Graphs of the square of the time period ( T^2 ) versus length ( l ) are plotted for two simple pendulums located on two different planets, A and B. The graphs are straight lines passing through the origin. The line for planet A makes an angle of 30^ with the length axis, while the line for planet B makes an angle of 60^ with the length axis. The ratio of the acceleration due to gravity on planet A to that on planet B

Step-by-step solution

The time period of a simple pendulum is given by T = 2 l g . Squaring both sides, we get T^2 = ( 4 ^2 g )l . This represents a straight line equation y = mx where y = T^2 , x = l , and the slope m = 4 ^2 g . Thus, the acceleration due to gravity g is inversely proportional to the slope of the T^2 versus l graph. For planet A, the slope m_A = (30^ ) = 1 3 . For planet B, the slope m_B = (60^ ) = 3 . The ratio of acceleration due to gravity is: g_A g_B = m_B m_A = 3 1 3 = 3 . Answer: 3