The variation of length ( L ) with temperature ( T ) for two different metal rods, A and B, is plotted on a graph with L on the y-axis and T on the x-axis. The resulting graph consists of two parallel straight lines. If the y-intercepts (representing the lengths at 0^ C ) for rod A and rod B are 20 cm and 50 cm respectively, what is the ratio of their coefficients of linear expansion ( _A : _B )?

Step-by-step solution

The equation for the linear expansion of a rod is given by L = L₀ (1 + T) = L₀ + L₀ T . When length L is plotted on the y-axis against temperature T on the x-axis, the slope of the straight line is dL dT = L₀ . Since the lines for rod A and rod B are parallel, their slopes must be equal: L_ 0A _A = L_ 0B _B Rearranging to find the ratio of the coefficients of linear expansion: _A _B = L_ 0B L_ 0A Given the y-intercepts (initial lengths at T = 0 ) are L_ 0A = 20 cm and L_ 0B = 50 cm : _A _B = 50 20 = 2.5 Answer: 2.5