Lessons:
- Projectiles and their Motion
- Gravitational Potential and Potential Energy
- Satellites, Orbits and Gravitation
- Electrical Potential and Electrical Potential Energy
Syllabus Statements:
You recall from our work in year 12 the suvat equations:
- v = u + at
- v2= u2 + 2as
- s = ut + 1/2 at2
But under what conditions can we use these equations?
For constantly accelerated motion in a straight line.
What about projectiles then? How can we apply these equations to projectile problems?
By considering the perpendicular components of the motion.
These can be selected as is appropriate to the problem.
You have already done lots of this but to consolidate and revise...
Do an experiment using a projectile to determine the acceleration due to gravity.
What measurements will you make? How will you ensure a fair test? How will you ensure accuracy and reliability?
Homework: Complete question 4.32 (Hutching pp51).
Hutchings pp35-48, Kirk pp 52-55
Gravitational Potential and Potential Energy
We have discussed and understand the concepts of gravitational field, field strength and force.
Gravitational potential requires us to consider the energy that a body gains or loses when it is moved inside a gravitational field. That is to say the work done on it or by it.
If we consider a unit mass at an infinite distance from a planet, then its potential energy is negligible (because the gravitational force acting on it is negligible).
As we move the unit mass into the gravitational field, it gains potential as the gravitational field does work on it.
W = F x, and force acting is gravitational force, F = GMm/x2
The total work done on the unit mass in moving it from infinity to a distance, r, from the planet is...
But because energy must be given to a unit mass in a gravitational field to move it to infinity, where it has zero GPE, then it must lose energy coming from infinity.
Thus gravitational potential is negative.
Gravitational potential at a point is the work done in moving a unit mass from infinity to that point.
If we consider a non unit mass, m, then we work out the gravitational potential energy of that mass...
Task:
Equate this with our simple equation (mgh) for potential energy and verify the value of g.
Now plotting the graph of gravitational potential against distance from the surface of the Earth...
We differentiate V with respect to r to find the potential gradient, then we get gravitational field strength...
You will recall from work on Electric fields, the idea of an equipotential line.
This is a line which can be draw that connects points that all have the same potential. In the case of a gravitational field, equipotential lines connect points of equal gravitational potential.
You will also recall that in order to deduce the direction of electric field lines you draw a line at right angles to the equipotential line. The same is true of gravitational fields.
Gravitational Potential and Escape Speed
In the last question you used conservation of energy to equate potential energy of the rock initially with the kinetic energy of the rock as it hit the Sun's surface.
We can use this process in reverse to determine how much kinetic energy an object must have to be able to move to infinity in a gravitational field. In other words how fast it must go to escape a planets gravitational field. This is escape speed.
Task:
Do complete derivations for these two equations for escape speed.
Htchings pp120-138, Kirk pp 158-166
Satellites, Orbits and Gravitation
Since the centripetal force acting on a satellite in orbit is provided by gravitational attraction.
Gives us the equation for the KE of an orbiting satellite.
You already know the equation for its GPE, and can add the two to give the total energy, which is...
You should now use these equations to draw on one set of axes graphs of how the potential, kinetic and total energies of a satellite change with its orbital radius.
Discussion Points:
1. What is the relationship between KE and GPE?
2. What is the relationship between this and escape speed?
3. How does this relate to the concept of a "Potential well"?
Kepler's 3rd Law
K3 follows on from this work, but depends on us considering the orbit of a satellite as being circular.
Equating centripetal force and gravitational force as before...
But substituting for velocity...
If a satellite experiences a force towards the centre of the Earth due to gravity, then why is it not accelerating towards the centre of the Earth?
It is! But why does it never gets there, because it is moving in a circle.
There is no opposing force to weight so objects falling towards the centre of the Earth are said to be in 'free fall'. Object describing this motion feel weightless.
A demonstrattion to help explain...
Now consolidate this lot with questions
Hutchings pp107-118, Kirk pp 158-166
Electrical Potential and Electrical Potential Energy
We have discussed and understand the concepts of Electric field, Electric field strength (E) and Electric force.
Electric potential requires us to consider the energy that a charge gains or loses when it is moved inside an electric field. That is to say the work done on it or by it.
If we consider a unit charge at an infinite distance from a charge, then its potential energy is negligible (because the electric force acting on it is negligible).
As we move the unit charge into the electric field, it gains potential as the electric field does work on it.
W = F x, and force acting is electric force, 
The total work done on the unit charge in moving it from infinity to a distance, r, from the charge is...
- The sign of the electric potential depends on the nature of the charges involved.
- Similar charges have positive potentials, opposite have negative.
- Electric potential at a point is the work done in moving a unit charge from infinity to that point.
- If we consider a non unit charge, q, then we work out the electric potential energy of that charge.
Important note...
The electric field for different arrangements of charges differs, and therefore does not always adhere to the equation .
To this end the equation derived above only applies to single point charges and outside a spherical area of evenly distributed charge, Q.
You need to be able to cope with situations like this, and when there are more than one point charges present.
Task:
Compile a table of similarities and differences between electic and gravitational fields. You will find these in any good text book or web site.
Exercises:
Potential Gradient
This is the way that potential changes with distance moved in a field, dV/dr.
It can be seen from our previous derivation that this is equal to electric field strength at that point...
dV/dr = E
Equipotentials
Following on from this, all point at the same potential (eqipotentials) experience the same electrical field strength.
Lines joining points of equipotential are called equipotential surfaces. Electric fields always act perpendicular to equipotential surfaces.
Recall the equipotential lines we drew on conducting paper.
Exercise:
Try to draw lines of equipotential for the following situations...
| A point charge | A charged sphere | Two point charges of the same charges. | Two point charges of different charges. |
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- Parallel charged plates
- 3 point charges (2 positive and 1 negative) in the same vicinity.
- a positive point charge in the vicinity of a negatively charged plate.
Kirk pp 158-166
anrophysics 2007