the world in a different light
Here’s a quick video to demonstrate the movement of a ball initially moving in a circle before its centripetal force (contact force by the circular wall) is removed. The ball is observed to move in a tangent to the circle, in accordance with Newton’s 1st Law, since there is no longer a net force acting on it.
Using the GeoGebra app above, I intend to demonstrate the relationship between total energy, kinetic energy and gravitational potential energy in a rocket trying to escape a planet’s gravitational field.
By changing the total energy of the rocket, you will increase the initial kinetic energy, thus allowing it to fly further from the surface of the planet. The furthest point to which the rocket can fly can be observed by moving the slider for “distance”. You will notice that the furthest point is where kinetic energy would have depleted.
Gravitational potential energy of an object is taken as zero at an infinite distance away from the source of the gravitational field. This means gravitational potential energy anywhere else takes on a negative value of $\dfrac{-GMm}{r}$. Therefore, the total energy of the object may be negative, even after taking into account its positive kinetic energy as total energy = kinetic energy + gravitational potential energy.
The minimum total energy needed for the rocket to leave the planet’s gravitational field is zero, as that will mean that the minimum initial kinetic energy will be equal to the increase in gravitational potential energy needed, according to the equation $\Delta U = 0 – (-\dfrac{GMm}{R_P})$, where $R_P$ is the radius of the planet.
Since $\dfrac{1}{2}mv^2 = \dfrac{GMm}{R_P}$, escape velocity, $v = \sqrt{\dfrac{2GM}{R_P}}$.
This GeoGebra app shows how angular velocity ω is the rate of change of angular displacement (i.e. $\omega=\dfrac{\theta}{t}$) and is dependent on the speed and radius of the object in circular motion (i.e. $v=r\omega$).
Students can explore the relationships by doing the following:
Keeping r constant and varying ω.
Keeping ω constant and varying r.
Keeping v constant by varying r and ω.
This GeoGebra app shows the relationship s = rθ.
One activity I get students can do is to look at the value of θ when the arc length s is equal to the radius r. This would give the definition of the radian, which is the angle subtended at the centre of a circle by an arc equal in length to its radius.
Mathematics defines the constant π as the ratio of a circle’s circumference to its diameter. This can also be shown in the app, although you need to drag the moving point to a point just short of one full revolution.
I am taking the opportunity (since my students are all doing home-based learning) to teach them how to use spreadsheets to do calculations and to obtain a best-fit line. While they can still submit graph work using PDF scanning apps such as Office Lens and Camscanner into Google Classroom for me to mark, they can make use of the spreadsheet-generated graph to check their results.
Even though for exams, we still require them to plot the points on paper and obtain the gradient and intercept from points on the best-fit line, nobody is going to do so when they start working. So I might as well teach them now.
Due to the lack of face-to-face time, I made this step-by-step video showing them how to do so.
For lab work, students often have to estimate a line of best fit for their data points manually. It takes a bit of practice to get it right. With this app, students can generate data points with varying types of scatter and predict their own best-fit line before comparing it with a computer generated one based on the least mean square method.