Newton's Laws of Motion

A football left on the grass stays put. Kick it on ice and it keeps gliding. Newton’s first law turns that everyday observation into a precise statement, and in doing so tells us which observers we can trust.

Why the laws are subtle#

Newton’s laws connect the forces on a body with the way it moves. To state them perfectly we would first need exact definitions of mass, force and acceleration, yet those definitions themselves lean on the laws. So the laws are partly definitions. Rather than untangle that history, we take a practical route: assume we know how to assign a mass to a body, a size and direction to a force, and how to measure acceleration in a chosen frame (as in the previous chapters), and then learn how to use the laws.

The first law#

The “if and only if” works both ways. Know that the forces cancel, and you know the particle is not accelerating. Know that it is not accelerating, and you know the forces cancel.

Which observer is right?#

Rest and acceleration only mean something relative to a frame, and different frames measure different accelerations. Can the first law hold in all of them? Try it.

Suppose the lift’s cable snaps. The cabin and everything in it fall with acceleration 9.8 m/s². Two careful observers apply the first law to the hanging lamp:

• Inside the lift, the lamp is at rest. Conclusion: the forces on it cancel, so $T = W$.
• On the ground, the lamp accelerates at 9.8 m/s². Conclusion: the forces do not cancel, so $T \ne W$.

Both cannot be true. One of the two frames is a “bad” frame in which the first law simply fails. (Here the cord goes slack, so $T = 0$ and the ground observer is right.)

Read this way, the first law becomes a definition of an inertial frame. Why call it a law, then? Because nature provides one: for everyday purposes the Earth is an inertial frame. A book resting on your desk has forces that (to excellent accuracy) cancel. A very precise measurement would reveal a tiny imbalance, since the Earth spins and orbits, but for routine problems the Earth frame works. Newton noticed exactly this and stated it as his first law. Unless we say otherwise, we always work in an inertial frame.

Other inertial frames#

Let S be inertial and let S′ move relative to S with constant velocity. From the previous chapter, accelerations measured in the two frames are related by

and $\vec a_{S',S} = 0$ because S′ moves uniformly. So $\vec a_{P,S'} = \vec a_{P,S}$: both frames measure the same acceleration, and since the first law holds in S, it holds in S′ as well.

The first law says what happens when the forces cancel. The second says exactly what happens when they don’t.

The second law#

Measured from an inertial frame, the acceleration of a particle equals the vector sum of all the forces on it divided by its mass:

Here $\vec F$ is the net force (the resultant of every force acting on the particle) and the acceleration points the same way as that net force. The first law is the special case $\vec F = 0$. This is a law of nature: once inertial frames are defined, it tells us how any force changes any motion.

A vector equation#

$\vec F = m\vec a$ holds component by component, which is how we use it in practice:

This also gives the newton its precise meaning: 1 N is the net force that gives a 1 kg mass an acceleration of 1 m/s², so 1 N = 1 kg·m/s².

Most mistakes in force problems happen before any algebra: a force left out, a force counted twice, a force that acts on the wrong body. A short recipe prevents all three.

Newton’s laws are about a particle. Real objects are huge collections of particles, but the laws still work for an extended body as long as every part of it has the same acceleration. With that in mind, here is a four-step algorithm for turning any situation into equations.

Step 1: decide the system#

Choose what you will apply the laws to: a block, two blocks stacked together, two blocks joined by a taut string, a piece of rope. The only rule is that every part of the system must share the same acceleration.

For example, if block B rides on block A without slipping, and A is tied by a taut string to block C, then A, B and C all move together: (A), (B), (A + B), (A + B + C) are all fine systems. But a block hanging vertically and a block sliding horizontally, though joined by a string over a pulley, move the same distance in different directions, so their accelerations differ and they cannot be lumped together (at least not yet).

Step 2: list the forces on the system#

Now, and only now, list the forces acting on the system by bodies outside it. Forces the system exerts on other things do not belong in the list. Asking “which forces act here?” before choosing the system has no answer, as the widget shows.

Step 3: draw the free-body diagram#

Represent the system by a single point in a fresh diagram and draw every listed force as an arrow starting from that point, with its direction and (if known) its size. This picture is the free-body diagram. Most problems involve forces in one plane, which you draw in the plane of the page.

Step 4: choose axes and write the equations#

Pick the x-axis along the direction in which the system accelerates (or is likely to), and the y-axis perpendicular to it. Then resolve every force and write

If the system is in equilibrium, any perpendicular pair of axes will do. If you need more equations, return to step 1 with a different system. Accelerations of connected bodies are often related by kinematics (a taut string gives equal magnitudes), so use the same symbol for them. Then solve.

An equilibrium example

You push on a wall and the wall pushes back just as hard. Yet the wall doesn’t move you backwards across the room. The third law is simple to state and surprisingly easy to misuse.

The third law, again#

If body A exerts a force on body B, then B exerts a force of equal size and opposite direction on A. The crucial detail is in the subscripts: the two forces act on different bodies. So when you apply the first or second law to a chosen system (step 2 of the recipe), at most one member of any third-law pair can appear in your list.

A table pushes a book up with a normal force $N$; the book pushes the table down with $N$. The first belongs in the book’s equation, the second in the table’s. The Earth pulls a falling apple down with $mg$, and the apple pulls the Earth up with $mg$. These don’t cancel: the first accelerates the apple, the second accelerates the Earth (by an immeasurably small amount, because the Earth’s mass is so huge).

What tension really means#

Imagine a block hanging from the ceiling by a string, and slice the string in your mind at some point A. The part below A and the part above A are in contact there and pull on each other with equal and opposite forces (third law again).

When a bus brakes, you lurch forward as if shoved, though nothing touched you. Pseudo forces let us describe that feeling honestly and still use F = ma from inside the bus.

Newton’s law from an accelerating frame#

Let frame S′ accelerate with constant $\vec a_0$ relative to an inertial frame S, without rotating. A particle P of mass $m$ has acceleration $\vec a$ measured in S′. From the chapter on changing frames,

S is inertial, so $m\vec a_{P,S}$ equals the net real force $\vec F$. Therefore

That is not Newton’s second law: an extra term $-m\vec a_0$ has appeared. Since we are so used to writing “net force = mass × acceleration,” we keep that habit by agreeing to treat $-m\vec a_0$ as if it were one more force on the particle.

Working from an inertial frame, $\vec a_0 = 0$ and no pseudo force is needed. Both approaches give the same physics; choose whichever makes the problem easier.

Feeling heavy, feeling light#

A bathroom scale measures the normal force it exerts on you, not your mass. In a lift accelerating upward at $a$, Newton’s second law from the ground frame gives $N - mg = ma$, so $N = m(g + a)$: you feel heavier. From inside the lift you would say the pseudo force $ma$ pushes you down and adds to your weight. Same answer.

A horse pulls a cart forward. By the third law the cart pulls the horse back with an equal force. So the forces cancel and nothing should move. Something is wrong with that argument. What?

The paradox#

Call the horse’s pull on the cart $F_1$ (forward) and the cart’s pull on the horse $F_2$ (backward). The third law guarantees $F_1 = F_2$. Their sum is zero. Why does the cart accelerate forward at all?

The flaw is that we added forces acting on two different bodies. The recipe says: choose a system first, then list the forces on it, all of them and only them.

Take the cart as the system#

On the cart act the horse’s forward pull $F_1$ and a backward force $f'$ from the road (friction and rolling resistance at the wheels). $F_2$ acts on the horse, so it has no business here. The cart accelerates forward if $F_1 > f'$, with $a = (F_1 - f')/M_c$.

Take the horse as the system#

On the horse act the cart’s backward pull $F_2$ and, crucially, the force of the road on its hooves. The horse pushes back on the road; the road pushes forward on the horse with a force $f$ (static friction). The horse accelerates forward if $f > F_2$.

The forces $f$ and $f'$ adjust themselves so that the horse and cart end up with the same acceleration and move together, as they must while the harness stays taut.

Inertia#

In an inertial frame a body accelerates if and only if a net force acts on it. Loosely, a body does not change its state of rest or of uniform straight-line motion unless something forces it to. We call this reluctance to change velocity inertia.