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…
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.
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Q1: How can a physical pendulum be used to measure acceleration due to gravity?
A physical pendulum measures gravity by relating its period of oscillation to gravitational acceleration. If the pendulum's length and period are precisely known, they can be used to calculate g using the derived formula. For example, a one-meter rod with a 1.64-second period yields g = 9.8 m/s². This method is highly accurate and has been used thousands of times in geophysical prospecting to measure free-fall acceleration at different Earth locations.
Q2: What is the difference between a simple pendulum and a physical pendulum?
A simple pendulum is modeled as a point mass on a string, while a physical pendulum is any real object with distributed mass that oscillates like a pendulum. Both use gravity as the restoring force, but gravity acts on the bob's center in a simple pendulum and on the center of mass in a physical pendulum. A coffee mug hanging on a hook exemplifies a physical pendulum, oscillating until friction dissipates the energy.
Q3: Why does the period of a pendulum depend on length and gravity but not mass?
The period of a simple pendulum depends only on its length and acceleration due to gravity, remaining independent of mass and maximum displacement. This relationship arises from the physics of oscillatory motion, where gravitational force and inertia scale proportionally with mass, canceling its effect on period. This independence makes pendulums reliable tools for measuring gravity at specific locations.
Q4: How does the parallel-axis theorem apply to calculating a pendulum's rotational inertia?
The parallel-axis theorem relates rotational inertia about different axes. For a uniform rod rotating about one end, the theorem uses the rod's rotational inertia about its center of mass and the distance between the pivot and center of mass. This calculation is essential for deriving the period formula and ultimately determining acceleration due to gravity from pendulum measurements.
Q5: What role does the center of mass play in physical pendulum oscillations?
In a physical pendulum, gravity acts on the center of mass, creating the restoring torque that drives oscillations. The distance between the pivot point and the center of mass directly affects the period of oscillation. For a uniform rod suspended from one end, this distance equals half the rod's length, which is substituted into the period formula to calculate gravitational acceleration.
Q6: What factors determine the accuracy of measuring gravity with a pendulum?
Accurate gravity measurements require precise knowledge of the pendulum's length and period of oscillation. The method is highly accurate when these parameters are carefully measured. Physical pendulums have been used extensively in geophysical prospecting, with thousands of measurements made to determine free-fall acceleration at different locations on Earth's surface.
Q7: How is the formula for acceleration due to gravity derived from pendulum motion?
The gravity formula is derived by starting with the physical pendulum's period equation, then squaring both sides and substituting the moment of inertia and the distance between pivot and center of mass. Rearranging yields g in terms of measurable quantities: period and pendulum geometry. This algebraic manipulation transforms the period relationship into a direct calculation of gravitational acceleration.