The pendulum in an antique clock consists of a thin, uniform...

The pendulum in an antique clock consists of a thin, uniform rod of length 30.00 $\mathrm{cm}$ and negligible mass, on which is mounted a solid disk of mass $692.2 \mathrm{~g}$ and diameter $6.350 \mathrm{~cm},$ as shown in Fig. 13.37. A nut of negligible mass allows the disk to be moved up or down slightly to adjust the clock's timekeeping. By how much does the period of the pendulum change when the nut is moved upward 1.000 $\mathrm{mm}$ (i.e., when the small distance $h$ shown in the figure increases by $1 \mathrm{~mm}$ )?

Key Concepts

Small-Angle Approximation

The small-angle approximation simplifies the analysis of pendulum motion by assuming that for small angles (typically less than about 10 degrees), the sine of the angle is approximately equal to the angle itself (in radians). This approximation allows the complex equation of motion for the pendulum to be linearized, resulting in a simple harmonic oscillator model. It makes the derivation of the period formula tractable and is widely used in physics problems involving oscillations.

Parallel Axis Theorem

The parallel axis theorem is used to calculate the moment of inertia of a body about any axis, given its moment of inertia about a parallel axis that passes through its center of mass and the distance between the two axes. This theorem is critical when analyzing physical pendulums because the pivot point is often not located at the center of mass. By applying the parallel axis theorem, one can adjust the moment of inertia to reflect the contribution from the mass distribution relative to the actual axis of rotation.

Physical Pendulum

A physical pendulum is an extended body that oscillates about a pivot point under the influence of gravity. Unlike the idealized simple pendulum which assumes all mass is concentrated at a point, the physical pendulum takes into account the distribution of mass in the object. Its period of small-amplitude oscillations is determined by both the moment of inertia and the distance between the pivot point and the center of mass, through the formula T = 2??(I/(mgd)).

Moment of Inertia

The moment of inertia is a measure of an object's resistance to angular acceleration about a given axis. For a physical pendulum, it quantifies how the mass is distributed relative to the pivot point. A higher moment of inertia indicates that more mass is located farther from the pivot, which generally leads to a longer period of oscillation. Calculating the moment of inertia is essential for accurately determining the dynamics of a swinging pendulum.

Transcript

Please give Ace some feedback