PART ONE
A cello string vibrates in its fundamental
mode with a frequency of 239 1/s. The vibrating segment is 76.2 cm long and has a
mass of 0.48 g.
Find the tension in the string.
Answer in units of N.

PART TWO (REQUIRES PICTURE)
Find the frequency of the string when it
vibrates in six segments.
Answer in units of 1/s.

1) 83.6 N

2) 1434 Hz

Explanation:

1)

The fundamental frequency of the standing waves on a string is given by the equation

where:

L is the length of the string

T is the tension in the string

is the mass per unit length, which can be rewritten as

, where m is the mass of the string

For the string in this problem, we have:

L = 76.2 cm = 0.762 m is the length of the string

is the mass

is the fundamental frequency

Re-arranging the equation for T, we find the tension in the string:

2)

In this problem, we see that the string vibrates in six segments.

A standing wave on a string has different modes of vibrations, characterized by different frequencies:

- In the  fundamental mode (), the string vibrates in one segment

- In the second harmonic (), the string vibrates in 2 segments

... and so on

The frequency of the nth-harmonic is given by

where is the fundamental frequency.

In this problem, we know that

And the string is vibrating in 6 modes, so we are finding the frequency of the 6th harmonic, which is