4. A uniform rod of mass M and length L is pivoted at a point a distance L/3 from the top end, as shown on the right. The rod is pulled back so that it makes a small angle with the vertical and is then released. (a) Using integral calculus, show that the rotational inertia for the rod around its pivot is ML/9. L/3 (b) i. Using Newton's 2nd Law in rotational form, write but do NOT solve a differential equation in terms of M, L, and physical constants, as appropriate, that can be used to determine the angular displacement theta as a function of time t. ii. Using the differential equation from part (b)i, show that the period of oscillation for the rod is given by the expression T = 2 2L 3g (intel) CORE 17 8th Gen An experiment is performed with thin, uniform metal rods of different lengths, each pivoted around a point one third the length of the rod from the top end and used as a physical pendulum, as shown in the figure on the previous page. Each rod is pulled back so that it makes a small angle with the vertical and is then released, and the period of oscillation is measured. The data are recorded in the table below. Length (m) 0.5 1.0 1.5 2.0 2.5 Period (s) 1.2 1.6 2.0 2.3 2.6 (c) Indicate below which quantities should be graphed to yield a straight line whose slope could be used to calculate a numerical value for the acceleration due to gravity g. Vertical axis: Use the remaining rows in the table above, as needed, to record any quantities that you indicated that are not given. (d) Plot the straight line data points on the graph below. Clearly scale and label all axes, including units if appropriate. Draw a straight line that best represents the data. Horizontal axis: 1 1 1 I 1 1 1 I t I 1 1 1 1 1 1 (e) Using your straight line, determine an experimental value for g.