff}(r) = V(r) + \frac{L^2}{2mr^2}$.
9. Binet's orbital equation uses the substitution $u = \frac{1}{r}$. What is the standard form of Binet's equation?
Explanation: Binet's equation relating orbit geometry to the force law is given by $\frac{d^2u}{d\theta^2} + u = -\frac{m}{L^2 u^2} F\left(\frac{1}{u}\right)$.
10. If a particle moves under an attractive inverse-cube central force $F(r) = -\frac{k}{r^3}$, what kind of orbit can it potentially trace depending on initial conditions?
Explanation: An inverse-cube force law yields orbits known as Cotes' spirals, hyperbolic spirals, or reciprocal spirals when evaluated through Binet's equation.
11. Which of the following statements is a common misconception about central forces?
Explanation: Inverse-square forces are merely a specific subclass of central forces; central forces can follow any distance dependence such as Hooke's law ($F \propto -r$). Statement 2 is the misconception.
12. For a particle moving under an attractive inverse-square central force $F(r) = -\frac{k}{r^2}$ (like gravity), what is the true potential energy function $V(r)$ taking zero at infinity?
13. What is the total mechanical energy $E$ of a particle of mass $m$ in a central field in terms of radial velocity $dot{r}$, angular momentum $L$, and potential $V(r)$?
Explanation: Total energy is kinetic plus potential energy. Splitting kinetic energy into radial and transverse components gives $E = \frac{1}{2}mdot{r}^2 + \frac{L^2}{2mr^2} + V(r)$.
14. In a two-body central force problem, the problem can be reduced to a single-body problem by replacing the mass $m$ with:
Explanation: Two-body central force motion separates into center-of-mass motion and relative motion, where the relative motion is governed by the reduced mass $\mu = \frac{m_1 m_2}{m_1 + m_2}$.
15. Kepler's Second Law of planetary motion (equal areas swept in equal intervals of time) is a direct consequence of:
Explanation: Areal velocity $\frac{dA}{dt} = \frac{L}{2m}$ is constant because angular momentum $L$ is conserved under the central gravitational force of the Sun.
PYQ Patterns
These are pattern-based practice questions. Authentic year labels are only used when verified from a source.
1. If a particle moves in a central force field described by $F(r) = -\frac{k}{r^3}$, the differential equation of the orbit using $u = \frac{1}{r}$ becomes $\frac{d^2u}{d\theta^2} + k' u = 0$. What is the value of $k'$?
2. What is the condition for a circular orbit of radius $r_0$ to be stable under a general central attractive force $F(r)$?
Explanation: Stability of a circular orbit requires the effective potential to have a local minimum at $r_0$, leading to the condition $\left[ \frac{r}{F}\frac{dF}{dr} + 3 \right]_{r_0} > 0$, which simplifies to $\frac{d}{dr}(\ln F) > -3$.
3. A particle of mass $m$ moves in a circular orbit of radius $r_0$ under the central force $F(r) = -\frac{k}{r^2}$. What is its angular momentum $L$?
Explanation: For a circular orbit under $F = k/r^2$, centripetal force equals attractive force: $\frac{mv^2}{r_0} = \frac{k}{r_0^2} \implies v = \sqrt{\frac{k}{m r_0}}$. Angular momentum $L = m v r_0 = m \sqrt{\frac{k}{m r_0}} r_0 = \sqrt{m k r_0}$.
4. If the potential energy function of a particle in a central field is given by $V(r) = \frac{1}{2} k r^2$ (isotropic harmonic oscillator), what is the corresponding central force $F(r)$?
Explanation: Using $F(r) = -\frac{dV}{dr}$, differentiating $\frac{1}{2}kr^2$ with respect to $r$ yields $-kr$. Thus $F(r) = -kr$ (Hooke's law).
5. The areal velocity of a planet is $4.5 \times 10^{15} \text{ m}^2\text{s}^{-1}$. What is its angular momentum per unit mass ($L/m$)?
Explanation: Areal velocity is given by $\frac{dA}{dt} = \frac{L}{2m}$. Therefore, $\frac{L}{m} = 2 \frac{dA}{dt} = 2 \times 4.5 \times 10^{15} = 9.0 \times 10^{15} \text{ m}^2\text{s}^{-1}$.
6. Which of the following central force laws produces closed non-precessing orbits for all bound states (Bertrand's Theorem)?
Explanation: According to Bertrand's theorem, the only central force laws that yield stable, closed orbits for all bound particles are the inverse-square law ($F \propto r^{-2}$) and Hooke's law ($F \propto r$).
Timed Topic Test
Attempt these questions under a ten-minute limit using a separate timer.
1. A particle moves under a central force such that its areal velocity is constant. This implies that:
Explanation: Constant areal velocity ($\frac{dA}{dt} = \frac{L}{2m} = \text{const}$) is a direct mathematical outcome of the conservation of angular momentum $L$.
2. What is the curl of any valid central force field $\mathbf{F}(\mathbf{r}) = f(r)hat{r}$?
Explanation: All central forces are conservative, which mathematically means $\nabla \times \mathbf{F} = 0$.
3. In the effective potential $V_{eff}(r) = V(r) + \frac{L^2}{2mr^2}$, what role does the term $\frac{L^2}{2mr^2}$ play?
Explanation: The term $\frac{L^2}{2mr^2}$ acts as an effective repulsive potential barrier (centrifugal potential) that prevents a particle with non-zero angular momentum from colliding with the origin at $r = 0$.
4. Using Binet's equation, if the orbit of a particle under a central force is a circle passing through the origin given by $r = 2a \cos\theta$, what is the force law $F(r)$?
Explanation: For $r = 2a \cos\theta$, $u = \frac{1}{r} = \frac{1}{2a}\sec\theta$. Differentiating twice gives $\frac{d^2u}{d\theta^2} = u(2u^2 a^2 - 1)$ or similar steps yielding an inverse-fifth power force law $F(r) \propto -r^{-5}$.
5. If the total energy $E$ of a particle in an attractive central potential equals the minimum of the effective potential ($E = V_{eff}(r_0)$), the particle will execute:
Explanation: When $E = V_{eff, min}$, radial velocity $dot{r} = 0$ and $r$ remains constant at $r_0$, describing a stable circular orbit.
6. Which coordinate system is most appropriate for solving problems involving central forces due to spherical symmetry?
Explanation: Spherical symmetry makes polar or spherical polar coordinates ideal because the force depends only on the radial distance $r$, simplifying equations of motion.
7. What is the dimension of areal velocity $\frac{dA}{dt}$?
Explanation: Areal velocity is area per unit time, so its dimensional formula is $[\text{length}^2 / \text{time}] = [L^2 T^{-1}]$.
8. For an attractive inverse-square force $F(r) = -\frac{k}{r^2}$, what is the condition on total energy $E$ for the orbit to be bound (elliptical)?
Explanation: For bound orbits (ellipses), the total mechanical energy $E$ is negative ($E < 0$). $E = 0$ gives a parabola, and $E > 0$ gives a hyperbola.
9. In a central force field, the angular velocity $dot{\theta}$ can be expressed in terms of angular momentum $L$, mass $m$, and distance $r$ as:
Explanation: Since magnitude of angular momentum is $L = m r^2 dot{\theta}$, solving for $dot{\theta}$ yields $dot{\theta} = \frac{L}{mr^2}$.
10. Which of the following statements about central forces is incorrect?
Explanation: Central forces are spherically symmetric, meaning their magnitude depends strictly on $r$ and not on angular coordinates like $\theta$. Hence, statement 3 is incorrect.