Acceleration Due to Gravity in Physics

The acceleration due to gravity, g, was determined by dropping a metal bearing and measuring the free-fall time with a pendulum of known period. The measured value is 9.706 m/s2 with a standard deviation of 0.0317, which does not fall within the range of known terrestrial values. Centrifugal forces and altitude variations cannot account for the discrepancy. The calculation is very sensitive to the measured drop time, making it the likely source of error.

Theory

The acceleration due to gravity is the acceleration experienced by an object in free-fall at the surface of the Earth, assuming air friction can be neglected.

It has the approximate value of 9.80 m/s2, although it varies with altitude and location. The gravitational acceleration can be obtained from theory by applying Newton’s Law of Universal Gravitation to find the force between the Earth and an object at its surface. Newton’s Law of Universal Gravitation for the force between two bodies is

where m1 and m2 are the masses of the bodies, r12 is the distance between the centers of mass of the bodies, and G is the Universal Gravitational Constant which has a current accepted value of 6.

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673 × 10-11 Nm2/kg2. The force between the Earth and a mass, m, would be where ME and RE are the mass and radius of the Earth, respectively. For a particular location, G, ME, and RE are constant and may be grouped under a single constant, g.

For obvious reasons, g is sometimes called the local gravitational constant.

It will be numerically equivalent to the acceleration due to gravity on a spherical, non-rotating planet. The real acceleration due to gravity will be different than the above due to “centrifugal” and Coriolis effects. The values that follow were taken from the CRC Handbook of Chemistry and Physics, 75th ed. and illustrate the variability of the value. As expected, the value is lower at the equator due to centrifugal force.

In this experiment, g was measured using kinematics. A metal bearing was dropped from a known height and the time was measured. The kinematic equation that gives position as a function of time is We will apply this equation to a “drop” (v0 = 0) of height, h, as shown below. Making these substitutions, we obtain

Rearranging

Experimental

The solenoid electromagnet was a simple coil of #18 wire with an iron core. The power source for both solenoids was a standard LabVolt regulated power supply. The steel bearing had a diameter of 1.6 cm and a mass of 28.4 g. The physical pendulum consisted of an aluminum rod which is weighted at the bottom. A stopwatch was used to record the period of the pendulum. The distance was measured with a standard meter stick.

The period of the pendulum was measured by measuring the time for five oscillations and dividing. The experiment was arranged as shown in Figure 2a. The pendulum was pulled away from equilibrium and held in place by an electromagnet. The bearing was held in place by another solenoid wired to the same power supply. A piece of spark tape was attached to the inside surface of the pendulum. When the power supply was shut off, the bearing and pendulum were released simultaneously. The bearing contacts the pendulum as shown in Figure 2b, leaving a mark on the tape. The distance was then measured. After several calibration runs, ten experimental runs were performed. The results were obtained utilizing the same time and ten measured distances.

Results and Discussion

The acceleration due to gravity was measured to be 9.706 m/s2 with a standard deviation of 0.0317. The values quoted in the theory section show that this measurement is well outside the expected range. The difference between the equator and the poles is only about 0.05, and these values differ from 9.80 by only 0.02 to 0.03. The values from the literature account for centrifugal force, but not altitude. A quick calculation would show that this is also negligible. If we recalculate the value of g from the theory section by adding 10 km to the Earth’s radius, we obtain a value that differs by only 0.03. The calculation of g from our measurements is very sensitive to time since it is squared in the calculation. We can recalculate g using the distance from run 1 to see how it might affect the answer. If we vary the time for a quarter oscillation of the pendulum by just 0.03 s, we obtain

Time, s (at h=1.609 m)
Calculated g, m/s2
0.572
9.84
0.575
9.73
0.578
9.63

A difference of 0.003 s would correspond to a difference of 0.003 ×4×5=0.06 seconds in the measurement of five oscillations. The measurement of time clearly deserves more attention in future experiments. Human reaction time is already a few tenths of a second. Future experimental designs should seek to measure the time more accurately. More oscillations would make this measurement more accurate, but damping might become as issue. This suggests that an electronic method should be used. A good future experiment would be to measure the altitude variation of g, but this would require better accuracy than the current experiment.