A porter at a railway station holds a loaded handcart on the sloping ramp to the platform. If his rope becomes slack, the cart creeps down the ramp. If he pulls very hard, the cart starts up the ramp. For every pull in between, the cart does not move at all. The physics does not give one answer here; it gives a range, and a range is an inequality.
Words that are inequalities
Physics states many of its rules as limits. Static friction is at most \(\mu_s N\). The tension in a string is at least zero, because a string cannot push. A body stays on a curved track only if the track still presses on it. Each phrase becomes a sign: "at most" is \(\le\), "at least" is \(\ge\), and "more than" is \(>\). An equation gives one value; an inequality gives a whole range of allowed values.
The rules, and the one that reverses
An inequality is solved like an equation, with one difference. Adding or subtracting the same number on both sides keeps the inequality. Multiplying or dividing both sides by the same positive number keeps it too. Multiplying or dividing by a negative number reverses it: \(<\) becomes \(>\). Fig 5.1 shows why.
Treat an inequality like an equation, doing the same operation to both sides. Adding, subtracting, and multiplying or dividing by a positive quantity keep its direction. Multiplying or dividing by a negative quantity reverses it. Never multiply by a quantity whose sign you do not know.
\[ a < b \;\Rightarrow\; b - a > 0 \]
Why. subtract \(a\) from both sides. \(b - a\) is a positive number.
\[ k(b - a) < 0 \]
Why. a negative number times a positive number is negative.
\[ kb - ka < 0 \;\Rightarrow\; ka > kb \]
Read it. expand, then add \(ka\) to both sides. The order of \(ka\) and \(kb\) is the reverse of that of \(a\) and \(b\).
Solve \(3 - 2x \ge 7\).
Forces that cannot change sign
Many conditions in physics come from forces that can act in one direction only. A string can pull but not push, so \(T \ge 0\). A floor or a track can push but not pull, so \(N \ge 0\). Static friction adjusts itself, but only up to a limit, so \(f_s \le \mu_s N\). Write the force from the equations of motion, apply its condition, and solve the inequality.
At the top of a vertical circle, the tension in a string is \(T = mv^2/r - mg\), from the chapter Motion in a Vertical Circle. The condition \(T \ge 0\) becomes \(mv^2/r - mg \ge 0\). Add \(mg\) to both sides: \(mv^2/r \ge mg\). Multiply by \(r/m\), a positive number, so the sign stays: \(v^2 \ge gr\). Both sides are positive, so taking square roots keeps the sign too: \(v \ge \sqrt{gr}\). For a bucket of water swung on an arm of \(0.9\,\text{m}\), the speed at the top must be at least \(\sqrt{9} = 3\,\text{m/s}\).
A sine is never more than one, and this too is a condition. Light going from glass into air obeys \(\sin r = \mu\sin i\). A refracted ray exists only if \(\mu\sin i \le 1\), that is, \(\sin i \le 1/\mu\). For larger angles there is no refracted ray, and the light is totally reflected. The limit \(\sin C = 1/\mu\) is the critical angle of the chapter on refraction.
A stone is whirled in a vertical circle on a string \(2.5\,\text{m}\) long. Take \(g = 10\,\text{m/s}^2\). What is the least speed, in m/s, it must have at the top?
A \(20\,\text{kg}\) box rests on a floor with \(\mu_s = 0.5\). Take \(g = 10\,\text{m/s}^2\). A shopkeeper pushes it horizontally with \(60\,\text{N}\), and it does not move. What is the friction on the box?
Light inside glass of refractive index \(1.5\) meets the surface with air. For which angle of incidence is there no refracted ray at all?
Two limits at once
The absolute value \(|x|\) is the size of \(x\) without its sign: \(|-3| = |3| = 3\). A condition on a size, \(|x| \le a\), says that \(x\) may be positive or negative but no larger than \(a\) either way. So \(|x| \le a\) means \(-a \le x \le a\). Shifted by \(c\), \(|x - c| \le a\) means \(c - a \le x \le c + a\): a band of width \(2a\) centred on \(c\).
Friction on a slope gives exactly this. For the porter's cart, the friction needed to hold it is \(f = mg\sin\theta - P\), up the ramp. Its size must not exceed the limit \(\mu_s mg\cos\theta\). So the cart holds only if \(|mg\sin\theta - P| \le \mu_s mg\cos\theta\). Move the slider in the figure below and watch the needed friction change sign in the middle of the band.
A loaded handcart of total mass \(50\,\text{kg}\) stands on a ramp at \(37°\). Treat it as a block with \(\mu_s = 0.5\) against the ramp. Take \(g = 10\,\text{m/s}^2\), \(\sin 37° = 0.6\), \(\cos 37° = 0.8\). Find the least and the greatest pull along the ramp that hold it.
Weight component down the ramp: \(500 \times 0.6 = 300\,\text{N}\). Limiting friction: \(0.5 \times 500 \times 0.8 = 200\,\text{N}\).
The cart holds only if \(|300 - P| \le 200\), that is \(-200 \le 300 - P \le 200\).
Subtract \(300\) throughout: \(-500 \le -P \le -100\). Multiply throughout by \(-1\), which reverses both signs: \(500 \ge P \ge 100\).
Any pull from \(100\,\text{N}\) to \(500\,\text{N}\) holds the cart. At \(100\,\text{N}\) it is about to slide down, and at \(500\,\text{N}\) about to slide up.
A \(10\,\text{kg}\) block rests on a \(37°\) incline with \(\mu_s = 0.25\). Take \(g = 10\,\text{m/s}^2\), \(\sin 37° = 0.6\), \(\cos 37° = 0.8\). What is the least pull along the slope, in newtons, that holds it?
Which values of \(x\) satisfy \(|x - 5| \le 2\)?
Two steps break inequalities. Multiplying by a quantity of unknown sign, such as \(\cos\theta\) past \(90°\) or \(m_1 - m_2\), may reverse the inequality without warning, so find the sign first. Squaring both sides is safe only when both are positive: \(-3 < 2\), but \(9 > 4\). Watch the ends of a band as well. At \(P = 100\,\text{N}\) the cart is on the point of sliding, which is the equality, not a third kind of motion.
Conditions run through the whole book. Friction gives the limiting friction, the band of pulls on an incline and blocks that move together or slip. It also decides when a ladder against a smooth wall is safe. Circular motion gives the least and greatest speed on a banked road, a car leaving a crest, the rotor and the well of death. The vertical circle gives the least speed at the top and the bottom and the three ranges of launch speed. Energy gives turning points and forbidden regions and the escape condition. Optics gives the critical angle and the grazing ray. Thresholds also run through modern physics: the photoelectric threshold, the energy to excite an atom and Bragg's law.
- Forgetting to reverse the sign after dividing by a negative number.Every multiplication or division by a negative number reverses \(<\) and \(>\). Check the answer with one test value.
- Taking the limit as the value.Static friction is at most \(\mu_s N\). It equals \(\mu_s N\) only on the point of sliding.
- Keeping one side of an absolute value.\(|x - c| \le a\) gives two limits, \(c - a \le x\) and \(x \le c + a\). Write both.