Scaling of String Power

Classical Mechanics Level 1

An oscillating string carries a power $P$ per unit time. If the length of the string is doubled (while keeping the mass of the string fixed) and the oscillations of the string are twice as rapid (while keeping the wave velocity constant), but the amplitude of oscillation is cut by a factor of 3, what is the new power carried by the oscillating string?

\frac{2P}{9}

\frac{P}{3}

\frac{2P}{3}

\frac{P}{4}

1 solution

Matt DeCross
May 14, 2016

The initial power per unit time carried by the string is:

$P = \frac12 \mu v \omega^2 A^2.$

The wave velocity is kept constant, so $v$ does not change. If the length is doubled, the linear mass density $\mu$ is cut by a factor of 2, and so is the power. If the frequency doubles, then the power quadruples by the above; similarly if $A \to \frac{A}{3}$ , this multiplies the power by a factor of $\frac19$ . Keeping track of all the factors, one has:

$P \to \frac12 (4) \frac19 P= \frac{2}{9}P .$

An advice, it is unclear what you mean by doubling the length of the string. Better say density decrease by a half.

Lin Su - 4 years, 1 month ago

I have added the clarification that the mass of the string is to be kept fixed when doubling the length. Part of the point of the problem is to translate that into the condition on the density.

Matt DeCross - 4 years, 1 month ago

Scaling of String Power

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