Gravitational pot. teachers notes
Wells and 'ills
A 1kg ball is at the bottom of the 10m deep, round hole shown in the image.
The ball is pushed out of the hole at constant speed by exerting a force in the horizontal direction.
- Using Excel construct an iterative model to plot a graph of the workdone vs the horizontal displacement.
- Show that the force exerted at any point is equal to the gradient of this graph.
By copying the image into LoggerPro you can analyse the curve to find that it's equation is y = 0.5x2 -10
To calculate the works done as the ball is pushed up the slope you need to know the gradient at each point. differentiating the function gives the gradient dy/dx = x
Here is my Excel model:
This line has the function y = 5x2 the gradient is 10x which is the same as the force .
Lines of equipotential are lines joining points where the ball would have equal potential energy.
- Draw the lines of equipotential for the hole as they would be seen from above.
I did the equipotentials in Geogebra.
This is how:
x2 + y2 =r2 is the equation of a circle of radius r.
Sequence[x² + y² = r2, r, 0, 100, 10] will plot a set of circles where r = 0 to 100 in steps of 10.
you want the circles to be plotted every time the potential energy = certain values
PE = 5r2 so r2 = PE/5
Sequence[x² + y² = PE / 5, PE, 0, 100, 10] will plot a when PE = 0, 10, 20 etc
Potential
Potential is defined as the work done per unit mass taking a small test mass from infinity to the point in question.
- Show that the potential a distance r from the earth is GM/r.
The diagram below shows the lines joining points of equal potential energy for a new type of field.
- Show that the equation relating the force and distance is F = 5/x3
Using LoggerPro image analysis I got the x values then added the PE manually.
The bst fit curve is 2.5/x2 which is the integral of 5/x3
3D Potential well
- plot a line of dots in 3D using the sequence function.
- make one coordinate of the dots vary according to the inverse of the distance from the origin
- make another coordinate equal to some variable, a
- turn on trace
- vary a with a slider
OK, here is the code: Sequence[(-1 / sqrt(i² + a²), i, a), i, -5, 5, 0.1]