Paragraph: A metallic rod of length 1 m has one end free and other end rigidly clamped. Longitudinal stationary waves are set up in the rod in such a way that there are total six antinodes present along the rod. The amplitude of an antinode is 4 10⁻⁶ ~m . Young's modulus and density of the rod are 6.4 10¹⁰ ~N / m ^2 and 4 10^3 Kg / m ^3 respectively. Consider the free end to be at origin and at t =0 particles at free

Step-by-step solution

Speed of wave v= y =4 10^3 l= 5 2 + 4 = 4 11 Frequency v= v = 4 10^3 4 11 1 =11 10^3 ~Hz ; Wave Number K= 2 = 11 2 (i) Equation of standing wave in the rod aligned & S=A coskx ( t + ) where A =4 10⁻⁶ ~m at x =0, t =0 & ~S = A ~A = A cosk (0) =1 = 2 & S=4 10⁻⁶ ( 11 2 x ) (22 10^3 t ) aligned aligned & (ii) Strain = d s d x =-22 10⁻⁶ ( 11 2 x ) (22 10^3 t ) stress =Y strain & stress =140.8 10^4 (22 10^3 t ) ( 11 2 x+ ) aligned (iii) Strain at t =1 ~s and x = 2 = 1 2 m | d s d x |_ x= 2 ^ l=1 =22 10⁻⁶ ( 11 4 )=11 2 10