Central Force Definition
Any force is called a central force when it is always directed towards a fixed point, and its magnitude is dependent on the distance r of the particle from a fixed point (O).
Mathematically, this force can be described as:
\[F=f(r)\hat{r}\]
Here,
F = Central conservative force,
\[\hat{r}=\frac{\vec{r}}{\mid \vec{r}\mid }\] can be described as a unit vector in the direction of r.
F represents conservative central force,
f (r) represents vector magnitude | r |
As well as the conservative force can be represented further as
\[\vec{\triangledown }\times \vec{F}(r)=0\]
When force is positive, it can be considered an attractive force directed towards O. However, when fris negative, the force is deemed to be negative. The same can be illustrated in graphical form below:
(Image will be uploaded soon)
Where,
\[F(r)=-\frac{dU}{dr}\]
Here,
F(r) is the magnitude of a central force, and
U(r) is the potential energy (that is time-independent)
Now, for a particle under central force making a uniform circular motion, it should have centripetal force in the following manner:
\[\frac{mv^2}{r}=F(r)\]…..(a)
Here,
r = the initial radius
v = the speed that satisfies the equation (a) of centripetal force
Relevance of the Central Force Motion
The relevance of the central force motion of a particle:
Many natural phenomenons define the central force in Physics. Some of them are observed in:
Planets that work around the sun
Natural satellites that move around the earth
Two charged particles in terms of the movement against each other
Different Motions in a Central Force
There are broadly two different motion types observed in a central force field. They are:
Bounded Motion: In this case, the distance between two bodies or objects keeps a similar value, and never gets more than the set figures. Examples of such motion are the motion of the planets around the sun.
Unbounded Motion: In this case, the distance between the two bodies or objects is infinite in its initial and final stages. An example of such a movement is the scattering of alpha particles in the Rutherford experiment.
Properties of Central Force
There are several distinct properties that occur in a central force; some of them are:
The overall motion of the particle can take place in a plane curve, and that can be understood from the equation:
The overall motion of the particle can take place in a plane curve, and that can be understood from the equation:
F = F (r) = ma, where m refers to mass and a refers to the acceleration
Also, r × a = 0,
\[r\ast \frac{dv}{dt}=0\]
\[\frac{d}{dt}(r\ast v)=0\]
r × v = h = constant vector;
Where r and v lie in the same plane and h stays perpendicular to the same plane for every value of t. Therefore, the path stays in a plane.
Taking from the above equation, the angular momentum of the conserved particle is as follows;
m (r x v) = mh
L = mh is a constant, and the angular momentum is constant.
The position vector r with respect to the central force covers equal areas in equal time periods; thus, having a constant areal velocity. In the following graph, in a small time interval dt, the radius r covers an area of dA. Therefore, the covered area can be likened to that of half the area of a parallelogram with the sides r and dr.
\[dA=\frac{1}{2}\mid r\times dr\mid \]
\[dA=\frac{1}{2}\mid r\times vdt\mid \]
\[\frac{dA}{dt}=\frac{1}{2}\mid r\times v\mid \]
\[\frac{dA}{dt}=\frac{h}{2}\]
(Image will be uploaded soon)
Motion in a Central Force Field - Equations
One of the most widely used coordinate methods to represent particle motion under a central force is through the application of a polar coordinate system. If the central force is represented in the r direction, the equation can be defined as:
\[F=F(r)\]
\[L=\frac{1}{2}mr^2-V(r)\]
\[F=F(r)=f(r)\hat{r}\] and \[F=-\triangledown V(r)\] where V(r) is the central potential.
We know that, the equation of motion for objects moving a central field using “lagrangian formalism”. The lagrangian of system is basically the difference in the kinetic and potential energy of the system, it is given by:
\[L=K.E-P.E\]
\[L=\frac{1}{2}mr^2-V(r)\]
Potential Energy of a Central Force
For a particle moving from points to P1 to P2, the central force can act on its origin, and the path of the particle is taken as a combination of radial and curved path segments. Therefore, the primary force can act on the direction of the radial segments and lie in perpendicular to the curved parts' displacement. Thus, the total work done via the central force across the curved section can be considered zero, and the whole work done can be defined as:
(Image will be uploaded soon)
If the central force is a conservative force, then the magnitude F(r) of a central force can always be expressed as the derivative of a time-independent potential energy function U(r) i.e., \[F(r)=-\frac{dU}{dr}\]. To show this, it suffices that the work W done by the force depends only on initial and final positions, not on the path taken between them.
\[W=\int_{r_1}^{r_2}F.dr=\int_{r_1}^{r_2}f(r)\hat{r}dr=\int_{r_1}^{r_2}Fdr=U(r_2)-U(r_1)\]
Total Energy of the Particle in Motion
As F is a conservative force, the total energy can be described as:
\[E=\frac{1}{2}mv^2+U(r)\]
\[v^2r^2+r^2\Theta ^2\]
Where \[v^2\] can be defined as \[[(\dot{r})^2+r^2(\dot{\Theta })^2]\]
\[E=\frac{1}{2}m((\dot{r})^2+r^2(\dot{\Theta })^2)+U(r)\]
\[E=\frac{1}{2}m(h^2(\frac{du}{dt})^2\frac{1}{u^2}(hu^2)^2)+U(r)\]
This equation gives the expression for total Energy of the Particle in Motion under the effect of central force.
FAQs on Central Force
1. What is a central force in Physics and how can its direction and dependence be mathematically described?
A central force always acts along the line joining a particle and a fixed point (centre), and its magnitude depends only on the distance between them. Mathematically, it is expressed as F = f(r)̂r, where r is the distance vector and f(r) is a function of r only.
2. Give two real-world examples of central forces observed in nature.
Examples of central forces include:
- Gravitational force between the Sun and a planet
- Coulomb force between two charged particles
3. What is the difference between bounded and unbounded motion in a central force field?
Bounded motion occurs when an object remains at a finite distance from the centre (e.g., a planet orbiting the Sun). Unbounded motion means the object can move infinitely far from the centre (e.g., alpha particle scattering). The type depends on the total energy of the particle relative to the potential energy in the field.
4. How is angular momentum conserved in central force motion?
For a particle under a central force, the angular momentum (L = m(r × v)) remains constant because the force always points towards (or away from) the centre, causing zero torque. This conservation leads to the motion being confined to a plane.
5. Why does motion under a central force always occur in a plane?
Since the central force imparts zero torque about the centre, the angular momentum vector is conserved, fixing the motion in a single plane perpendicular to angular momentum.
6. How is the concept of areal velocity related to central force motion?
Under a central force, a particle sweeps out equal areas in equal intervals of time (constant areal velocity). This is expressed as dA/dt = h/2, where h is the specific angular momentum. This is a statement of Kepler’s second law.
7. Are all central forces necessarily conservative? Explain with reasoning.
Not all central forces are conservative. A central force is conservative only if it is spherically symmetric and its work depends solely on the initial and final positions, not on the path taken. For example, gravitational and electrostatic forces are both central and conservative.
8. What is the relationship between central force and potential energy function U(r)?
A conservative central force can be derived from a potential energy function as F(r) = -dU/dr. This means the force at any point equals the negative gradient of the potential energy with respect to separation distance r.
9. How does the Lagrangian formalism apply to a particle in a central force field?
A particle moving under a central force has a Lagrangian given by L = K.E. - P.E. = (1/2)m(v^2) - V(r). This formalism helps derive the equations of motion using generalized coordinates, often simplifying the analysis when switching to polar coordinates.
10. What would happen if a central force depended on velocity instead of position?
If a force depended on velocity rather than only position, it would not be classified as a central force in the strict sense. Such forces could introduce torque, failing to conserve angular momentum, and would not guarantee planar motion or a conservative field.
11. Explain how Newton’s law of universal gravitation fits the definition of a central force.
Newton’s gravitational force is a classic example of a central force. It acts along the line joining two masses and its magnitude depends only on the distance between them, specifically as F = G(m1*m2)/r^2, satisfying the criteria of a central force.
12. What is the significance of the equation F(r) = -dU/dr in central force motion?
This equation expresses that a conservative central force can be obtained from a potential energy function. The negative sign indicates the force always acts to decrease the potential energy, aligning with the direction from higher to lower potential.
13. How is the total mechanical energy of a particle expressed in a central force field?
The total energy is given by E = (1/2)m[(dr/dt)^2 + r^2(dθ/dt)^2] + U(r). This sum includes kinetic energy (radial and angular components) and potential energy due to the central force.
14. Compare gravitational and Coulomb forces as central forces. What are the key similarities and differences?
Both gravitational and Coulomb forces are:
- Central (act along the line joining two points)
- Conservative and inversely proportional to r2
