Experimental Skills Formula Sheet

Every formula, least count, zero-error rule, and experiment result from this chapter, condensed for last-minute revision. Use it alongside the detailed topic pages.

Measuring Instruments

Vernier Caliper

$$\boxed{\text{LC} = 1\,\text{MSD} - 1\,\text{VSD}}$$$$\boxed{\text{LC} = \frac{1\,\text{MSD}}{\text{Number of VS divisions}}}$$$$\boxed{\text{Reading} = \text{MSR} + \text{VC} \times \text{LC}}$$

Quantity	Formula	Notes
Vernier scale division	$1\,\text{VSD} = \dfrac{9\,\text{mm}}{10} = 0.9\,\text{mm}$	Standard 10-division vernier
Least count (standard)	$\dfrac{1\,\text{mm}}{10} = 0.1\,\text{mm} = 0.01\,\text{cm}$	10 VS divisions span 9 mm
Least count (precise)	$\dfrac{1\,\text{mm}}{50} = 0.02\,\text{mm} = 0.002\,\text{cm}$	50 VS divisions span 49 mm
Least count (with MSD = 0.5 mm)	$\dfrac{0.5\,\text{mm}}{50} = 0.01\,\text{mm}$	50 VS = 49 MS
Inch vernier LC	$\dfrac{0.025\,\text{inch}}{25} = 0.001\,\text{inch}$	MSD = 0.025 inch, 25 VS divisions span 24 MS
MSR	Last complete division before VS zero	Lower value, not nearest
VC	Vernier coincidence — a whole number	$0, 1, \dots, 9$ for 10-division

Screw Gauge (Micrometer)

$$\boxed{\text{Pitch} = \frac{\text{Linear distance moved}}{\text{Number of rotations}}}$$$$\boxed{\text{LC} = \frac{\text{Pitch}}{\text{Number of circular scale divisions}}}$$$$\boxed{\text{Reading} = \text{LSR} + \text{CSR} \times \text{LC}}$$

Quantity	Value / Formula	Notes
Standard pitch	$0.5\,\text{mm}$	Distance moved per full rotation
LC (pitch 0.5 mm, 50 div)	$\dfrac{0.5}{50} = 0.01\,\text{mm} = 0.001\,\text{cm}$	Standard practical precision
LC (pitch 0.5 mm, 100 div)	$\dfrac{0.5}{100} = 0.005\,\text{mm}$	Theoretical; read to 0.01 mm in practice
LSR (PSR)	Last visible division on sleeve	Check if graduations are 0.5 mm or 1 mm
CSR	Circular scale division at reference line	Thimble reading

Instrument Comparison

Instrument	Measures	Range	Least Count
Metre scale	Length	0–100 cm	1 mm (0.1 cm)
Vernier caliper	Length, diameter, depth (internal/external)	0–15 cm	0.01 cm
Screw gauge	Thickness, wire diameter	0–2.5 cm	0.001 cm
Spherometer	Radius of curvature	—	0.001 cm
Stopwatch	Time	—	0.01 s
Thermometer	Temperature	—	0.1 °C

Zero Error and Correction

$$\boxed{\text{Actual Reading} = \text{Observed Reading} - \text{Zero Error (with sign)}}$$

Type	Position (vernier / screw)	Error value	Correction
Positive	VS zero right of MS zero / CSR below ref line	$+(\text{div}) \times \text{LC}$	Subtract error
Negative	VS zero left of MS zero / CSR above ref line	$-(n - \text{div}) \times \text{LC}$	Add $\lvert\text{error}\rvert$

• Negative vernier ($n=10$): e.g. 8th from end coincides $\Rightarrow -(10-8)\times 0.01 = -0.02\,\text{cm}$.
• Negative screw ($n=100$): e.g. CSR = 96 coincides $\Rightarrow -(100-96)\times 0.01 = -0.04\,\text{mm}$.

Mechanics Experiments

Simple Pendulum (finding g)

$$\boxed{T = 2\pi\sqrt{\frac{L}{g}}} \qquad \boxed{g = \frac{4\pi^2 L}{T^2}}$$

Item	Detail
Effective length	$L = \text{string length} + \text{bob radius } r$
Time period	$T = \dfrac{\text{time for 20 oscillations}}{20}$
Graph	$T^2$ vs $L$ → straight line through origin
Slope	$\dfrac{4\pi^2}{g}$
Amplitude	Keep small (< 10°) so $T = 2\pi\sqrt{L/g}$ holds

Young’s Modulus (Searle’s Apparatus)

$$\boxed{Y = \frac{FL}{A\,\Delta L} = \frac{4FL}{\pi d^2 \Delta L}}$$

Plot $F$ vs $\Delta L$; measure wire diameter $d$ to get area $A = \pi d^2/4$.

Surface Tension (Capillary Rise)

$$\boxed{S = \frac{\rho g r h}{2\cos\theta}}$$

For water–glass, $\cos\theta = 1$.

Coefficient of Viscosity (Stokes’ Law)

$$\boxed{\eta = \frac{2r^2(\rho - \sigma)g}{9 v_t}}$$

Measure terminal velocity $v_t$ of a sphere falling through the liquid.

Specific Heat Capacity (Calorimeter)

$$\boxed{m_1 c_1 (T_1 - T) = m_2 c_2 (T - T_2)}$$

Sonometer (string vibrations)

$$\boxed{f = \frac{1}{2L}\sqrt{\frac{T}{\mu}}}$$

Quantity	Relation / Graph
Tension	$T = Mg$; $\mu$ = mass per unit length
At constant $T$	$f \propto \dfrac{1}{L}$ → $f$ vs $1/L$ straight line through origin
At constant $L$	$f \propto \sqrt{T}$ → $f^2$ vs $T$ straight line through origin

Sound Experiments

Speed of Sound (Resonance Tube)

$$\boxed{v = 2f(l_2 - l_1)}$$

End correction: $e = \dfrac{l_2 - 3l_1}{2}$, where $l_1, l_2$ are first and second resonance lengths.

Electricity Experiments

Ohm’s Law (V–I Characteristics)

$$\boxed{V = IR} \qquad R = \frac{V}{I} = \text{slope of } V\text{ vs } I$$

Ohmic conductor → straight line through origin. Diode/LED → curved (non-ohmic).

Metre Bridge (unknown resistance)

$$\boxed{\frac{R}{S} = \frac{l}{100-l}} \qquad R = S \cdot \frac{l}{100-l}$$

Item	Detail
Principle	Wheatstone bridge
Balance point	Galvanometer shows zero deflection
Best accuracy	Balancing length $l \approx 40$–$60\,\text{cm}$ (near 50 cm)
Specific resistance	$\rho = \dfrac{\pi d^2 R}{4L}$

Resistivity of a Wire

$$\boxed{\rho = \frac{RA}{L}}, \quad A = \pi r^2$$

$R$ from $V/I$ graph slope; radius $r$ from screw gauge; units $\Omega\!\cdot\!\text{m}$.

Potentiometer (EMF and internal resistance)

$$\boxed{\frac{E_1}{E_2} = \frac{l_1}{l_2}} \qquad \boxed{r = \left(\frac{l_1 - l_2}{l_2}\right) R}$$

$l_1$ = balancing length with key open; $l_2$ = balancing length with key closed across $R$. Advantage over a voltmeter: draws no current at balance, so it measures true EMF.

Galvanometer

$$\boxed{K = \frac{E}{(R + G)\theta}} \quad \text{(figure of merit)}$$

Conversion	Formula
To ammeter	Shunt $S = \dfrac{I_g G}{I - I_g}$
To voltmeter	Series $R_h = \dfrac{V}{I_g} - G$

Optics Experiments

Convex Lens — u–v Method

$$\boxed{\frac{1}{f} = \frac{1}{v} - \frac{1}{u}}$$

Item	Detail
Sign convention	$u$ negative (object left), $v$ positive (real image right), $f$ positive
Graph	$1/v$ vs $1/u$ → straight line, slope $= -1$
Intercepts	Both equal $1/f$

Convex Lens — Displacement (Bessel’s) Method

$$\boxed{f = \frac{D^2 - d^2}{4D}}$$

$D$ = fixed object-to-screen distance, $d$ = separation between the two lens positions. Condition: $D > 4f$. No parallax error.

Concave Mirror — Focal Length

$$\boxed{\frac{1}{f} = \frac{1}{v} + \frac{1}{u}} \qquad f = \frac{uv}{u+v}$$

For real images, both $u$ and $v$ are negative. Distant-object method: $f = R/2$.

Prism — Angle and Refractive Index

$$\boxed{A = \frac{r_1 + r_2}{2}} \qquad \boxed{\delta = i + e - A}$$$$\boxed{n = \frac{\sin\!\left(\dfrac{A + \delta_m}{2}\right)}{\sin\!\left(\dfrac{A}{2}\right)}}$$

Item	Detail
Minimum deviation	Occurs when $i = e$ (symmetric path)
Graph	$\delta$ vs $i$ → U-shaped, minimum at $\delta_m$

Refractive Index (Glass Slab, travelling microscope)

$$\boxed{n = \frac{\text{Real depth}}{\text{Apparent depth}}}$$

Young’s Double Slit — Wavelength of Light

$$\boxed{\lambda = \frac{\beta\, d}{D}}$$

$\beta$ = fringe width $= \dfrac{\text{distance for } n \text{ fringes}}{n}$, $d$ = slit separation, $D$ = slit-to-screen distance.

Error Analysis

Error Propagation

For $Z = A^a B^b C^c$:

$$\boxed{\frac{\Delta Z}{Z} = \lvert a\rvert\frac{\Delta A}{A} + \lvert b\rvert\frac{\Delta B}{B} + \lvert c\rvert\frac{\Delta C}{C}}$$

Types of Errors

Category	Examples	How to minimize
Systematic	Zero error, index error, backlash, least-count limit	Check/correct zero error, calibrate, use smaller-LC instruments
Random	Parallax, reaction time, fluctuating readings	Take $\geq 3$ readings, take mean, view scale at eye level
Personal / gross	Misreading, bias, careless mistakes	Follow procedure, double-check, stay objective

Significant Figures

• All non-zero digits are significant.
• Zeros between digits are significant.
• Leading zeros are not significant.
• Trailing zeros after a decimal are significant.

Mean Absolute Error

$$\lvert\Delta x_i\rvert = \lvert x_i - \bar{x}\rvert \qquad \overline{\lvert\Delta x\rvert} = \frac{1}{n}\sum_{i=1}^{n}\lvert\Delta x_i\rvert$$

Report a measurement as $x = (\bar{x} \pm \overline{\lvert\Delta x\rvert})$; random error is typically of the order of the least count.

Graphs and Data Presentation

$$\boxed{\text{Slope} = \frac{y_2 - y_1}{x_2 - x_1}}$$

Choose points far apart on the best-fit line. Rules: pick a suitable scale, plot points clearly, draw a best-fit line/curve, label axes with units, and give a title.

Experiment	Plot	Shape
Simple pendulum	$T^2$ vs $L$	Straight line through origin
Lens (u–v)	$1/v$ vs $1/u$	Straight line, slope $-1$
Ohm’s law	$V$ vs $I$	Straight line through origin
Sonometer	$f$ vs $1/L$	Straight line through origin
Prism	$\delta$ vs $i$	U-shaped, min at $\delta_m$
Diode	$I$ vs $V$	Curved (exponential)

High-Yield Reminders

Related Pages

• Vernier Caliper — least count, reading, zero error in detail
• Screw Gauge — pitch, least count, backlash
• Common Experiments — apparatus, procedures, viva questions
• Units and Measurements — error theory and significant figures
• Current Electricity — electrical experiments
• Optics — lens, mirror, prism, YDSE theory