In most school physics laboratories in India, two long wires hang side by side from a beam near the ceiling. A small frame joins their lower ends, with a bubble level on it. One wire carries slotted weights; the other carries a fixed load that nobody changes. Only one wire is being tested. The second one looks like a waste of wire.
A stretch too small to see
Recall three ideas from the chapter on Stress, Strain and Hooke's Law. The stress in a wire is the force per unit area, \(F/A\). The strain is the fractional change in length, \(\Delta L/L\). Below the elastic limit, stress is proportional to strain, and the constant ratio is Young's modulus, \(Y\).
So \(Y = FL/(A\,\Delta L)\). The load \(F\), the length \(L\) and the area \(A\) are easy to measure. The extension \(\Delta L\) is the hard part. A steel wire \(2\,\text{m}\) long and \(0.2\,\text{mm}^2\) in section stretches only \(0.5\,\text{mm}\) under a \(1\,\text{kg}\) load. To know \(Y\) within a few per cent, that half millimetre must be read to a hundredth of a millimetre.
A hundredth of a millimetre is smaller than the sag of a beam when someone leans on it. It is also smaller than the change in the wire's own length when the room warms by a degree. A single wire read against a fixed scale would show all of these as "extension". The experiment must remove them before it can measure anything.
The apparatus
In Searle's apparatus, two identical wires hang from the same rigid support. The reference wire carries a fixed dead load that keeps it straight. The test wire carries a hanger for slotted weights. A frame hangs from each wire, and a spirit level is hinged to the reference frame. Its other end rests on the tip of a micrometer screw fixed to the test frame.
Follow one reading in Fig 1.1. A load is added, the test wire stretches, and its end of the level drops. The bubble runs towards the higher end. The student turns the micrometer until its tip lifts the level back to flat. The screw has now moved up by exactly the extra stretch of the test wire. So the change in the micrometer reading is the extension.
The screw in this apparatus has a pitch of \(0.5\,\text{mm}\), and its head carries \(50\) divisions. One division is therefore \(0.5/50 = 0.01\,\text{mm}\), the least count. With the wire of the figure, each kilogram needs exactly one turn of the screw.
A small change is measured best against a twin that shares every disturbance except the one being measured. The disturbances then cancel, and only the wanted change remains.
This is why the second wire is not a waste. Suppose the room warms. Both wires lengthen together, both frames drop together, and the level stays flat. Suppose the beam sags a little under the added load. Again both wires drop together. The level tilts only when the test wire stretches more than the reference wire, and that happens only because of the added load.
You use the same trick when you judge your height against a friend standing beside you. If both of you step onto a platform, the comparison does not change. Only a real difference in height shows.
The fixed load on the reference wire has its own job. A wire off the reel has kinks and bends. A load of a kilogram or two pulls them straight before the readings start, so the kinks cannot slowly straighten during the experiment. For the same reason the test wire starts with the empty hanger as its dead load, and every extension is measured from there.
In Searle's experiment, why is the extension measured against a second wire instead of against a fixed scale on the wall?
Which of these errors does the reference wire cancel?
From the readings to Y
Rearrange \(Y = FL/(A\,\Delta L)\) as \(F = (YA/L)\,\Delta L\). The load is proportional to the extension, and the constant of proportion is \(YA/L\). So a graph of load against extension is a straight line through the origin, with slope \(YA/L\). The slope, multiplied by \(L/A\), gives \(Y\).
In Searle's experiment the test wire is \(2\,\text{m}\) long and \(0.2\,\text{mm}^2\) in section. A load of \(2\,\text{kg}\) stretches it by \(1\,\text{mm}\). Take \(g = 10\,\text{m/s}^2\). Find Young's modulus of the wire.
Convert to SI first: \(F = 2 \times 10 = 20\,\text{N}\), \(A = 0.2\,\text{mm}^2 = 2 \times 10^{-7}\,\text{m}^2\), \(\Delta L = 10^{-3}\,\text{m}\).
\(Y = \dfrac{FL}{A\,\Delta L} = \dfrac{20 \times 2}{2 \times 10^{-7} \times 10^{-3}} = 2 \times 10^{11}\,\text{Pa}\), the value for steel.
A square millimetre is \(10^{-6}\,\text{m}^2\), not \(10^{-3}\,\text{m}^2\). This conversion is the commonest slip in the experiment.
A real student never trusts one reading. She adds the loads one at a time and reads the micrometer after each. Then she removes them one at a time and reads again. Each extension is taken as the mean of its loading and unloading readings. Then she plots the means and draws the best straight line through them.
The wire of Fig 1.2 is \(2.00\,\text{m}\) long with a diameter of \(0.40\,\text{mm}\). Her loading readings for \(1\), \(2\), \(3\) and \(4\,\text{kg}\) were \(0.79\), \(1.60\), \(2.41\) and \(3.20\,\text{mm}\); her unloading readings were \(0.81\), \(1.62\), \(2.41\) and \(3.20\,\text{mm}\). Take \(g = 10\,\text{m/s}^2\). Find \(Y\).
The loads are \(10\), \(20\), \(30\) and \(40\,\text{N}\). The mean extensions are \(0.80\), \(1.61\), \(2.41\) and \(3.20\,\text{mm}\).
Slope of the line \(= 40\,\text{N} \div 3.20\,\text{mm} = 12.5\,\text{N/mm} = 1.25 \times 10^4\,\text{N/m}\). Then \(Y = \text{slope} \times L/A\).
\(A = \pi (0.20 \times 10^{-3})^2 = 1.257 \times 10^{-7}\,\text{m}^2\). So \(Y = 1.25 \times 10^4 \times 2.00 / 1.257 \times 10^{-7} = 1.99 \times 10^{11}\,\text{Pa}\).
\(Y \approx 2.0 \times 10^{11}\,\text{Pa}\), the value for steel.
Why load and unload, and why wait
The unloading readings are a test of the wire. If the wire has stayed within its elastic limit, it returns along the same line, and the two sets agree. If the unloading readings stay larger, the wire has taken a permanent set, and the experiment has spoilt it. Averaging the two sets also cancels the small slack, or backlash, in the thread of the screw.
After each change of load the student waits a minute before reading. The new weight sets the hanger swinging, and the wire takes a moment to settle at its new length. A reading taken at once catches the wire in motion.
A graph beats any single reading for three reasons. The line averages the small random errors of all the points. A constant error, such as a kink that straightened before the first reading, shifts every point equally and leaves the slope unchanged. And a curve bending away from the line warns at once that the elastic limit has been crossed.
In a Searle's experiment the best line through the readings passes through \(60\,\text{N}\) at \(3.0\,\text{mm}\). The wire is \(1.5\,\text{m}\) long and \(0.15\,\text{mm}^2\) in section. Find Young's modulus, in units of \(10^{11}\,\text{Pa}\).
A student finds that every unloading reading is about \(0.3\,\text{mm}\) larger than the loading reading for the same load. What has happened?
Long, thin and gently loaded
The extension is \(\Delta L = FL/(AY)\). For a given load it grows with the length \(L\) and falls with the area \(A\). A wire two metres long and a fraction of a millimetre thick gives an extension of a few millimetres. A short thick rod would give a few thousandths of a millimetre, far below the least count of any screw.
The loads must stay well below the elastic limit, and so far below the breaking load. Beyond the limit Hooke's law fails, the line bends, and the readings no longer give \(Y\). The wire also takes a permanent set and cannot be used again.
Which reading limits the answer
Every measured quantity carries an uncertainty of about one least count. Its effect on \(Y\) is that uncertainty as a fraction of the quantity. The length, \(2.000\,\text{m}\) read to \(1\,\text{mm}\), is known to \(0.05\) per cent. The largest extension, \(3.20\,\text{mm}\) read to \(0.01\,\text{mm}\), is known to about \(0.3\) per cent.
The diameter is the weak point. A diameter of \(0.40\,\text{mm}\), read with a screw gauge to \(0.01\,\text{mm}\), is uncertain by \(2.5\) per cent. The area goes as the diameter squared, so its error doubles to \(5\) per cent. That is why the diameter is measured at several places along the wire, in two directions at right angles, and averaged.
For the student's wire of Fig 1.2, rank the contributions to the error in \(Y\), largest first. Every reading is uncertain by one least count.
Why does Searle's experiment use a long, thin wire instead of a short, thick rod of the same steel?
George Searle taught practical physics at the Cavendish Laboratory in Cambridge for more than forty years. He designed simple apparatus that let a student measure small quantities honestly, with every source of error in view. In 1908 he gathered his methods in a book called Experimental Elasticity. His two-wire apparatus for Young's modulus is still in school laboratories across India, more than a century later.
Return to the two wires in the school laboratory. The second wire is the whole trick. It shares the warming of the room and the sag of the beam, so that the level sees only the load. Four or five loads, read up and down, give a line whose slope is \(YA/L\). The steel of the test wire comes out near \(2 \times 10^{11}\,\text{Pa}\).
- Using the area in mm² or the extension in mm directly in \(Y = FL/(A\,\Delta L)\).Convert first: \(1\,\text{mm}^2 = 10^{-6}\,\text{m}^2\) and \(1\,\text{mm} = 10^{-3}\,\text{m}\).
- Using the diameter in place of the radius in \(A = \pi r^2\).\(A = \pi d^2/4\). Halve the diameter before squaring.
- Taking the extension from the micrometer reading itself.The extension is the change in the reading from the dead-loaded start.
- Thinking the reference wire stretches with the load.The reference wire keeps its fixed load. It only shares the disturbances.