Newton's first law

The default state of an object is constant velocity. Same speed, same direction. Standing still is just the special case where the velocity happens to be zero.

Read it carefully. The law does not say "no force, no motion." It says no net force, no change in motion. An object that's already moving keeps moving, with the velocity it already has, until something forces it to do otherwise.

This contradicts everyday intuition. In daily life, things slow down when you stop pushing them. Push a book across a table and let go: it stops. Coast a bike and stop pedaling: it slows. The pattern looks like motion requires a force. It doesn't. What's actually happening is that friction, drag, and rolling resistance are quietly applying forces in the opposite direction, sapping speed. Take those forces away, and the book and the bike just keep going.

The first law also kills a related habit: thinking that "at rest" and "moving steadily" are different physical situations. They aren't. They are both equilibrium. Both have zero net force. The difference is just what value the constant velocity happens to take.

"Net force is zero" is such a useful condition that it gets a name: translational equilibrium. Everything in this lesson is one of two activities. Either you're identifying that an object is in equilibrium and concluding its velocity is constant, or you're using the equilibrium condition to solve for an unknown force.

Stated as an equation, the condition is just the vector sum of all forces on the system equals zero:

That single equation is shorthand for two independent statements in two dimensions, three in three dimensions. The vector sum is zero only when every component is zero. So in the plane,

Both of these have to hold at once. Neither one alone is equilibrium.

Two scenarios that both qualify as equilibrium, and that students sometimes treat as different physical cases:

Real problems live in two dimensions, sometimes three. The vector sum has to be zero across all of them at once. Equivalently, the forces have to balance separately along every axis.

This means forces can be balanced in one direction and unbalanced in another. When that happens, velocity changes only in the unbalanced direction. The balanced direction sees no acceleration at all.

Consider a crate that is being pushed forward and slightly downward at an angle, sliding east at constant velocity along a rough floor. The crate is in equilibrium horizontally and vertically, and you can see why if you draw the four forces and resolve them.

The push has both a horizontal component (forward) and a vertical component (downward). Resolve everything onto $x$ and $y$, write equilibrium twice, and you get two clean equations:

The horizontal balance fixes the friction force at exactly $F\cos\theta$. The vertical balance fixes the normal force at $mg + F\sin\theta$, which is bigger than $mg$ alone, since the push is helping press the crate into the floor. Either equation alone would not be enough. You need both, because equilibrium means balance in every direction.

Newton's first law is true in some places and false in others. The "places" here are reference frames: the vantage points from which you measure motion. A frame in which the first law works is called inertial. A frame in which it fails is non-inertial.

Picture a tennis ball sitting on a smooth tray inside a car that is rolling east at a steady 30 m/s. The ball doesn't roll. Two observers watch it.

The driver sits inside the car. From the driver's frame, the ball is at rest. The driver feels no forces unbalancing the ball, agrees that the net force on it is zero, and the ball's velocity (which is zero in this frame) stays zero. Newton's first law works in the driver's frame.

A bystander stands on the sidewalk. From the bystander's frame, the ball is moving east at 30 m/s, the same as the car. The bystander also sees no horizontal forces on the ball, agrees that the net force is zero, and the ball's velocity (which is 30 m/s east in this frame) stays 30 m/s east. Newton's first law works in the bystander's frame too.

Both frames are inertial. They disagree about the ball's velocity, but they agree on the physics: zero net force, constant velocity. That's what "inertial" means. The actual numerical value of velocity depends on the frame; the law itself does not.

Now the driver hits the brakes. The car decelerates. From the bystander's frame, nothing changes about the ball: still no horizontal force on it, still moving at 30 m/s east. But the car is now slowing beneath it. The ball appears to slide forward across the tray, eventually flying off, just because the car is leaving the ball behind.

From the driver's frame, the same event looks completely different. The ball was sitting still, and then suddenly leapt forward across the tray with no one touching it. There is no force the driver can name that pushed it forward. Newton's first law has failed inside the car. The driver's frame, while braking, is non-inertial.

A sign of mass $m$ hangs from two strings. Each string makes the same angle $\theta$ with the vertical. The sign is at rest. Find the tension in each string in terms of $m$, $g$, and $\theta$.

Two quick checks before moving on. When $\theta = 0$ the strings hang straight down, $\cos\theta = 1$, and each string carries $T = mg/2$. That's right: vertical strings split the weight evenly.

As $\theta$ approaches $90^\circ$ the strings approach horizontal, $\cos\theta$ approaches zero, and $T$ blows up. That's also right: a perfectly horizontal string has no vertical component to support the weight, so the tension would have to be infinite to compensate. In practice strings always sag, which is the universe's way of refusing to be horizontal.

The hardest part of the first law is trusting it when your intuition says otherwise. Recognize them in your own thinking and you'll avoid most of the test traps that exist on this topic.

Ten scenarios. Each gives you a situation; you pick the right physics call from a list of chips. Wrong picks come with explanations of why they're tempting, so a near miss is almost as useful as a hit.