At a railway station a porter pushes a luggage trolley. When the trolley is empty, a light push sets it moving quickly. When it is loaded with suitcases, the same push hardly gets it going. Two porters pushing the loaded trolley together get it moving as briskly as one porter moved the empty one.
What a net force does
The first law says that a net force changes the velocity. It does not say by how much. The answer comes from experiment. Push a trolley on a smooth floor with a steady force and measure its acceleration. Then push the same trolley with twice the force. Then load the trolley so that its mass is doubled, and push again.
Compare the dots in Fig 3.1. In the first state the lower trolley covers twice the distance of the upper one in every interval. Its acceleration is twice as large, \(2\,\text{m/s}^2\) against \(1\,\text{m/s}^2\). Twice the force gives twice the acceleration. In the second state the lower trolley carries a load that doubles its mass to \(4\,\text{kg}\). Now twice the force on twice the mass gives the same acceleration as the upper trolley.
So the acceleration is proportional to the net force, \(a \propto F\), and inversely proportional to the mass, \(a \propto 1/m\). Put together, \(a \propto F/m\). The unit of force is chosen so that the constant of proportionality is exactly one. That choice gives the second law in its familiar form.
\[ F = ma \]
The newton
The newton is the net force that gives a body of mass \(1\,\text{kg}\) an acceleration of \(1\,\text{m/s}^2\). So \(1\,\text{N} = 1\,\text{kg·m/s}^2\). A newton is a small force. An apple of mass \(100\,\text{g}\) has a weight of about \(1\,\text{N}\). Lifting a one-litre bottle of water needs about \(10\,\text{N}\).
The acceleration of a body is directly proportional to the net force acting on it, is inversely proportional to its mass, and is in the direction of the net force.
Mass measures inertia
A loaded trolley is harder to speed up than an empty one. It is also harder to stop. The second law turns this into a number. The mass of a body measures its inertia: the larger the mass, the smaller the acceleration for a given net force. Mass is a scalar, measured in kilograms, and it is the same wherever the body is taken.
An auto-rickshaw with its passengers has a mass of \(600\,\text{kg}\). It starts from rest and reaches \(9\,\text{m/s}\) in \(6\,\text{s}\) on a straight level road. Find the net force on it, assuming the acceleration is uniform.
Straight road, starting from rest: \(u = 0\), \(v = 9\,\text{m/s}\), \(t = 6\,\text{s}\), \(m = 600\,\text{kg}\).
Kinematics gives \(a = (v - u)/t\). The second law gives \(F_{\text{net}} = ma\).
\(a = (9 - 0) / 6 = 1.5\,\text{m/s}^2\). \(F_{\text{net}} = 600 \times 1.5 = 900\,\text{N}\).
The net force is \(900\,\text{N}\), forward. The engine's push is larger than this, because friction and air resistance act backward; \(900\,\text{N}\) is what is left over.
A net force of \(12\,\text{N}\) acts on a trolley of mass \(4\,\text{kg}\). What is its acceleration in \(\text{m/s}^2\)?
The same net force acts on two bodies. Body B has half the mass of body A. How does the acceleration of B compare with that of A?
The second law is a vector law
Force and acceleration are both vectors. The second law says more than "the sizes are proportional". It says that the acceleration points in the same direction as the net force. Several forces may act on the body; the law uses their vector sum.
\[ \sum \vec F = m\,\vec a \]
The mass \(m\) is a positive scalar, so multiplying \(\vec a\) by \(m\) changes its length but not its direction. The acceleration is always along the net force. It need not be along the velocity. A ball thrown upward has an upward velocity and a downward acceleration, because the net force on it, its weight, points down.
One law per axis
A vector equation holds component by component. Choose two perpendicular axes, \(x\) and \(y\). The \(x\) components of the forces add up to \(m a_x\), and the \(y\) components add up to \(m a_y\). This is the same idea as the independence of perpendicular motions in the last unit. Each axis gets its own equation, and the two are solved together.
A puck of mass \(2\,\text{kg}\) rests on smooth ice. Two strings pull it horizontally, one with \(3\,\text{N}\) towards the east and one with \(4\,\text{N}\) towards the north. Find the size and direction of its acceleration.
Take \(x\) east and \(y\) north. Along \(x\): \(a_x = 3 / 2 = 1.5\,\text{m/s}^2\). Along \(y\): \(a_y = 4 / 2 = 2\,\text{m/s}^2\).
The size is \(a = \sqrt{1.5^2 + 2^2} = 2.5\,\text{m/s}^2\). The direction makes \(\tan\phi = 2 / 1.5 = 4/3\) with the east, so \(\phi = 53°\) north of east.
Check with the net force: \(\sqrt{3^2 + 4^2} = 5\,\text{N}\), and \(5 / 2 = 2.5\,\text{m/s}^2\), along the same direction.
A stone thrown upward is still rising. Air resistance is neglected. What is the direction of its acceleration?
Mass against weight
In daily speech we "weigh" a bag of sugar and give the answer in kilograms. Physics keeps the two words apart. The mass of the bag is the amount of inertia it has, in kilograms. Its weight is the force with which the Earth pulls it, in newtons. The second law connects the two. A falling body has only its weight acting, and it accelerates at \(g\). So \(W = mg\).
An astronaut of mass \(60\,\text{kg}\) weighs \(600\,\text{N}\) on the Earth. On the Moon her mass is still \(60\,\text{kg}\), and a push that gave her \(1\,\text{m/s}^2\) on the Earth gives her \(1\,\text{m/s}^2\) there too. But her weight on the Moon is only \(60 \times 1.6 = 96\,\text{N}\), so she can jump much higher.
A box of mass \(10\,\text{kg}\) is taken to the Moon, where \(g = 1.6\,\text{m/s}^2\). Which statement is correct?
Now the trolleys at the station. One porter's push on the loaded trolley gives a small acceleration, because the mass is large. Two porters give twice the net force to the same mass, and so twice the acceleration. If the empty trolley has half the mass of the loaded one, two porters on the loaded trolley match one porter on the empty trolley exactly, as Fig 3.1 showed.
Edmond Halley visited Newton in Cambridge in 1684 and asked him about the path of a planet. Newton had the answer but not the paper. Over the next three years he wrote the Philosophiae Naturalis Principia Mathematica, which Halley paid to publish in 1687. Near the start of it stand three short "axioms, or laws of motion". The second, in Newton's words, says that the change of motion is proportional to the motive force impressed.
- Using one force, such as the engine's push, in F = ma.The second law uses the net force, the vector sum of all the forces on the body.
- Assuming the acceleration is along the velocity.The acceleration is along the net force. A rising stone moves up and accelerates down.
- Giving weight in kilograms.Weight is a force, in newtons: \(W = mg\). Kilograms measure mass.