For a physical pendulum, which moment of inertia should be used in \(T=2\pi\sqrt{I/(mgd)}\)?

In the physical-pendulum formula, what distance does \(d\) represent?

A physical pendulum has \(I=0.80\,\mathrm{kg\,m^2}\), \(m=2.0\,\mathrm{kg}\), and \(d=0.40\,\mathrm{m}\). Find its small-angle period.

A rigid body is pivoted exactly at its center of gravity, so \(d=0\). Does gravity provide a small-angle restoring torque about the pivot?

A uniform rod of length \(1.2\,\mathrm{m}\) pivots about one end. Use \(I=\frac{1}{3}mL^2\) and \(d=L/2\) to find the period.

A rigid body has \(m=1.5\,\mathrm{kg}\), \(d=0.20\,\mathrm{m}\), and small-angle period \(1.6\,\mathrm{s}\). Find \(I\) about the pivot.

A physical pendulum has \(I=0.45\,\mathrm{kg\,m^2}\), \(m=1.2\,\mathrm{kg}\), and \(d=0.30\,\mathrm{m}\). Find its small-angle angular frequency.

If \(I\) about the pivot is doubled while \(m\) and \(d\) stay fixed, by what factor does the physical-pendulum period change?

A point mass at distance \(L\) from a pivot has \(I=mL^2\) and \(d=L\). Show that the physical-pendulum period reduces to the simple-pendulum period.

Define the equivalent simple-pendulum length \(L_{\mathrm{eq}}\) for a physical pendulum by matching \(T=2\pi\sqrt{L_{\mathrm{eq}}/g}\). Derive \(L_{\mathrm{eq}}\).

A uniform disk of radius \(R\) pivots about a point on its rim. Use \(I_{\mathrm{cm}}=\frac{1}{2}mR^2\) and the parallel-axis theorem to find its small-angle period.

A physical pendulum has \(I=0.12\,\mathrm{kg\,m^2}\), \(m=0.80\,\mathrm{kg}\), and \(d=0.15\,\mathrm{m}\). Find its angular frequency and period.

A thin hoop of radius \(R\) pivots about a point on its rim. Use \(I_{\mathrm{cm}}=mR^2\) to derive its small-angle period.

A physical pendulum with \(m=2.0\,\mathrm{kg}\), \(I=0.64\,\mathrm{kg\,m^2}\), and \(d=0.25\,\mathrm{m}\) has period \(2.6\,\mathrm{s}\). Estimate the local \(g\).

A uniform rod of length \(L\) pivots at a point a distance \(a\) from its center. Use \(I=I_{\mathrm{cm}}+ma^2\), with \(I_{\mathrm{cm}}=\frac{1}{12}mL^2\), to derive the small-angle period.

A physical pendulum has equivalent length \(L_{\mathrm{eq}}=I/(md)\). Show that its angular frequency can be written as \(\omega=\sqrt{g/L_{\mathrm{eq}}}\).

For the rod in the previous setup, derive the pivot offset \(a\) that minimizes the period, and state the minimum period in terms of \(L\) and \(g\).

Compare a uniform rod pivoted about one end with a simple pendulum whose length is the rod length \(L\). Derive the ratio of their small-angle periods and interpret why they differ.

A physical pendulum has known \(m\) and \(d\), but unknown pivot moment of inertia \(I\). You measure small-angle periods \(T_1\) and \(T_2\) in gravitational fields \(g_1\) and \(g_2\). Derive a consistency condition on the measurements and an expression for \(I\).

A rigid body has center-of-mass moment of inertia \(I_{\mathrm{cm}}\). It is pivoted a distance \(a\) from its center of mass. Derive the pivot distance that minimizes the small-angle period, and express the minimum period in terms of \(I_{\mathrm{cm}}\), \(m\), and \(g\).