(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...

(a) Experiment: effect of tension on the frequency of a vibrating string

1. Stretch one uniform string over the two bridges of a sonometer. Keep the vibrating length \(l\) between the bridges constant throughout, and use the same string so that its mass per unit length remains constant.
2. Pass the string over a pulley and attach a known mass \(m\). The tension is the weight of the mass: \(T=mg\).
3. Set a tuning fork of known frequency \(f\) vibrating and place its stem on the sonometer box. Adjust the hanging mass until the string resonates with the tuning fork. Resonance is recognised because the string vibrates with a large amplitude; a small paper rider placed on the string is thrown off.
4. Record the frequency of the tuning fork and the corresponding tension. Repeat for several different tuning forks and tensions.
5. Calculate \(\sqrt{T}\) for each reading and plot \(f\) against \(\sqrt{T}\). A straight line through the origin shows that, for constant length and mass per unit length, \(f \propto \sqrt{T}\).

(b) Standing waves on a string fixed at both ends

Nodes are points that do not move. Antinodes are points of maximum vibration.

Fundamental frequency: there is one loop. The fixed ends are nodes, and the centre is an antinode. Thus \(l=\frac{\lambda}{2}\).

Second overtone: this is the third harmonic, so there are three loops. There are four nodes and three antinodes. Thus \(l=\frac{3\lambda}{2}\).

(c) Formation of virtual images

Virtual image in a concave mirror: the object must be between the pole of the mirror and the principal focus \(F\). The reflected rays diverge, but their backward extensions meet behind the mirror. The image is virtual, upright and magnified.

Virtual image in a converging lens: the object must be between the lens and its near principal focus. The rays leaving the lens diverge, and their backward extensions meet on the same side of the lens as the object. The image is virtual, upright and magnified.

Examination reminder: For both a concave mirror and a converging lens, a virtual image is formed only when the object is closer to the optical device than its focal length. Show the actual rays as solid lines and the backward extensions as dashed lines.