(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:<br /><br />\[ T = 2\pi \sqrt{\frac{L}{g}} \]<br /><br />where \( T \) is the period, \( L \) is the length of the pendulum, and \( g \) is the acceleration due to gravity.<br /><br />Given:<br />- \( T = 4 \) seconds<br />- \( g = 10.0 \, \text{m/s}^2 \)<br /><br />We need to solve for \( L \):<br /><br />\[ 4 = 2\pi \sqrt{\frac{L}{10}} \]<br /><br />First, divide both sides by \( 2\pi \):<br /><br />\[ \frac{4}{2\pi} = \sqrt{\frac{L}{10}} \]<br /><br />Square both sides to eliminate the square root:<br /><br />\[ \left(\frac{4}{2\pi}\right)^2 = \frac{L}{10} \]<br /><br />\[ \frac{16}{4\pi^2} = \frac{L}{10} \]<br /><br />Multiply both sides by 10 to solve for \( L \):<br /><br />\[ L = 10 \times \frac{16}{4\pi^2} \]<br /><br />\[ L = \frac{160}{4\pi^2} \]<br /><br />\[ L = \frac{40}{\pi^2} \]<br /><br />Calculating this gives approximately:<br /><br />\[ L \approx 4.05 \, \text{meters} \]<br /><br />So, the length of the pendulum should be approximately 4.05 meters.<br /><br />**Perpendicular Axes Theorem:**<br /><br />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:<br /><br />\[ I_z = I_x + I_y \]<br /><br />**Proof:**<br /><br />Consider a planar object in the XY-plane. For a small mass element \( dm \) at a point with coordinates \((x, y)\), the moment of inertia about the Z-axis is given by:<br /><br />\[ dI_z = r^2 \, dm = (x^2 + y^2) \, dm \]<br /><br />Integrating over the entire object, we have:<br /><br />\[ I_z = \int (x^2 + y^2) \, dm = \int x^2 \, dm + \int y^2 \, dm \]<br /><br />This can be separated into:<br /><br />\[ I_z = I_x + I_y \]<br /><br />where \( I_x = \int y^2 \, dm \) and \( I_y = \int x^2 \, dm \).<br /><br />Thus, the perpendicular axes theorem is proved.