(c) Determine the length 'of a simple pendulum that will swing back and forth in simp! harmonic motion with a period of 4 seconds (Use the value of acceleration due to gravity g=10.0m/s^2 (2 marks) State and prove the perpendicular axes theorem for the moment of inertia.

(c) To determine the length of a simple pendulum that will swing with a period of 4 seconds, we can use the formula for the period of a simple pendulum: where is the period, is the length of the pendulum, and is the acceleration due to gravity.Given:- seconds- We need to solve for : First, divide both sides by : Square both sides to eliminate the square root: Multiply both sides by 10 to solve for : Calculating this gives approximately: So, the length of the pendulum should be approximately 4.05 meters.**Perpendicular Axes Theorem:**The perpendicular axes theorem states that for a planar object lying in the XY-plane, the moment of inertia about an axis perpendicular to the plane (the Z-axis) is equal to the sum of the moments of inertia about the X and Y axes. Mathematically, it is expressed as: **Proof:**Consider a planar object in the XY-plane. For a small mass element at a point with coordinates \((x, y)\), the moment of inertia about the Z-axis is given by: Integrating over the entire object, we have: This can be separated into: where and .Thus, the perpendicular axes theorem is proved.