The pendulum experiment (step 1.4) What is a thought experiment? (steps 1.4 and 1.9)

A thought experiment is a way for scientists to conclude on hypotheses or theories without materially conducting an experiment. Thought experiments are extremely relevant regarding studies on different unreachable celestial bodies or for example black holes. Basic process of thought experiments, some examples:

Falling bodies according to Galileo (step 1.4) The Aristotelian theory that a heavy body falls faster than a lighter one is illustrated through the thought experiment of having a heavier ball and a lighter ball attached to one another and observing their fall. Following Aristotle’s theory, the lighter ball would slow down the fall of the heavy body, and therefore the composite body would fall more slowly than the heavy body on its own. However, there is a contradiction as the composite body is heavier than the heavy ball on its own and thus should actually fall faster! The absurdity of the results enabled Galileo to refute Aristotle’s theory.

Newton’s cannonball (step 1.9)

Newton imagined an experiment where a cannonball would be thrown from the top of a mountain so high that it would be over the atmosphere, in order to eliminate friction which would slow the cannonball down. Without gravity, the cannonball would continue in a straight line. However, on Earth, the gravitational force pulls objects towards the center of the Earth so the cannonball falls in a parabolic trajectory on the Earth’s surface after some time. The faster the cannonball would be thrown, the further away it will land. What happens is that, as more distance is covered by the cannonball, the Earth’s curve increases and there is the illusion (from the cannonball’s point of view) that the surface “lowers”. However, since the cannonball is constantly falling, it remains at the same distance away from the Earth. Thus, Newton thought that there could be a certain speed where the cannonball could be thrown fast enough so that it would never fall back on the surface of the Earth. He thought that the cannonball would continue to
revolve around the earth and fall into orbit. From this thought experiment he was capable to deduct that the moon was ‘continuously falling’ towards the Earth.

Cf. Cannonball exercise

Exercise: the cannon ball experiment (step 1.9)

This exercise is about finding the magnitude of the velocity that is necessary to put a cannonball in orbital motion around the Earth, at an altitude equal to h. To that effect, a cannonball is fired from a height h above the ground at a velocity v° parallel to the ground.

A few assumptions are made:

•    The Earth is a perfect sphere of radius R = 6370 km.
•    The ball is in free fall (the only external force acting on the ball is its weight).
•    The gravitational field has a constant magnitude g and is pointing towards the center of the Earth.

The motion of the ball for small duration Δt is a combination of (see Figure):

• A horizontal motion of magnitude X.
• A vertical motion of magnitude Y. 1°/ a) Show that X = v° × Δt.

b) Show that X = (R+h) × sin(α)

c) Use the previous answers to write v° in terms of R, h, α, and Δt.

2°/  a) Show that after Δt, the ball will only remain at the same height above the ground if the following condition is met:

(R + h) × cos (α) + Y = R + h.

b) Use the second law of Newton to show that   Y = ½ g (Δt)2.

c) Use the previous answers to write cos(α) in terms of R, h, g, and Δt.

If Δt is not too big (less than one minute), the distance travelled is very small compared to the circumference of the Earth. Therefore, we can assume that the angle α is very small relative to 1, in which case a very good approximation is: cos(α) = 1 – α2/2 and sin(α) = α.

3°/ Show that α² = g Δt² / (R + h).

4°/ a)  Use the answers to 1) and 3) to show that vº = √[g(R+h)].

b) Deduce that, as long as h is much smaller than the Earth radius,  v° depends only on g and the Earth radius R, and compute v° using  g = 9.8m.s-2  .

5°/ Compare the value you found to the one mentioned in the video.