(a) Describe an experiment to show how the frequency of the note emitted by a vibrating string depends on the tension in the string (b) Draw diagrams showin...

Answer Details

(a) Experiment: how the frequency of the note depends on the tension

Apparatus: a sonometer (a hollow wooden box carrying a thin, uniform stretched wire), two movable bridges, a fixed peg, a frictionless pulley at one end, a scale-pan carrying known masses, a set of tuning forks of known frequency, and a small paper rider.

Procedure: The wire is stretched over the two bridges so that the vibrating (effective) length \(l\) between them is kept constant throughout, and the same uniform wire is used so that the mass per unit length \(\mu\) is constant. One end is fixed to the peg; the other passes over the pulley and carries the scale-pan. A known mass \(m\) is placed on the pan, giving a tension \(T = mg\) in the wire. A tuning fork of known frequency \(f\) is struck and its stem pressed on the box. The mass (tension) is adjusted until the wire vibrates in unison with the fork in its fundamental mode; at resonance a small paper rider placed at the middle of the wire is violently thrown off. The tension \(T\) and the fork frequency \(f\) are recorded. The experiment is repeated with four more forks of different frequency, each time altering the mass to obtain resonance, while \(l\) and \(\mu\) are held fixed.

Sample results:

A graph of \(f\) against \(\sqrt{T}\) is plotted:

The graph is a straight line passing through the origin of slope \(\approx 31.6\ \text{Hz N}^{-1/2}\). This shows that the frequency is directly proportional to the square root of the tension, \( f \propto \sqrt{T}\), in agreement with \( f = \dfrac{1}{2l}\sqrt{\dfrac{T}{\mu}}\).

(b) Vibrating string fixed at both ends

(i) Fundamental frequency (first harmonic): the string vibrates in a single loop, with a node (N) at each fixed end and one antinode (A) at the middle.

(ii) Second overtone (third harmonic): the string vibrates in three loops, giving four nodes (N) and three antinodes (A).

(c) Formation of a virtual image (ray diagrams)

(i) Concave mirror: with the object O placed between the pole P and the principal focus F, the two reflected rays diverge. One incident ray parallel to the axis reflects through F; a second ray directed to the pole P reflects symmetrically about the axis. Producing the reflected rays backwards behind the mirror (broken lines) locates a virtual, erect and magnified image I.

(ii) Converging lens: with the object O placed between the lens and its principal focus F (the magnifying-glass position), the refracted rays diverge. One incident ray parallel to the axis refracts through the far focus F'; a second ray through the optical centre passes straight on. Producing the refracted rays backwards on the same side as the object (broken lines) locates a virtual, erect and magnified image I.