A student plots a graph of the square of the time period ( T^2 ) on the y -axis versus the length ( L ) of a simple pendulum on the x -axis for two different planets, A and B. The resulting graphs are straight lines passing through the origin. The line for planet A makes an angle of 30^ with the L -axis, while the line for planet B makes an angle of 60^ with the L -axis. The ratio of the acceleration due to gravity o

Step-by-step solution

The time period T of a simple pendulum of length L is given by: T = 2 L g Squaring both sides, we get: T^2 = ( 4 ^2 g ) L This represents a straight line passing through the origin ( y = mx ) when T^2 is plotted against L . The slope of this line is: m = = 4 ^2 g From this, we can see that the acceleration due to gravity g is inversely proportional to the slope : g 1 For planets A and B, the ratio of their accelerations due to gravity is: g_A g_B = _B _A Given _A = 30^ and _B = 60^ : g_A g_B = 60^ 30^ = 3 1 3 = 3 T