April30

Hi, my name is Jamie and I’m the scribe for today. (Sorry for late posting; computer wouldn’t work)

Today in class, Ms K talked about the booklet on Gravitational Potential Energy/Well, Escape Velocity, Total Energy and Binding Energy.
Now, I will just summarize what we have covered up today; including the work sheet on Gravitational Energy.

Gravitational Potential Energy

According to Newton’s Law of Universal Gravitation, force of gravitational attraction between 2 masses (m₁ and m₂), at any separation distance (R), is given by an equation:

F=Gm₁m₂/R² (where F is a vector)

To increase the separation of 2 masses from R₁ to R₂ requires work to overcome the force of attraction (ex. stretching a spring) which increases the gravitational potential energy. After remembering the relationship between gravitational force and separation (shown below as a graph), the resultant equation for the change in potential energy is:

∆E_gpe = (-Gm₁m₂/R₂) – (-Gm₁m₂/R₁)
^Potential Energy at R₂ ^ Potential Energy at R₁
The 1st term depends on R₂ and the 2nd term on R₁ (each term is an expression for the gravitational potential energy or E_gpe at that separation)


At any separation distance R, E_gpe, between m₁ and m₂, the equation is:

E_gpe = -Gm₁m₂/R (where E_gpe is a scalar; G is constant; R is distance between 2 masses)
Each side of the equation is proportional.


Gravitational Potential Well

* E_gpe = -Gm₁m₂/R always produces negative values.
* As R increases (masses get farther apart), E_gpe increases by becoming less negative.
* As R approaches infinity, PE approaches zero. Zero value of E_gpe between 2 masses occurs when they are infinity apart.

The two objects that have force of attraction between them resulting negative E_gpe is called a Potential Well.

Example:



Pretend the m₁ is the earth and m₂ is the rocket. For the diagram above, R₁ represents the earth’s radius. If rocket is at rest on earth, then there is no KE on the rocket, only E_gpe which is equal to – Gm₁m₂/R₁.

Total energy of rocket (E total), where is just the E_gpe , is the sum of E_gpe and KE:

E total = E_gpe + KE

Assuming that the rocket rises above Earth’s surface to a height of R₂ where it has less E_gpe and some KE. The E total of rocket remains the same constant value.

E total = KE + E_gpe = KE + (-Gm₁m₂/R₂)


Escape Velocity

We must know the earth’s Potential Well when trying to calculate minimum velocity (Escape Velocity) the rocket must have to escape. Rocket’s initial KE must exceed depth of the potential well at earth’s surface, making total energy positive. Meaning, the rocket must reach an infinite distance where E_gpe = 0 before coming to rest.

(USING THE RECENT EXAMPLE…) At Earth’s surface: E total = - Gm_em_r/R_e (where “e” stands for earth and “r” for rocket”)

Energy is then change to KE if the rocket: KEr = ½ m_rv_r²

* ½ m_rv_r² = - Gm_em_r/R_e

Therefore, V_r = √2Gm_e/R_e

A rocket launched from earth w/ a velocity greater than 1.12 x 10⁴ m/s will move away from earth (losing KE and gaining E_gpe ).
* Since KE is greater than depth of its E_gpe , its total energy always remains positive.


Total Energy and Binding Energy

For rocket to escape from earth’s potential well, its KE must exceed the E_gpe and if this doesn’t happen, it is said to be bound to earth.
Binding Energy is amount of additional KE it needs to escape.

For rocket at rest on earth’s surface, binding energy is identical in magnitude to E_gpe at earth’s surface (rocket’s not moving).

E total = KE + E_gpe = 0 + (- Gm_em_r/Re) = - Gm_em_r/Re
E binding = Gm_em_r/Re

* If rocket is in orbit at any radius Ro in potential well of earth, then the centripetal force that keeps the rocket is the circular orbit is provided by force if gravitational attraction between earth and rocket.

If rocket has mass m_r and an orbital velocity of v_r:

Fc = Fg OR m_rv_r²/Ro = Gm_em_r/Ro²

Total energy is:

E total = KE + E_gpe = 1/2 m_rv_r² + (- Gm_em_r/Ro) = ½ Gm_em_r/Ro +
- Gm_em_r/Ro = ½ Gm_em_r/Ro = ½ E_gpe

* Total energy of satellite in a circular orbit at any radius of orbit Ro is negative and equal to ½ the value of E_gpe at this radius.
* Satellite is bound to earth and its binding energy is:

E binding = ½ Gm_em_r/Ro







To be continued……