Centrifugal term in mechanical energy in gravitation

In summary, the effective potential in Newtonian gravitation is given by equation (1), where μ is the reduced mass of the system planet-star. The term U_{centrifugal} = L^2/2μr^2 is often called the angular momentum barrier and prevents the particle from getting too close to the origin. This is due to the angular momentum being constant, causing the tangential kinetic energy to increase at a greater rate than the decrease in radius. This leads to a total kinetic energy that is greater than what is allowed by conservation of energy. In general, the centrifugal term prevents the planet from colliding with the star by creating a barrier at the minimum distance of approach.
  • #1
Soren4
128
2
I'm studying effective potential in Newtonian gravitation. The mechanical energy of a body can be written
$$E=\frac{1}{2}\mu {\dot{r}}^2+\frac{L^2}{2\mu r^2}-\gamma \frac{m M}{r^2} \tag{1}$$

Where [itex]\mu[/itex] is the reduced mass of the system planet-star.

Consider now the term $$U_{centrifugal}=\frac{L^2}{2\mu r^2}$$
I don't understand this explanation found on Morin.
The [itex]L^2/2mr^2[/itex] term in the effective potential is sometimes called the angular momentum barrier.It has the effect of keeping the particle from getting too close to the origin.Basically, the point is that [itex]L ≡ mr^2\dot{ θ}[/itex] is constant, so as r gets smaller, [itex]\dot{θ}[/itex] gets bigger. But [itex]\dot{θ}[/itex] increases at a greater rate than r decreases, due to the square of the [itex]r[/itex] in [itex]L=mr^2\dot{ θ}[/itex]. So eventually we end up with a tangential kinetic energy, [itex]mr^2\dot{ θ}^2/2[/itex], that is greater than what is allowed by conservation of energy.

Why [itex]mr^2\dot{ θ}^2/2[/itex] is greater than maximum KE allowed by conservation of energy in this case?

In general how does [itex]U_{centrifugal}[/itex] prevent the planet to collide with the star (provided the planet has non zero angular momentum)?
 
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  • #2
Soren4 said:
Why mr2˙θ2/2mr2θ˙2/2mr^2\dot{ θ}^2/2 is greater than maximum KE allowed by conservation of energy in this case?

do not compare the two terms , rather one should compare the total potential and the kinetic energy - and for a bound state the total energy should be negative that id the potential energy should be greater than the kinetic energy.-or plot the graph of potential + centrifugal term and see that the bound state energy is -ve.

Soren4 said:
In general how does UcentrifugalUcentrifugalU_{centrifugal} prevent the planet to collide with the star (provided the planet has non zero angular momentum)?

if you look at the above potential energy plot you will see the barrier as its positive and goes to infinity as r goes to minimum distance of approach. say r0.
 

Related to Centrifugal term in mechanical energy in gravitation

What is the centrifugal term in mechanical energy in gravitation?

The centrifugal term in mechanical energy in gravitation refers to the outward force experienced by an object rotating around a central point. This force arises due to the object's inertia and is directed away from the center of rotation.

How does the centrifugal term affect mechanical energy in gravitation?

The centrifugal term adds to the total mechanical energy of the system, increasing it as the object rotates faster. This increase in energy can be seen in the object's higher velocity and increased distance from the center of rotation.

What are the units of the centrifugal term in mechanical energy in gravitation?

The units of the centrifugal term are typically in joules (J), as it is a measure of energy. However, it can also be expressed in other units such as ergs or foot-pounds.

How is the centrifugal term related to the gravitational force?

The centrifugal term is directly related to the gravitational force, as it is dependent on the mass of the object, its distance from the center of rotation, and the strength of the gravitational force.

Can the centrifugal term be negative?

Yes, the centrifugal term can be negative if the direction of rotation is opposite to the direction of the gravitational force. In this case, the centrifugal term would decrease the total mechanical energy of the system.

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