What is gravitational potential energy? What is the implication of this definition?
- the GPE of two bodies is the work that was done in bringing the bodies to their present position from when they were infinitely far apart.
- All GPE values are negative as work is done by as mass as it moves closer to another mass.
What are two formulas for gravitational field strength?
- g = F/m DB
- g = -GM/r² DB
What is a formula for GPE, and how is it derived?
- Work done = GPE = integral between r and infinity of the force on a small test mass dr.
W = Ep = -GMm/r DB
What is gravitational potential? How can it be calculated?
(Vg) at a point P in a gravitational field is the work done per unit mass in bringing a small point mass m from infinity to point P.
Vg = W/m
Vg = -GM/r (Jkg-1) DB
How can you calculate the work done in moving a mass, m, from P to Q in a gravitational field?
W = change in Ep = m∆V not in DB
When an external agent does work to move the mass at a constant small speed (so KE of mass is negligible).
What does a graph of distance against gravitational potential (y axis) look like?
- A declining negative curve with an asymptote at both axis. As Vg → 0, r → ∞
- Called a ‘potential well’.
How can you find the electric field created by an arrangement of point charges?
Add the vectors of the electric fields created by each of the individual charges. Often some will cancel out.
What is electric potential? How can you calculate it?
(Ve) at point P is the amount of work done per unit charge as a small positive test charge q is moved from infinity to the point P.
Ve = W/q = kQ/r (JC-1)
What is electric potential energy?
- the GPE of two charges is the work that was done in bringing the charges to their present position from when they were infinitely far apart.
What are two formulas for electric potential energy, Ep?
Ep = qVe = kq₁q₂/r
How can you calculate the work done in moving a charge from A to B in an electric field?
W = change in Ep = q∆V not in DB
When an external agent does work to move the mass at a constant small speed (so KE of mass is negligible).
How can you find the potential at point P of an electric field created by an arrangement of point charges?
- Potential is a scalar quantity, it is the sum of the individual electric potentials.
What does the graph of distance against electric potential (y axis) of a positively charged spherical object radius R look like?
- Potential is constant everywhere inside the sphere (when r is less than R),
- Outside, the potential declined as 1/r
What is an equipotential surface?
A surface that connects all points around a mass or charge that have the same potential. Often spherical ‘shells’. Perpendicular to field lines.
What is special about a graph of distance against potential? How is it derived?
The gradient of the graph is the same as the magnitude of the field strength (either gravitational or electric)
Found by equating two equations m∆V and Fdistmoved for work done moving a (mass) from one equipotential surface to a neighbor.
How can you derive an equation for the kinetic energy of an object orbiting a planet? (need to know how to do)
The gravitational force provides the centripetal force,
∴ GMm/r² = mv²/r
so v² = GM/r
Multiply both sides by m/2 to get Ek = GMm/2r
How can you derive the total energy of an object orbiting a planet?
Et = Ek + Ep Et = GMm/2r - GMm/r Et = -GMm/2r
How can you find the velocity of an object orbiting a planet?
v =sqrt(GM/r) DB
What does the graph of energy against orbital radius look like for Et, Ek and Ep?
All are curves beginning at R radius of the planet that tend towards the x axis.
- Et is a reflection of Ek in the x axis(-ve)
- Ep = 2Et, also negative.
What will happen if an object is launched from the surface of a planet with a total energy…
< 0
= 0
> 0
- It will go into an elliptical or circular orbit
- It will follow a parabolic path and never return
- It will follow a hyperbolic path to infinity.
What will happen if a spaceship in orbit fires its rockets
- opposite the direction of travel
- in the same direction as travel.
- The total energy will increase, the radius of the orbit will increase, Ek decreases, Ep increases.
- The radius of the orbit decreases, Ek increases, Ep decreases.
COUNTERINTUITIVE! Imagine climbing up or down the potential well of the energy graphs.
What is escape velocity?
The minimum speed that a mass must have in order to escape a gravitational field ie. reach infinity. Applies to ballistic motion only.
How can you derive escape velocity?
Note that the total energy of the system must be 0 at launch (conservation - when it reaches infinity it will have no energy left)
Ek + Ep = 0
1/2 mvesc² - GMm/R = 0
rearrange. DB
Why does an astronaut feel weightless?
The astronaut and spacecraft are both falling freely - they have the same acceleration.
