Q7. The Period T_(1) of the oscillation of a simple Pendulum depend. only on the mass m of the bob, the length L of the thread and the accelenturing. due to gravity of the place concerned if T=k m^2 L^2 g^2 where x, y, z, k are unknown numbers and that k=2 pi , determine the values of the Unknown numbers and hence write down the linear equation for the simple Pendulum (6mks)

To determine the values of the unknown numbers \(x\), \(y\), and \(z\) in the equation \(T = km^x l^y g^z\), we need to use the known constants and the physical principles of the pendulum.<br /><br />Given that \(k = 2\pi\), we can rewrite the equation as:<br /><br />\[ T = 2\pi m^x l^y g^z \]<br /><br />Now, let's analyze the physical meaning of each term in the equation:<br /><br />- \(T\) represents the period of the pendulum.<br />- \(m\) represents the mass of the pendulum.<br />- \(l\) represents the length of the pendulum.<br />- \(g\) represents the acceleration due to gravity.<br /><br />The period of a simple pendulum is known to be proportional to the square root of the length of the pendulum and inversely proportional to the square root of the acceleration due to gravity. Mathematically, this relationship is given by:<br /><br />\[ T = 2\pi \sqrt{\frac{l}{g}} \]<br /><br />Comparing this with the given equation \(T = 2\pi m^x l^y g^z\), we can see that the mass \(m\) does not appear in the known relationship for the period of a simple pendulum. Therefore, we can conclude that \(x = 0\).<br /><br />Next, we compare the terms involving \(l\) and \(g\). We know that the period is proportional to the square root of the length and inversely proportional to the square root of the acceleration due to gravity. This gives us:<br /><br />\[ \sqrt{l} \rightarrow l^{\frac{1}{2}} \]<br />\[ \sqrt{g} \rightarrow g^{\frac{1}{2}} \]<br /><br />Therefore, we can conclude that \(y = \frac{1}{2}\) and \(z = -\frac{1}{2}\).<br /><br />Substituting these values back into the equation, we get:<br /><br />\[ T = 2\pi m^0 l^{\frac{1}{2}} g^{-\frac{1}{2}} \]<br /><br />Simplifying further, we have:<br /><br />\[ T = 2\pi \sqrt{\frac{l}{g}} \]<br /><br />This is the known relationship for the period of a simple pendulum. Therefore, the values of the unknown numbers are \(x = 0\), \(y = \frac{1}{2}\), and \(z = -\frac{1}{2}\).<br /><br />The linear equation for the simple pendulum in terms of the given variables is:<br /><br />\[ T = 2\pi \sqrt{\frac{l}{g}} \]