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A sonometer string \( A B \) of length \( 1 \mathrm{~m} \) is stretched by a load and the tension \( T \) is adjusted so that the string resonates to a frequency of \( 1 \mathrm{kHz} \). Any point \( P \) of the wire may be held fixed by use of a movable bridge that can slide along the base of sonometer.
(1) If point \( P \) is fixed so that \( A P: P B:: 1: 4 \), then the smallest frequency for which the sonometer wire resonates is 5 \( \mathrm{kHz} \).
(2) If \( P \) be taken at midpoint of \( A B \) and fixed, then when the wire vibrates in the third harmonic of its fundamental, the number of nodes in the wire (including \( A \) and \( B \) ) will be totally seven.
(3) If the fixed point \( P \) divides \( A B \) in the ratio \( 1: 2 \), then the tension needed to make the string vibrate at \( 1 \mathrm{kHz} \) will be \( 3 T \). (neglecting the terminal effects)
(4) The fundamental frequency of the sonometer wire when \( P \) divides \( A B \) in the ratio \( a: b \) will be the same as the fundamental frequency when \( P \) divides \( A B \) in the ratio \( b: a \).
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