15.11: Measuring Acceleration Due to Gravity

Consider a coffee mug hanging on a hook in a pantry. If the mug gets knocked, it oscillates back and forth like a pendulum until the oscillations die out.

A simple pendulum can be described as a point mass and a string. Meanwhile, a physical pendulum is any object whose oscillations are similar to a simple pendulum, but cannot be modeled as a point mass on a string because its mass is distributed over a larger area. The behavior of a physical pendulum can be modeled using the principles of rotational motion and the concept of the moment of inertia. For both a simple and a physical pendulum, the restoring force is the force of gravity. With a simple pendulum, gravity acts on the center of the pendulum bob, while in the case of a physical pendulum, the force of gravity acts on the center of mass (CM) of the object.

The period (T) of a simple pendulum depends on its length and acceleration due to gravity (g). The period is entirely independent of other factors, such as mass and maximum displacement. Given the dependence of T on g, if the length of a pendulum and the period of oscillation is precisely known, they can be used to measure the acceleration due to gravity. This method for determining gravity can be very accurate.

A physical pendulum can also be used to measure the free-fall acceleration due to gravity at a particular location on Earth's surface, thousands of measurements of which have been made during geophysical prospecting.

An object in a gravitational field experiences acceleration due to the gravitational force.

Using a physical pendulum, the value of acceleration due to gravity can be measured.

Consider a uniform rod of length L and mass m suspended from one end. The distance between the pivot point and the rod's center of mass is half the rod's length.

Recall the equation for the rotational inertia of a rod, rotating about a perpendicular axis through its center of mass.

Using the parallel-axis theorem, the rotational inertia about a perpendicular axis through one end of the rod is determined.

Recall the time period of the physical pendulum. Squaring both sides and substituting the value of the distance between the pivot point and the center of mass and the moment of inertia, the formula for the acceleration due to gravity is determined.

If the length of the rod is one meter and it has a time period of 1.64 seconds, the value of g is calculated to be 9.8 meters per second square.