Question

One end of a thin, long uniform string having linear mass density λ is fixed to a wall and other end with a block which can move along a vertical rail. The string is horizontal and has a tension F. If block starts to execute SHM along the rail with amplitude a and angular frequency ω, calculate 

(i) energy transferred to string during one complete oscillation of the block and 

(ii) maximum rate of transfer of energy to the string. 

Answer 1

Due to harmonic oscillation of the block, transverse wave is produced which travel along the string.

Velocity of transverse wave in the string, v = √(T/λ)

Assuming wave propagation direction to be positive x-direction and vertically upward direction to be positive y-direction and that the block starts oscillating at t = 0 from mean position, its displacement at time t will be

Yblock = asin(ωt)  ......(1)

Since, the end of string, connected with the block also oscillates with it, therefore displacement of this end of string is also same as that of block. Since. wave propagates with velocity v along the string, therefore, equation of wave travelling along the string will be

Since, the string is thin, therefore, linear mass density λ is small or velocity of wave propagation will be very high. Hence, slope, tanθ of string will be very small.

Negative sign indicates that vertical component of tension in string at end is downwards. It means string pulls the block downward as shown figure in or block exerts an upward force equal to F_v. Power is transferred from block to string due to this force which is equal to where u is velocity of the block.