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A mass \(m\) moves in a circular orbit (centered ón the origin) in the field of an attractive central force with potential energy \(U=k r^{n} .\) Prove the virial theorem that \(T=n U / 2\).

Short Answer

Expert verified
Kinetic energy \(T\) is \(\frac{n}{2}\) times the potential energy \(U\).

Step by step solution

01

Understand the Problem

We are given a system where a mass \(m\) moves in a circular orbit around a central force with potential energy \(U = k r^n\). We are tasked with proving that the kinetic energy \(T\) is equal to \(\frac{n}{2} U\), which is a form of the virial theorem in this context.
02

Write Down Known Equations

In a central force problem, the force \( F \) is given by the negative gradient of the potential energy, \( F = - \frac{dU}{dr} \). The potential energy is given as \( U = k r^n \). The virial theorem for circular orbits states \( 2T + nU = 0 \). Here, we need to show \( T = \frac{n}{2} U \).
03

Apply the Condition of Circular Orbit

For a circular orbit, the centripetal force required (\( F_c = \frac{mv^2}{r} \)) is provided by the force from the potential energy, \( F = -\frac{dU}{dr} \). Hence, \[ \frac{mv^2}{r} = -\frac{d}{dr}(kr^n) \].
04

Calculate the Force

Calculate \( \frac{dU}{dr} \): \( \frac{d}{dr}(kr^n) = knr^{n-1} \). The force \( F \) is then \( F = -knr^{n-1} \).
05

Equate Forces for Circular Motion

Set the centripetal force equal to the negative potential energy force (since it's attractive): \[ \frac{mv^2}{r} = knr^{n-1} \]. Simplifying gives \[ mv^2 = knr^n \].
06

Express Kinetic Energy

The kinetic energy \( T \) is \( \frac{1}{2}mv^2 \). Substitute \( mv^2 = knr^n \) into the kinetic energy equation: \[ T = \frac{1}{2}knr^n \].
07

Relate Kinetic and Potential Energy

Using \( U = kr^n \), the kinetic energy can be expressed in terms of \( U \): \( T = \frac{1}{2}knr^n = \frac{1}{2}n(kr^n) = \frac{n}{2}U \). This establishes that \( T = \frac{n}{2}U \), proving the statement.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Central Force
A central force is a force that is directed along the line joining two bodies and depends solely on the distance between them. In physics, this type of force is crucial when studying objects that move under the influence of gravity, electric fields, or other similar forces. Some important characteristics of central forces include:
  • They always act along a line joining the center of one object to the center of the other.
  • Central forces are conservative, meaning that the work done by or against them depends only on the initial and final positions.
  • The force can be attractive (like gravity) or repulsive (like the force between similar charges).
In the context of the exercise, the central force is derived from the potential energy formula, \( U = k r^n \). The force itself is calculated by taking the negative gradient of the potential energy, which determines how the object will move in response to that force.
Circular Orbit
A circular orbit occurs when an object moves in a perfect circle around another object, with the force of attraction (like gravity) acting as the centripetal force keeping it in motion. This is a special case of the more general elliptical orbit, where the centripetal force is supplied by the central force derived from potential energy.
  • For a stable circular orbit, the centripetal force required to keep the object moving in a circle is provided entirely by the central force.
  • In our exercise, this relationship is shown by equating the centripetal force to the derivative of the potential energy.
  • Mathematically, the necessary centripetal force for a circular orbit is given by: \( F_c = \frac{mv^2}{r} \).
This compelling balance of forces ensures that the object continues to move along its circular path, neither falling into the center nor flinging off into space.
Kinetic Energy
Kinetic energy is the energy of motion. Any object that is moving has kinetic energy, given by the formula: \( T = \frac{1}{2} mv^2 \). It depends on both the mass of the object and its velocity squared. In our problem, kinetic energy plays a pivotal role in understanding the relation established by the virial theorem.
  • Kinetic energy is crucial in the context of orbital mechanics because it helps determine how fast the object is moving along its path.
  • In this scenario, the kinetic energy is directly tied to the potential energy through the virial theorem.
  • By analyzing the forces and motions involved, we proved that the kinetic energy can be expressed as a fraction of the potential energy: \( T = \frac{n}{2} U \), where \( n \) is derived from the potential energy relation.
This relationship is significant in many areas of physics, especially when studying systems that are bound by forces, like planets around a star.
Potential Energy
Potential energy is the stored energy in a system due to its position or configuration. In the case of central forces like gravity, potential energy can change as the relative positions of interacting bodies change. For this exercise, potential energy is defined by the expression \( U = k r^n \).
  • With central forces, potential energy is usually a function of distance between two objects.
  • The potential energy in the exercise plays a key role in determining the dynamics of the system, influencing how objects move in response to each other.
  • The virial theorem connects potential energy and kinetic energy in stable orbits, showing how energy balances within such systems: \( 2T + nU = 0 \).
Understanding potential energy and its implications helps us predict how the system behaves over time, particularly when we can express it in terms of other quantities of interest like kinetic energy.

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Most popular questions from this chapter

A particle of mass \(m_{1}\) and speed \(v_{1}\) collides with a second particle of mass \(m_{2}\) at rest. If the collision is perfectly inelastic (the two particles lock together and move off as one) what fraction of the kinetic energy is lost in the collision? Comment on your answer for the cases that \(m_{1} \ll m_{2}\) and that \(m_{2} \ll m_{1}\).

In one dimension, it is obvious that a force obeying Hooke's law is conservative (since \(F=-k x\) depends only on the position \(x,\) and this is sufficient to guarantee that \(F\) is conservative in one dimension). Consider instead a spring that obeys Hooke's law and has one end fixed at the origin, but whose other end is free to move in all three dimensions. (The spring could be fastened to a point in the ceiling and be supporting a bouncing mass \(m\) at its other end, for instance.) Write down the force \(\mathbf{F}(\mathbf{r})\) exerted by the spring in terms of its length \(r\) and its equilibrium length \(r_{\mathrm{o}} .\) Prove that this force is conservative. [Hints: Is the force central? Assume that the spring does not bend.]

Which of the following forces is conservative? (a) \(\mathbf{F}=k(x, 2 y, 3 z)\) where \(k\) is a constant. (b) \(\mathbf{F}=k(y, x, 0) .\) (c) \(\mathbf{F}=k(-y, x, 0)\). For those which are conservative, find the corresponding potential energy \(U,\) and verify by direct differentiation that \(\mathbf{F}=-\nabla U\).

(a) Consider a mass \(m\) in a uniform gravitational field \(\mathbf{g},\) so that the force on \(m\) is \(m \mathbf{g},\) where \(\mathbf{g}\) is a constant vector pointing vertically down. If the mass moves by an arbitrary path from point 1 to point \(2,\) show that the work done by gravity is \(W_{\mathrm{grav}}(1 \rightarrow 2)=-m g h\) where \(h\) is the vertical height gained between points 1 and 2. Use this result to prove that the force of gravity is conservative (at least in a region small enough so that \(\mathrm{g}\) can be considered constant). (b) Show that, if we choose axes with \(y\) measured vertically up, the gravitational potential energy is \(U=m g y\) (if we choose \(U=0\) at the origin).

(a) Consider an electron (charge \(-e\) and mass \(m\) ) in a circular orbit of radius \(r\) around a fixed proton (charge \(+e\) ). Remembering that the inward Coulomb force \(k e^{2} / r^{2}\) is what gives the electron its centripetal acceleration, prove that the electron's KE is equal to \(-\frac{1}{2}\) times its \(\mathrm{PE}\); that is, \(T=-\frac{1}{2} U\) and hence \(E=\frac{1}{2} U\). (This result is a consequence of the so-called virial theorem. See Problem 4.41.) Now consider the following inelastic collision of an electron with a hydrogen atom: Electron number 1 is in a circular orbit of radius \(r\) around a fixed proton. (This is the hydrogen atom.) Electron 2 approaches from afar with kinetic energy \(T_{2} .\) When the second electron hits the atom, the first electron is knocked free, and the second is captured in a circular orbit of radius \(r^{\prime} .\) (b) Write down an expression for the total energy of the three-particle system in general. (Your answer should contain five terms, three PEs but only two KEs, since the proton is considered fixed.) (c) Identify the values of all five terms and the total energy \(E\) long before the collision occurs, and again long after it is all over. What is the KE of the outgoing electron 1 once it is far away? Give your answers in terms of the variables \(T_{2}, r,\) and \(r^{\prime}\).

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