Orbiting particle with given potential. Find the total energy. Need help

In summary, the conversation discusses a light particle of mass m orbiting around a massive attractive center with a potential of V(r) = Cr^2. By using equations for circular motion, it is shown that the total energy of the particle is E = 2Cr^2. The Bohr postulate for quantization and the result from part (a) are then used to derive an expression for the energy associated with allowed orbits in terms of the mass of the particle, n, and other constants.
  • #1
PanosP
1
0

Homework Statement



(a) A light particle of mass m orbits in a circular orbit around a massive attractive centre with a potential that is given by

V(r) = Cr^2

(b) Using the equations for circular motion show that the total energy of such a particle must be given by

E = 2Cr^2

Use the Bohr postulate for the quantization L = mvr = nh(bar) in combination with the answer to part (a) to arrive at an expression for the energy associated with the allowed orbits in terms of the mass of the particle n and other constants.


I have no idea how to even begin this!

ANY help would be deeply appreciated!

thanks!
 
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  • #2
Welcome to PF!

Hi PanosP ! Welcome to PF! :smile:

(try using the X2 tag just above the Reply box :wink:)
PanosP said:
(a) A light particle of mass m orbits in a circular orbit around a massive attractive centre with a potential that is given by

V(r) = Cr^2

(b) Using the equations for circular motion show that the total energy of such a particle must be given by

E = 2Cr^2

I have no idea how to even begin this!

Find the force from V(r), then use good ol' https://www.physicsforums.com/library.php?do=view_item&itemid=26" :smile:
 
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  • #3


I would be happy to provide some guidance on how to approach this problem. First, let's break down the problem into smaller parts. In part (a), you are given the potential V(r) = Cr^2 and asked to find the total energy of a light particle (mass m) in a circular orbit around a massive attractive centre. This means that the particle is moving in a circular path with a constant speed, which we can relate to the potential using the equations for circular motion.

The total energy of a particle in a potential is given by the sum of its kinetic energy (KE) and potential energy (PE). We can write this as E = KE + PE. In circular motion, the kinetic energy is given by KE = (1/2)mv^2, where v is the speed of the particle. We can also relate the speed to the radius of the orbit using the fact that the particle is moving in a circular path. This gives us v = (rω), where ω is the angular velocity.

Now, let's consider the potential energy. In this case, the potential is V(r) = Cr^2. We can relate the potential energy to the angular momentum (L) of the particle using the Bohr postulate, which states that L = mvr = nh(bar). This means that the angular momentum is quantized and can only take on certain values determined by the integer n and the reduced Planck's constant (h(bar)). We can write the potential energy as PE = (1/2)kL^2, where k is a constant.

Putting everything together, we can write the total energy as E = (1/2)mv^2 + (1/2)kL^2. Substituting in our expressions for v and L, we get E = (1/2)m(rω)^2 + (1/2)k(nh(bar))^2. Simplifying this expression, we arrive at the answer given in part (b), E = 2Cr^2.

To find the energy associated with allowed orbits, we can use the fact that the angular momentum is quantized and the expression for the potential energy in terms of L. We can write the allowed values of n as n = 1, 2, 3, ... and substitute them into the expression for PE. This will give us a series of values for PE, which we can then
 

1. What is an orbiting particle with given potential?

An orbiting particle with given potential refers to a particle that is moving in a circular or elliptical path around a central point, while experiencing a specific potential energy due to the forces acting upon it.

2. How is the total energy of an orbiting particle with given potential calculated?

The total energy of an orbiting particle with given potential is calculated by adding the particle's kinetic energy and potential energy together. This can be represented by the equation E = K + U, where E is the total energy, K is the kinetic energy, and U is the potential energy.

3. What factors influence the total energy of an orbiting particle with given potential?

The total energy of an orbiting particle with given potential is influenced by the mass and velocity of the particle, as well as the strength of the potential energy due to the forces acting upon it.

4. How does the total energy affect the motion of an orbiting particle with given potential?

The total energy of an orbiting particle with given potential determines the type of orbit the particle will have. A higher total energy will result in a more elliptical orbit, while a lower total energy will result in a more circular orbit.

5. What units are used to measure the total energy of an orbiting particle with given potential?

The total energy of an orbiting particle with given potential is typically measured in joules (J) or electron volts (eV), depending on the specific situation and the units used for the particle's mass and velocity.

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