Newton’s Second Law

Too Fast to Count

Let something go and it falls about 4.9 m in the first second, about 19.6 m by the end of the second. A clock that beats once a second cannot keep up.

Atwood gives an example in his book of 1784: a lead ball let go from the inner dome of St Paul’s in London fell about 83 m in 4.5 seconds. Free fall should have taken it about 99 m in that time; the air held back the rest.

His answer was to slow the fall: two equal boxes on the ends of a silk line over a very light wheel, with a little extra weight on one side. The machine here follows the figure in his book. A steel ball let go on the same beat as box A reaches the paper in about 0.6 s; by then A has moved less than 3 cm.

Free fall, first second: about 4.9 m · this machine: about 7.7 cm · the steel ball is the page’s own

The Extra 7.8 Grams

Each box with the weights inside it is 202 g, and the two balance. A flat brass bar of 7.8 g, the rider, lies on box A. Its weight is the only one not cancelled, and it pulls the whole set into motion.

It has more to move than itself. Both boxes have to go with it, and the wheel and the four friction wheels under its axle have to turn faster and faster. Atwood measured the wheels as worth 85.5 g at the wheel’s rim. In all about 498 g: 64 times the rider.

So the acceleration is a sixty-⁠fourth of free fall’s, a = g/64, about 0.153 m/s². A covers 7.7 cm in the first second, 30.6 cm in two, 69 cm in three: the distance grows as the square of the time, slowly enough to mark at each beat of the clock.

Rider 7.8 g · whole set about 498 g · a = g/64 ≈ 0.153 m/s² · red marks: A at each beat

When the Rider Comes Off

A ring is clamped to the scale, wider inside than the box and shorter than the rider. Box A passes through it; the rider is caught and stays behind.

From that moment the two sides weigh the same again, and nothing is left unbalanced. A neither stops nor speeds up: it goes on down at the speed it had. With no net force a thing stays at rest or keeps moving in a straight line at a steady speed: that is Newton’s first law, and Atwood wrote as much in his margin.

It also makes the speed measurable. In Atwood’s set-⁠up the ring sits at 30.6 cm; A reaches it on the second beat, moving at about 30.6 cm a second. The stage sits at 91.9 cm, and A lands on it exactly two seconds later, on the fourth beat.

Two seconds speeding up · two seconds at a steady speed · ring at 30.6 cm, stage at 91.9 cm

Double Force, Double Mass

Put on two riders, then three, and take half a rider’s weight, then a whole one, out of each box: the mass moved stays at 64 parts. The pull becomes two and three times as large, and so does the speed after one second: 15.3, 30.6 and 46.0 cm a second.

The other way round, the same force on twice the mass gives half the acceleration. Newton’s second law says: the acceleration is in proportion to the net force, in inverse proportion to the mass, and in the direction of the force, a = F/m. It is usually written F = m a.

Newton, writing the law down in 1687, said that the change of motion is in proportion to the force applied and happens along the line the force acts in. Atwood said that before him no book described experiments that put the laws of motion to a proper test; his machine measured force, mass, distance, time and speed one by one.

One, two, three riders · the same 64 parts of mass · after one second 15.3, 30.6, 46.0 cm a second

How Hard the Line Pulls

While the boxes hang still, the line pulls on each with exactly its weight. Once they move, it does not.

A is speeding up downward, so the upward pull on it is less than its weight: A with the rider is 210 g and weighs 2.06 N, and the line pulls it with about 2.03 N. B is speeding up upward, so the line pulls it with more than its weight: B weighs 1.98 N, and the line pulls it with about 2.01 N.

Taking the line’s pull for one side’s weight is the commonest mistake with problems like this. With a light wheel and no friction the pull is T = 2 m₁ m₂ g/(m₁ + m₂), always between the two weights. The wheel here has mass, so the two sides differ by a little more; that little makes the wheel turn faster and faster.

A: weighs 2.06 N, pulled with 2.03 N · B: weighs 1.98 N, pulled with 2.01 N (computed here)

Mass Is Not Weight

The wheels add mass here but no pull. Their weight rests on the axle and drags neither side down, yet turning them faster takes force. Atwood counted them as 85.5 g at the wheel’s rim, moved along with the boxes and the rider.

Mass says how hard it is to change a thing’s motion, in kilograms; weight is the Earth’s pull on it, a force in newtons: w = m g. On the Moon the masses would stay the same and the weights drop to about a sixth, and this machine would run with about a sixth of its acceleration.

Because weight is in proportion to mass, without air light and heavy things fall equally fast: twice the mass is pulled twice as hard and has twice as much to move, so the acceleration is the same. In 1971 the Apollo 15 astronaut David Scott dropped a hammer and a feather together on the Moon, and they landed together.

Mass moved: boxes 404 g + rider 7.8 g + wheels 85.5 g · the pull: the rider’s weight, about 0.076 N

Pulling Away

The same law runs a car. A Toyota Yaris hybrid (the less powerful version) has a kerb weight of 1,⁠145 to 1,⁠190 kg by the maker’s figures and goes from rest to 100 km/h in 9.7 seconds. That is an average acceleration of about 2.86 m/s², about 0.29 g, and an average net force of about 3,⁠300 to 3,⁠400 N.

Fill it with people and luggage up to the maker’s gross weight of 1,⁠615 kg, and the same net force needs 13.2 to 13.7 seconds. A loaded truck starts slowly, and needs longer to stop, for the same reason.

That is the net force: the wheels’ push forward less the drag of the air and the road. So the wheels push harder still. At a steady speed the two cancel and the net force is zero: keeping a speed takes no net force; changing it does.

0⁠–⁠100 km/h in 9.7 s · average 2.86 m/s² · the same net force, fully loaded: 13.2⁠–⁠13.7 s (computed here)

Stopping

Stopping is a change of motion too, and it also takes a net force: the quicker the stop, the larger the force. The US safety standard FMVSS 208 crashes cars into a rigid wall at up to 56 km/h with a belted dummy inside. The dummy’s chest acceleration may not go over 60 g, except for moments adding up to no more than 3 milliseconds.

The dummy is about 78 kg. Stopping it from 56 km/h within 30 cm takes an average of about 41 g, a force of about 31 kN; stretch the stop to 60 cm and it falls to about 21 g and 16 kN; to 1 m, about 12 g and 9 kN. a = v²/(2 d): twice the distance, half the force.

The front of the car crumpling, the belt stretching and the airbag filling all lengthen the stop, and the belt, the airbag and the seat share the force. Under the general rule of the US standard FMVSS 209, at a pull of 11,⁠120 N the webbing of a lap belt may stretch by at most 30 % and of a shoulder belt by at most 40 %.

Stopping from 56 km/h: 30 cm about 41 g · 60 cm about 21 g · 1 m about 12 g (computed here for these distances, not measured)

The Machine of 1784

George Atwood had been a fellow of Trinity College and gave lectures on experimental philosophy in the University of Cambridge; the machine was made for them. In 1784 he described it, with a series of experiments, in a book on the straight-⁠line motion and rotation of bodies. By his account it was made by the London instrument makers L. Martin and G. Adams.

The wheel’s axle does not turn in a fixed hole: it rests on the rims of four friction wheels and rolls instead of rubbing. Atwood tested it. With the two sides balanced, 2 grains, about 0.13 g, set the whole set moving: friction was less than a sixtieth of the rider.

He started each run with a flat rod held under the box, pulled away on a beat of the seconds pendulum. Before the book came out, Poli in Naples had ordered one from London; its maker, Ramsden, added a trigger that let the pendulum and the box go together. Machines of the 19th century all had such a trigger, and most had a fixed set of wheels or a simple pulley in place of the four friction wheels.

Friction under 2 grains (about 0.13 g) · seconds pendulum about 0.994 m, one second a swing

Taking the Machine Apart

Nine parts: the wheel, the friction wheels, the table and standards, the boxes and line, the ring, the stage, the scale, the seconds clock, and the column with its feet.

Point at one and it lifts out with its name; leave the pointer still and all nine are named.

Wheel · friction wheels · table and standards · boxes and line · ring · stage · scale · seconds clock · column and feet