Melde's experiment - class-XI

$Tp^2=constant$
$T _1P _1^2=T _2p _2^2$
$\dfrac{T _2}{T _1}=\dfrac{p _1^2}{p _2^2}$
$\Rightarrow \dfrac{p _1^2}{p _2^2}=4\Rightarrow p _2^2=\dfrac{p _1^2}{4}$
$\Rightarrow p _2^2=\dfrac{64}{4}=16$
$\Rightarrow p _2=4$

In Melde's experiment, the frequency formula gives the relation between tension T and the number of loops n. For transverse vibration (fork at right angles), n1 * sqrt(T1) = 2 * n2 * sqrt(T2) for longitudinal vibration, or generally frequency is proportional to sqrt(T) for a fixed length. With 4 loops at 100 g in transverse mode, the fundamental frequency condition relates the tensions such that changing to 1 loop in longitudinal mode requires increasing the tension to 400 g. Since the initial tension is 100 g, an additional weight of 300 g is needed.

In perpendicular mode, f = (p/2L) * sqrt(T/m). In parallel mode, f = (p/L) * sqrt(T/m). To keep the same number of loops p and same frequency f, the length L must change. Setting (p/2L1) * sqrt(T/m) = (p/L2) * sqrt(T/m) results in L2 = 2 * L1.

Frequency f is proportional to sqrt(T). In perpendicular mode, f = (p/2L) * sqrt(T/m). Thus, p * sqrt(T) = constant. Initially, 4 * sqrt(T) = 2 * sqrt(T + 3). Squaring both sides: 16T = 4(T + 3) -> 16T = 4T + 12 -> 12T = 12 -> T = 1g.

In the parallel position, the string completes one full vibration for every one vibration of the tuning fork, because the tension is varied at twice the frequency of the fork.

One cycle of up and down vibration for transverse waves on the string is two cycles of string tension increase and decrease. The tension is maximum both at the loops’ maximum up position and again at maximum down position. Therefore, in longitudinal drive mode, since the string tension increases and decreases once per tuning fork vibration, it takes one tuning fork vibration to move the string loop to maximum up position and one to move it to maximum down position. This is two tuning fork vibrations for one up and down string vibration, so the tuning fork frequency is half the string frequency.
Hence, the number of loops in longitudinal mode(tuning fork) is half of that of transverse mode(string).

From the formula f = (p/2L) * sqrt(T/m), increasing the number of loops p increases the frequency f, assuming other variables remain constant.

Melde's experiment demonstrates the relationship between frequency, tension, mass per unit length, and wavelength (via loop length) in a vibrating string.

Frequency f = (p/L) * sqrt(T/m). Since f is constant, p1/L1 * sqrt(T1/m1) = p2/L2 * sqrt(T2/m2). m is proportional to diameter squared (d^2). Given d2 = d1/2, m2 = m1/4. Given L2 = L1/2. Substituting: 4/L1 * sqrt(16/m1) = 2/(L1/2) * sqrt(T2/(m1/4)). Solving for T2 gives 32 g-wt.