91Ó°ÊÓ

For a simple pendulum, the period T is given by the formula: T = 2\pi\sqrt{\frac{l}{g}} where l is the length of the pendulum and g is the acceleration due to gravity (9.81 m/s²). Plug in the given length (1.000 m) and calculate the period: T = 2\pi\sqrt{\frac{1.000}{9.81}}.

The actual period of the grandfather clock is 1% faster than that of a simple pendulum. So, multiply the calculated period in step 1 by 0.99 to get the actual period: Actual T = 0.99 * T.

The pendulum's mass is distributed between the uniform thin rod and the point mass at its end. The moment of inertia of a uniform thin rod rotating about its end is given by I_rod = \frac{1}{3} m_rod * L^2. The moment of inertia of a point mass at the distance L from the axis is given by I_mass = m_mass * L^2.

The total moment of inertia of this pendulum is I_total = I_rod + I_mass. We know that for any physical pendulum, the period is given by T = 2\pi\sqrt{\frac{I}{mgr}}, where r is the distance from the axis to the center of mass of the pendulum. From step 2, we have the actual period, T, so we can use this to solve for I_total: I_total = \left(\frac{T^2 g r}{4\pi^2}\right). We need to find the center of mass, r. For the uniform thin rod, the center of mass is halfway along its length, at 0.5 m. The point mass's center of mass is at its location, 1.000 m. The combined pendulum’s center of gravity can be found by considering the two masses in parallel: r = \frac{m_rod * 0.5 + m_mass * 1.0}{m_rod + m_mass}. Plug in the moments of inertia and the center of mass values, and solve the equation for I_total. Once you have I_total, find the fraction of the total mass of the pendulum in the uniform thin rod, which is m_rod / (m_rod + m_mass). Finally, multiply this fraction by 100 to get the percentage.