Exercises — Newton's first law — inertia, operational definition of force

WHAT the law says: velocity constant $⟺$ net force zero. The puck moves in a straight line at fixed speed, so its velocity is not changing. WHY: "No one touches it" and "frictionless" together mean nothing is left over to change the motion. Therefore ${\vec{F}}_{\text{net}}=0$, and the speed stays $4\text{ m/s}$. Ten seconds later it is still $4\text{ m/s}$.

Apply the test: is the velocity vector constant?

• (a) speeding up → speed changes → velocity changes → force acts.
• (b) at rest → velocity constant (it's zero, which is still constant) → ${\vec{F}}_{\text{net}}=0$. ✓
• (c) constant speed but circular → direction changes → velocity changes → force acts (this is the classic trap; see below).
• (d) straight line, steady speed → velocity constant → ${\vec{F}}_{\text{net}}=0$. ✓ Answer: (b) and (d).

WHAT: the block is at rest, so $\vec{v}$ is constant, so by the first law ${\vec{F}}_{\text{net}}=0$. WHY vertical balance: the only two vertical forces are gravity ($19.6\text{ N}$ down) and the normal push $N$ (up). For their sum to be zero: $N-19.6=0\Rightarrow N=19.6\text{ N (up)}.$ The first law didn't tell us $N$ directly — it told us the forces must cancel, and from that we solved for $N$.

Constant velocity $⟹{\vec{F}}_{\text{net}}=0⟹$ net force $=0$. Horizontally the only forces are thrust (forward) and drag (backward). They must cancel: $\text{drag}=\text{thrust}=40,000\text{ N}.$ Many forces act, yet the net is zero — exactly the message of example 3 in the parent note.

WHAT force acts: while the cloth slides, kinetic friction drags the plate sideways with force $f=\mu mg=0.2\times 0.5\times 9.8=0.98\text{ N}.$ WHY only briefly: this force acts only during the tiny $\Delta t=0.05\text{ s}$ the cloth is beneath. The sideways acceleration is $a=f/m=\mu g=0.2\times 9.8=1.96{\text{ m/s}}^2$. Speed gained: $v=a\Delta t=1.96\times 0.05=0.098\text{ m/s}$. So the plate creeps at about $0.098\text{ m/s}\approx 10\text{ cm/s}$ — barely moving. Its inertia resisted the change because the force had almost no time to act. It practically stays put. See Friction.

WHAT: no horizontal force pushes the torso forward. By the first law, the torso keeps moving at its original $10\text{ m/s}$. WHY the lurch: the bus (and the passenger's feet, gripped to the floor) slow down at $5{\text{ m/s}}^2$, but the torso, with no forward-changing force, continues. Relative to the decelerating bus, the torso moves forward — the lurch. Forward force on the torso: none. The forward motion is inertia, not a force. (In the bus's own accelerating frame you'd invent a forward pseudo-force — a signal the bus is a non-inertial frame.)

(a) Speed is constant but the car turns, so its direction changes → velocity changes → ${\vec{F}}_{\text{net}}\ne 0$. (b) A body on a circle of radius $r$ at speed $v$ needs a net inward (centripetal) force of magnitude $F=\frac{m{v}^2}{r}=\frac{1200\times 15^2}{50}=\frac{1200\times 225}{50}=5400\text{ N},$ pointing toward the centre of the circle. (c) Static Friction between tyres and road provides it. Without friction (icy track) the car would obey the first law and shoot off in a straight line (tangent), not curve. Look at the figure: the velocity arrow is tangent (green), the net force arrow points inward (red).

(a) The two forces point in different directions and don't cancel, so ${\vec{F}}_{\text{net}}\ne 0$ → not equilibrium. (b) The forces are perpendicular (east ⟂ north), so use the right-triangle rule (Pythagoras) to add them: $|{\vec{F}}_{\text{net}}|=\sqrt{6^2+8^2}=\sqrt{36+64}=\sqrt{100}=10\text{ N}.$ (c) No — a nonzero net force means velocity changes, so the speed is not constant. The first law only says "constant velocity" when the net is exactly zero, which it isn't here.

(a) Ground frame — the honest view. The ball must accelerate forward at $2{\text{ m/s}}^2$ along with the train. The only horizontal thing that can push it forward is the horizontal part of the string tension. So the string tilts back so its tension has a forward component. Balancing gives, from the horizontal and vertical force equations, $\tan \theta =\frac{a}{g}=\frac{2}{9.8}=0.2041,$ where $\theta$ is the angle from vertical. So $\theta =\arctan (0.2041)\approx 11.5^{\circ }.$ Here $\arctan$ answers "which angle has this tangent?" — we know the ratio (opposite $a$ over adjacent $g$ on the little force triangle) and want the angle back. See the figure: the tilt (orange), the forward tension component (blue), gravity (red). (b) Train frame. The train is non-inertial (it accelerates). There the ball hangs still at an angle, so the passenger invents a backward pseudo-force $ma=0.5\times 2=1\text{ N}$ to explain the tilt. That force is not real — it is the price of using a dishonest (accelerating) frame. The first law is the litmus test that exposes it.

(a) No net force (nothing to push or pull it), so by the first law velocity is constant → acceleration $=0$. (b) Zero. This is the Galileo/Newton punchline: in the absence of friction and drag, no force is needed to maintain motion. Aristotle would have burned fuel forever; Newton burns none. (c) The pen was moving at the ship's velocity. With no force on it, it keeps that same velocity — so relative to the cabin it floats motionless (or drifts in a straight line at constant speed if given a nudge). This is inertia made visible.

(a) East and west cancel ($10-10=0$). Left over: $10\text{ N}$ north. Net force $=10\text{ N}$ north, not zero. (b) Yes — the first law never says "force zero means motionless." A moving body with nonzero net force simply has changing velocity. Motion and force are independent questions. (c) The object keeps its northwest velocity and now accelerates northward at $a=F/m=10/2=5{\text{ m/s}}^2$, curving its path. The first law's equilibrium is broken, so we hand off to Newton's second law — F=ma to find exactly how the velocity evolves.

The single question behind every problem ::: Is the velocity vector (speed and direction) constant? If yes, ${\vec{F}}_{\text{net}}=0$; if no, a net force acts. Why did the plane and the resting block both have ${\vec{F}}_{\text{net}}=0$? ::: Their velocities were constant; the many individual forces cancelled to zero net. Why did the orbiting satellite and circling car not? ::: Their directions changed, so velocity changed, demanding a net centripetal force. What always signals a non-inertial frame? ::: A free body appears to accelerate with no real force, forcing you to invent a pseudo-force of size $ma$.