It all began with a conversation on a bike ride about a feather and a lead ball falling. My sister had a vague memory of the famous story about Galileo and the leaning tower of Pisa. She recalled the principle that anything dropped from the top should reach the ground at the same time. She also knew there was some complication about the air dragging lighter things more than heavy things. Or was it flat-shaped things verses pointy things? Or was it just the surface area that mattered?
Figure 1 Galileo with symbols of his work
Her main query was about the principle itself: it seems to go against everyday experience – would light and heavy things really reach the ground together, if there was no air in the way?
Galileo and motion under gravity
Apparently, it was a pupil of Galileo’s who originated the story in a biography he wrote of his teacher. Today it’s considered more likely to have been a thought-experiment rather than a live demonstration.
The idea was that if two spheres of the same shape, but different masses were dropped from the top of the tower they would reach the bottom at the same time.
It does seem counterintuitive: surely heavier things fall faster? That was certainly the prevailing view of natural philosophers at the time, handed down from their authority: Aristotle.
Figure 2 Aristotle’s vs Galileo’s theory
Today we have astounding visual evidence to help us get over the illusion that everyday experience gives us all. There is a place where air, and hence air resistance, is entirely absent: the Moon. This 1971 video by astronaut David Scott demonstrates a feather and hammer falling precisely together.
Figure 3 Feather and hammer falling on the Moon
A clearer demonstration was made more recently by Professor Brian Cox using a different method. He used a vast chamber in Cleveland, Ohio in which almost all the air in it had been pumped out. It is used to test spacecraft.
Today, these visual demonstrations help us overcome our false intuitions. Back in Galileo’s day (late 16th/early 17th century) a clever, conceptual way was also worked out. We imagine a brick falling freely; then imagine another identical one falling next to it. Then we imagine tying them together. Would they suddenly fall faster because they were double the mass, tied with a bit of string? We have to conclude that the rate of falling doesn’t depend on the mass of things (or weight in everyday parlance, here on Earth). In the absence of air resistance, the rate of falling is a fixed thing, common to all falling objects on Earth.
“What was it that Galileo actually did, then if the Pisa story is a myth”, was the next obvious question, given that he’s known for his work on understanding gravity.
It’s true, he did fundamental experiments on how gravity makes things move. Rather than climbing up a tower and doing a tricky outdoor experiment he constructed an apparatus indoors, out of wood.
There’s a marvellous reconstruction of one of his experiments in the science museum (Museo Galileo) in Florence where he lived for much of his life (Figure 4).
Figure 4 Galileo’s inclined plane experiment
It’s hard without modern instruments like stopwatches, light sensors or cameras to follow the progress of a falling ball. Galileo got around this problem by setting up an inclined ramp to make the fall under gravity measurable. The ramp had a groove in it, down which metal balls could run freely. In effect, the descent from top to bottom was spread out along the ramp. His purpose was to time the descent of various balls to study how they moved. At the time, there were no stopwatches, so timing was itself a challenge. He overcame this by using a steady flow of water from a tank and weighing the amount of water that had passed during the descent. Cunning!
To minimise the effect of friction, in his own words:
[we made the] groove very straight, smooth, and polished, and having lined it with parchment, also as smooth and polished as possible, we rolled along it a hard, smooth, and very round bronze ball.
He rolled the ball from a quarter of the way up, then halfway, then along the full length of the ramp. He reported: “we always found that the spaces traversed were to each other as the squares of the times”. In plain English, this means if it travels 1 metre in the first second it travels 4 meters in the second second and 9 metres in the third second … and so on (Figure 5). In other words, it gets faster and faster. He had shown that an object falling under the influence of gravity is accelerating. Not only this, but it accelerates uniformly – picking up the same amount of speed every second.
A lovely video of this experiment by Jim Al Khalil explains it clearly.
When the experiment is repeated with balls of different weight, it makes no difference to the time taken. The time of descent doesn’t depend on the mass of an object. All objects falling freely under gravity (i.e. without friction or air resistance) accelerate to the same degree, regardless of their mass. We now know the size of this acceleration: speed increases by 9.81 meters/second every second – that’s about 22mph faster every second. That’s a lot! No wonder you get hurt when you fall from a height.
Figure 5 Images of a freely falling basket ball taken with a stroboscope
All this seems counterintuitive – it’s so easy to go along with Aristotle’s wrong idea that heavy things fall faster. My sister grappled with the implications of this challenge: “So gravity acts with the same force on all objects despite their different weights. It’s only air resistance in normal circumstances which slows down a feather compared to a hammer. Is that right?” she asked.
The dialogue was beginning to reveal where the difficulty in understanding lay. As she pointed out: “surely, if the feather was exactly the same shape as the ball, why would the air resistance be different?” She felt confused, as would most people, trying to fathom this.
The problem lies in her idea that “gravity acts with the same force on all objects”. As we know from experience, dropping a heavy saucepan on your toe causes a lot more pain then dropping a teaspoon on it! A heavy object exerts more force than a light one. It‘s not the force that is the same for all objects falling under gravity. As Galileo showed, it’s the motion – the way they speed up. He showed that gravity causes things to accelerate as they fall. And that acceleration is the same for all objects regardless of their mass.
Back to the leaning tower of Pisa. If there had been no air to resist the falling balls, heavy ones and light ones would all reach the ground together – that’s what we see happening on the Moon or in a vacuum chamber where there is no air: they accelerate equally, they arrive together. However, in normal situations, like dropping balls from a tower, their motion is resisted by the air, as they rush downwards.
If the balls are of the same size and shape this upward force of resistance (called F-AIR in Figure 6) will be much the same for light balls and heavy ones, as they are the same size and shape. But the force of gravity (F-GRAVITY) on the lighter ball is less than on the heavier one. So the air resistance has a much larger effect proportionately on the lighter one. That’s why in everyday places like Pisa, the lighter one takes longer to fall, in practice. .
Figure 6 heavy and light ball of same size and shape, falling
Today, space exploration has given us a vivid picture of a further aspect of motion under gravity. TV footage of astronauts walking on the Moon, rising high with every step, shows graphically how much weaker gravity is out there –six times weaker, in fact.
Figure 7 Astronauts jumping on the Moon
On the Moon things still fall at the same rate as each other, regardless of their mass, but they take longer about it than they would here on earth. As you might imagine, this is because the Moon is less massive than Earth. Each planet and moon has a different gravitational strength, dependent on its mass (the same goes for stars and galaxies).
This short video shows astronauts jumping on the Moon.
Newton and motion in general
If you’ve managed to stay with the argument thus far, you’ve already done a lot to fight against intuition – or “common sense”. It was revolutionary of Galileo to refute the thousand-year-old authority of Aristotle, who considered that heavier objects fall faster. By his experiment with a ramp, he had shown that gravity not only pulls things downwards, but it makes them go faster and faster as it does so. He had clarified the link between the force on something and the motion it causes. He had demonstrated a relationship, in the particular case of gravitational force, that Newton was able later (in 1687) to generalise and ultimately enshrine as a “law” of motion. It’s not just in the case of gravity that things go faster and faster: it’s a general principle. Forces cause acceleration.
Again, this seems utterly counter-intuitive at first sight. Surely if something is not accelerating, like a car at a steady 30 mph, you still need a force to keep it going steadily. That’s why you need an engine and have to pay for petrol to run it. Force is obviously involved, yet there’s no acceleration, the speed is a steady 30 mph.
What Newton’s law helps us do is untangle the various factors involved.
Of course, the engine provides a force to drive a vehicle forward; but equally, other forces – friction in the moving parts, wind and air resistance – hold it back (figure 8). When the forces are exactly balanced, there is no net force, and the vehicle moves without getting faster or slower. That’s what cruising at a steady speed is.
Figure 8 Horizontal forces on a moving vehicle
In a similar way, a parachutist may start their descent with a rapid acceleration, but once the air resistance has built up to the point where it exactly balances the weight pulling down, then the descent continues at steady “terminal” speed (in one sense of the word!).
So, Newton’s great law of motion states that an object will accelerate in proportion to the net force acting on it. An important corollary is that, if there is no net force on an object, it will not accelerate. But crucially, this doesn’t mean it must be stationary. It just means it moves with a constant steady speed, in a straight line. Unimpeded things just keep going. To speed it up or change direction a force is required.
A vivid demonstration of this law in action is available to us today, thanks to rocket technology. Two Voyager spacecraft were launched in 1977 to explore the planets. Initially, they burned fuel to develop the force needed to overcome the Earth’s gravitational pull and accelerate away. Later with the rocket fuel finished, they headed towards Jupiter and Saturn where the gravitational pull of the planets helped propel the Voyagers into interstellar space, beyond the residual pull of the Sun (see Figure 9).
Figure 9 trajectories of the two Voyager spacecraft
Today, long after they were expected to have finished their useful lives, they continue to travel beyond any significant gravitational force. They are freewheeling their way through endless space and will continue to do so forever. Their speeds are 38,000 and 34,000 mph relative to the Sun and are 14 and 12 billion miles away from their launch sites. Without power to drive them, they will continue to travel through interstellar space at a steady speed in a straight line, demonstrating for eternity that force is not required to maintain steady motion, only to accelerate or change direction.
Force and motion
It’s intuitive to associate force with speed. We think of a fast car as exerting more force than a less powerful one (which, of course it does, to overcome resistances), so we associate force with speed. On the other hand, you can be sitting in a train travelling at 80 mph or walking along the aisle of a plane at 400 mph with no sense of a force propelling you (as distinct from the train or plane). Your sitting and walking take place quite normally, as they would at home. No extra forces are pushing on your arms and legs as you hurtle along at speed.
When a plane takes off, however, or the brakes of a train are suddenly applied or a car lurches to the right or left, you do feel something. You’ve speeded up, slowed down or changed direction. These are all examples of acceleration – i.e. a change in speed or direction. What Newton’s law encapsulated, building on Galileo’s experiment, is that forces cause acceleration. It goes further, stating that the degree of acceleration is proportional to the force. Double the thrust from an engine means double the acceleration.
Acceleration is something that we do actually feel, whereas speed, we don’t. Fighter pilots and returning astronauts experience large “g” forces – the sensation of extreme acceleration on the body – as they go through dramatic changes of direction or speed. The rest of us have similar, but more moderate, experiences in our more mundane modes of transport. Sitting in a train seat at steady speed you move around and drink your coffee in much the same way as at home. Pulling away from a station, however, you feel yourself sinking back into your seat; or shuddering to a halt at a red signal, you might lurch forward. In an extreme case, you may even feel a bit queasy in the stomach. The vehicle you are in is accelerating (or decelerating) and your body has to follow suit. You may have to hold tight to keep your body aligned with the changing speed of the train. The looser parts inside your body get pulled around in much the same way.
So, unbalanced forces cause acceleration; balanced forces don’t. A car cruising along a motorway at 70mph is a case in point. The driver adjusts the accelerator till the engine is exerting just enough force to balance the air resistance, friction of the tyres on the road and friction in all the moving parts. It cruises at a steady speed. You may even get a momentary illusion when another faster vehicle overtakes slowly. If you ignore the surroundings rushing past, and focus on the passing vehicle it may appear to be moving gracefully ahead while you are at rest. Indeed, if all the surroundings were blanked out and the vehicle didn’t jolt, you’d have no way of telling whether you were moving or at rest. All your actions in a steadily moving frame of reference are the same as in a stationary one. This thought inspired Einstein to realise there is no such thing as being “at rest”. Everything in the universe is moving. All motion is relative: hence the name of his famous theory.
Once again, simply being curious and noticing what is around us can put us in touch with profound scientific ideas. From Galileo to my sister, thinking about what happens when things fall turns out to be a good starting point for grasping the subject of dynamics. This story has shown how carefully constructed experiments can help us see beyond where our instincts might at first lead us. That’s what learning is all about: breaking into new understanding. Hard to do at first, but ultimately liberating.
© Andrew Morris