Final answer:
Only statement (a) in the given question is true, as it correctly states that the frequency of vibration is 300 Hz. Statements (b) and (c) are false because the length of the string and the location of the nodes do not match the given wave equation parameters.
Step-by-step explanation:
The equation of the standing wave on a string clamped at both ends and vibrating in its third harmonic is given by y=0.4sin(0.314x)cos(600πt), where x and y are in cm and t in seconds. To evaluate the given statements, we examine the components of the wave equation.
(a) The angular frequency (ω) is given as 600π rad/s. The frequency (f) of vibration is the angular frequency divided by 2π, so f = ω / (2π) = 600π / (2π) = 300 Hz. Thus, statement (a) is true.
(b) For a string clamped at both ends, the wavelength (λ) of the standing wave in the third harmonic (n=3) can be given by λ = 2L/n, where L is the length of the string. The given wave number (k) is 0.314 cm⁻¹, and k is related to the wavelength by k = 2π/λ. Solving for L, we get L = n / (2k) = 3 / (2 * 0.314) = 4.78 cm, not 30 cm as stated. Therefore, statement (b) is false.
(c) Nodes occur at points where the sine term is zero, which is when x = mλ/n for m = 0, 1, 2,.... From the given wave number (k = 0.314 cm⁻¹), we have λ = 20 cm. Thus, nodes would occur at x = 0, 10 cm, and 20 cm, not 30 cm. Therefore, statement (c) is false.
Based on the analysis, only statement (a) is correct, making the answer option A: Only a is true.