Question

A horizontal wire is tied to supports at each end and vibrates in its second-overtone standing wave. The tension in the wire is 5.00 N, and the node-to-node distance in the standing wave is 6.48 cm.

(a) What is the length of the wire?
L =

Answer 1

The length of the wire vibrating in its second overtone with a node-to-node distance of 6.48 cm is 38.88 cm.

Step-by-step explanation:

The question involves concepts of physics, specifically waves and their properties in the context of standing waves on a string. When a string is vibrated at certain frequencies, standing waves with nodes and antinodes are produced. The second overtone means that the string vibrates in a mode where there are three segments of a wave between the supports, corresponding to three times the fundamental frequency.

Since the wire vibrates in its second overtone, this means it will have three antinodes and four nodes, including the nodes at the ends. If the node-to-node distance is 6.48 cm, this would correspond to half a wavelength. Therefore, the whole wavelength is twice this measurement, meaning one full wavelength is 12.96 cm. Given that there are three wavelengths fitting into the length of the string for the second overtone, the total length L of the string can be calculated by multiplying the length of one wavelength by three.

Length of wire (L) = 3 × one wavelength = 3 × 12.96 cm = 38.88 cm

Answer 2

To find the length of the wire vibrating in its second overtone, we multiply the node-to-node distance by 2 (to find a full wavelength) and then by 3 (for the second overtone). The length of the wire is 38.88 cm.

Step-by-step explanation:

We need to determine the length of the wire that is vibrating in its second-overtone standing wave. For a string fixed at both ends, the second overtone corresponds to the formation of three antinodes and four nodes (including the nodes at the fixed ends). The node-to-node distance given is 6.48 cm, so for three complete wavelengths, we have three times this node-to-node distance.
Step-by-step calculation:

1. Since one wavelength (λ) has two node-to-node segments, so the node-to-node distance is λ/2.
2. For the second overtone (also known as the third harmonic), there are three full wavelengths in the length of the wire.
3. Therefore, the length of the wire is 6.48 cm × 2 × 3.

Calculation:

L = 6.48 cm × 2 × 3 = 38.88 cm

The length of the wire is 38.88 cm.