SMA Physics • Physics Lab Class XI / XII • Waves & AC Mains ⚡ READY

Frequency of AC Mains with Sonometer

🏠 All Labs

Sonometer Resonance Sound Box, Scale & Movable Bridges

⚡ Method A: Wire carrying AC in B-field

$l$ = 63.2 cm | $f_0$ = 50.0 Hz

Paper Rider: Seated at Antinode

Suspended Mass ($M$) 1.000 kg

Tension ($T = Mg$) 9.80 N

Vibrating Length ($l$) 63.2 cm

Wire Frequency ($\nu_0$) 50.0 Hz

📏 Movable Bridge B Position ($x_B$): 73.2 cm

Bridge A: Fixed at $x_A = 10.0\text{ cm}$ Vibrating Length $l = x_B - x_A$: 63.2 cm

⚖ Suspended Slotted Weights (Hanger + Discs): 1.00 kg (9.80 N)

🧵 Wire Material & Linear Density ($m$):

Linear density $m = \pi r^2 \rho$: 2.403 × 10⁻³ kg/m $g = 9.80\text{ m/s}^2$

📋 Observation Table (AC Frequency Determination)

#	$M$ (kg)	$T = Mg$ (N)	$\sqrt{T}$ ($\text{N}^{1/2}$)	$l$ (cm)	$l$ (m)	$l^2$ ($\text{m}^2$)	$\frac{\sqrt{T}}{l}$	$f$ (Hz)	Action

📈 Graphical Analysis & Linear Regression

Graph Slope ($k$) --

Correlation ($R^2$) --

Freq from Slope -- Hz

Theoretical $f$ 50.0 Hz

📐 Virtual Screw Gauge: Measure Wire Diameter ($d = 2r$)

Rotate Thimble (Circular Scale): 0.600 mm

Pitch Scale Reading 0.50 mm

Circular Scale Div 10 div

Least Count ($LC$) 0.01 mm

Calculated Diameter 0.600 mm

Radius $r = d / 2 = \mathbf{0.300\text{ mm}} = 3.00 \times 10^{-4}\text{ m}$

Cross-sectional Area $A = \pi r^2 = \mathbf{2.827 \times 10^{-7}\text{ m}^2}$

Linear Mass Density $m = A \cdot \rho = \mathbf{2.403 \times 10^{-3}\text{ kg/m}}$

🎯 Unknown AC Mains Generator Challenge

An unknown AC signal generator powers the sonometer. Determine its true frequency by conducting resonance experiments, varying the suspended mass $M$, locating resonant lengths $l$, and evaluating $f$.

Your Experimental Frequency Estimate:

1. Aim of the Experiment

To determine the frequency of alternating current (AC) mains using a sonometer (by observing the resonant vibration of a stretched wire driven by electromagnetic force).

2. Apparatus Required

Sonometer: Hollow resonant wooden sound box with two sound holes, tuning pegs, and a 100 cm graduated scale.
Knife-edge Bridges: Two movable wedge-shaped wooden/brass bridges ($A$ and $B$) with sharp apexes to define vibrating wire length $l$.
Step-Down Transformer: Steps down 220V mains to safe 2V–6V AC.
Magnet / Electromagnet: Strong horseshoe permanent magnet (Method A) or AC electromagnet with soft iron core (Method B).
Sonometer Wire: Brass or copper wire (non-magnetic, for Method A) or soft iron wire (ferromagnetic, for Method B).
Slotted Weights with Hanger: 500g hanger + seven 500g slotted weights ($0.5\text{ kg to }4.0\text{ kg}$).
Light Paper Rider: Inverted V-shaped strip of paper placed at the center (antinode) of the vibrating segment.
Screw Gauge & Physical Balance: To measure the wire diameter and determine linear density $m$.

3. Step-by-Step Procedure

Tension Setup: Place a known mass $M$ (e.g., 1.0 kg) on the weight hanger suspended over the frictionless pulley. The tension is $T = Mg$.
Position Magnet: Position the horseshoe magnet (or electromagnet) at the exact midpoint between Bridge A and Bridge B.
Paper Rider Placement: Fold a light paper strip into an inverted V-shape and balance it carefully on the wire at the antinode (midpoint between bridges).
Close AC Switch: Turn on the AC power supply. Alternating current flows, producing a dynamic transverse magnetic driving force on the wire.
Resonance Search (Flutter & Fly-off): Slowly adjust movable Bridge B:
- When far from resonance, the wire hums quietly and the paper rider sits calmly.
- As the vibrating length approaches resonant length $l_0$, the wire vibration amplitude grows and the paper rider begins to flutter vigorously.
- At exact resonance, the amplitude peaks sharply and the paper rider is violently thrown off ("Flies Off!").
Record Resonant Lengths: Record length $l_1$ when approaching resonance by increasing length, and $l_2$ by decreasing length. The mean resonant length is $l = \frac{l_1 + l_2}{2}$.
Repeat for Various Loads: Repeat the observations for loads $M = 1.0, 1.5, 2.0, 2.5, 3.0\text{ kg}$.
Tabulate & Plot: Record all readings in the observation table and plot $\sqrt{T}$ against $l$ to determine the frequency from the slope.

4. Precautions

Ensure the sonometer wire is kink-free, uniform, and rests snugly on the sharp edges of the bridges.
Keep the pulley clean and lubricated to eliminate friction errors.
Use low voltage AC (2V–6V) from a step-down transformer to prevent excessive Joule heating of the wire ($I^2Rt$).
Place the paper rider precisely at the center between the two bridges where the fundamental antinode forms.

1. Transverse Stationary Waves on a Stretched String

When a stretched wire of length $l$, tension $T$, and mass per unit length (linear mass density) $m$ is fixed at both ends by two sharp bridges, transverse standing waves are established with nodes at the bridges and an antinode at the center.

The velocity of transverse waves along the wire is given by:

$$v = \sqrt{\frac{T}{m}}$$

For the fundamental mode of vibration, the length of the vibrating segment corresponds to half a wavelength ($\lambda / 2 = l \implies \lambda = 2l$). Hence, the natural fundamental frequency of the wire is:

$$\nu_0 = \frac{v}{\lambda} = \frac{1}{2l}\sqrt{\frac{T}{m}}$$

2. Dual Methods of Excitation & AC Frequency Relation

Method A: AC Current Through Wire in a Transverse Magnetic Field (Horseshoe Magnet)

When an alternating current $I = I_0 \sin(2\pi f t)$ of mains frequency $f$ passes through a non-magnetic wire (brass or copper) situated in a transverse magnetic field $B$ from a permanent horseshoe magnet, the magnetic Lorentz force acting on the wire is:

$$F(t) = B \cdot I(t) \cdot l = B I_0 l \sin(2\pi f t)$$

By Fleming's Left-Hand Rule, the force acts upwards during one half cycle and downwards during the next half cycle. Therefore, the frequency of the mechanical driving force is identical to the AC mains frequency $f$.

At resonance ($\nu_0 = f$):

$$f = \frac{1}{2l}\sqrt{\frac{T}{m}} \implies \frac{\sqrt{T}}{l} = 2f\sqrt{m} = \text{constant}$$

Method B: Ferromagnetic Wire Under an AC Electromagnet (Soft Iron Wire)

When an electromagnet fed by AC mains is held above a soft iron wire, the electromagnet attracts the soft iron wire whenever magnetic flux peaks, regardless of whether the current is in the positive or negative half-cycle.

Since AC current reaches maximum amplitude twice in every complete cycle, the soft iron wire is attracted twice per cycle. Thus, the wire vibrates at twice the mains frequency:

$$\nu_{\text{wire}} = 2f$$

Equating to the fundamental frequency of the wire:

$$2f = \frac{1}{2l}\sqrt{\frac{T}{m}} \implies f = \frac{1}{4l}\sqrt{\frac{T}{m}}$$

3. Graphical Method of Determination

Since $T = Mg$, we have $l \propto \sqrt{T}$. A graph of $\sqrt{T}$ on the y-axis against resonant length $l$ on the x-axis gives a straight line passing through the origin.

$$\text{Slope } k = \frac{\sqrt{T}}{l} = 2f\sqrt{m} \implies f = \frac{k}{2\sqrt{m}} = \frac{\text{Slope}}{2\sqrt{m}}$$

Alternatively, a graph of $T$ against $l^2$ yields a straight line with slope $S = \frac{T}{l^2} = 4 m f^2$, giving $f = \frac{1}{2}\sqrt{\frac{S}{m}}$.

4. Linear Mass Density Determination ($m$)

The linear mass density $m$ of the wire is determined by measuring its diameter $d = 2r$ with a micrometer screw gauge and using the density of the wire material $\rho$:

$$m = \frac{\text{mass}}{\text{length}} = \pi r^2 \rho = \frac{\pi d^2 \rho}{4}$$