Chapter 17, Problem 1CQ

To determine

(a)

What mode (m-value) is this?

Answer to Problem 1CQ

Solution: 4 antinodes

Explanation of Solution

Given:

A standing wave oscillating on a string at frequency f₀.

Formula used:

$f=nv/2l$

Where f is the natural frequency

n is the no. of antinodes

v is the velocity of wave

l is the length of the string

Let m-value be n then:

From the figure we can see that there 4 anti-nodes.

To determine

(b)

How many antinodes will there be if the frequency is doubled to 2f₀

Answer to Problem 1CQ

Solution:

8 antinodes

Explanation of Solution

Calculation:

Suppose L be the length of the string.

$L=n\lambda /2$

We have n = 4,

so:

$\lambda =L/2$

Let f₀ be the frequency.

$v=f_0\lambda$ = $f_0\times L/2$ = $f_{0}L/2$

When frequency is doubled then $f_0$ becomes 2 $f_0$:

Using $f=nv/2l$, we get:

2 $f_0$ = $nv/2L=(n\times f_0\times L)/(2\times 2\times L)$

From above calculation we will get:

n = 8

Conclusion:

Using the $f=nv/2l$, number of nodes and antinodes can be calculated when other things are provided.