Measurements and Experimentation

Every measurement is a number together with a unit, and physics agrees on a single system so results can be compared anywhere. The SI system fixes seven fundamental quantities, each with its own unit that cannot be derived from the others: length (metre), mass (kilogram), time (second), electric current (ampere), temperature (kelvin), luminous intensity (candela) and amount of substance (mole). Everything else is a derived quantity. Speed is length divided by time, so its unit (metre per second) is derived; area, volume, force, density and pressure are all derived in the same way by combining the fundamental units.

An ordinary scale can be read to about a millimetre; a vernier calliper sharpens this to a tenth of a millimetre. It has a fixed main scale and a sliding vernier scale. The trick is that 10 vernier divisions are made exactly equal to 9 main-scale divisions, so one vernier division is 0.9 mm. The least count is the difference, 1 mm − 0.9 mm = 0.1 mm. To read it you note the main-scale mark just before the zero of the vernier, then find which vernier mark lines up exactly with a main-scale mark; that number times the least count is added on.

For still smaller lengths, the diameter of a wire or the thickness of paper, we use a screw gauge. Turning the screw moves it forward along a fine thread. The pitch is the distance the screw advances in one full turn (usually 1 mm). The circular (head) scale is divided into 100 parts, so the least count is the pitch divided by the number of divisions, 1 mm ÷ 100 = 0.01 mm. A ratchet at the end clicks when the object is held just firmly enough, preventing over-tightening. The reading is the main-scale reading plus the circular-scale division times the least count.

Before trusting any instrument you check it for a zero error: when the jaws are closed (or the screw fully in), does the scale read exactly zero? If the zero of the vernier or circular scale lies ahead of the true zero, the error is positive and you subtract it from every reading. If it lies behind, the error is negative and you add its size. The rule is simple: correct reading = observed reading − zero error, always taking the error with its proper sign. Ignoring zero error is the most common reason a careful measurement still comes out wrong.

A simple pendulum is a small heavy bob hung by a light, inextensible thread. Pulled aside a little and released, it swings to and fro with a steady rhythm. The time for one complete swing is the time period T. Remarkably, T does not depend on the mass of the bob or (for small swings) on how far it is pulled aside; it depends only on the length L of the pendulum and on gravity g, through T = 2 × pi × square root of (L / g). Because the period is so reliable, pendulums were used in clocks for centuries.

The pendulum gives a neat way to measure g, the acceleration due to gravity. You measure the length L (thread length plus the bob's radius) and time, say, 20 oscillations with a stopwatch, then divide by 20 to get an accurate T. Rearranging the formula gives g = 4 × pi² × L / T². Better still, you repeat for several lengths and plot L against T². The graph is a straight line through the origin whose slope is g ÷ (4 × pi²), so the slope gives g without relying on a single reading.