# Section 4: Gravitational and Inertial Mass

A subtlety arises when we compare the law of universal gravitation with Newton's second law of motion. The mass that appears in the law of universal gravitation is the property of the particle that creates the gravitational force acting on the other particle; for if we double , we double the force on . Similarly, the mass in the law of universal gravitation is the property of the particle that responds to the gravitational force created by the other particle. The law of universal gravitation provides a definition of gravitational mass as the property of matter that creates and responds to gravitational forces. Newton's second law of motion, F=ma, describes how any force, gravitational or not, changes the motion of an object. For a given force, a large mass responds with a small acceleration and vice versa. The second law provides a definition of inertial mass as the property of matter that resists changes in motion or, equivalently, as an object's inertia.

Figure 8: Equality of gravitational and inertial mass.

Source: © Blayne Heckel. More info

Is the inertial mass of an object necessarily the same as its gravitational mass? This question troubled Newton and many others since his time. Experiments are consistent with the premise that inertial and gravitational mass are the same. We can measure the weight of an object by suspending it from a spring balance. Earth's gravity pulls the object down with a force (weight) of , where g is the local gravitational acceleration and the gravitational mass of the object. Gravity's pull on the object is balanced by the upward force provided by the stretched spring. We say that two masses that stretch identical springs by identical amounts have the same gravitational mass, even if they possess different sizes, shapes, or compositions. But will they have the same inertial mass? We can answer this question by cutting the springs, letting the masses fall, and measuring the accelerations. The second law says the net force acting on the mass is the product of the inertial mass, , and acceleration, a, giving us: or . But g is a property of the Earth alone and does not depend upon which object is placed at its surface, while experiments find the acceleration, a, to be the same for all objects falling from the same point in the absence of air friction. Therefore, is the same for all objects and thus for . We define the universal gravitational constant, G, to make .

The principle of the universality of free fall is the statement that all materials fall at the same rate in a uniform gravitational field. This principle is equivalent to the statement that . Physicists have found the principle to be valid within the limits of their experiments' precision, allowing them to use the same mass in both the law of universal gravitation and Newton's second law.

## Measuring G

Measurements of planets' orbits about the Sun provide a value for the product GMs, where MS is the mass of the Sun. Similarly, earthbound satellites and the Moon's orbit provide a value for GME, where ME is the mass of the Earth. To determine a value for G alone requires an a priori knowledge of both masses involved in the gravitational attraction. Physicists have made the most precise laboratory measurements of G using an instrument called a "torsion balance," or torsion pendulum. This consists of a mass distribution suspended by a long thin fiber. Unbalanced forces that act on the suspended mass distribution can rotate the mass distribution; the reflection of a light beam from a mirror attached to the pendulum measures the twist angle. Because a very weak force can twist a long thin fiber, even the tiny torques created by gravitational forces lead to measurable twist angles.

Figure 9: Schematic of a torsion balance to measure the gravitational constant, G.

Source: © Blayne Heckel. More info

To measure G, physicists use a dumbbell-shaped mass distribution (or more recently a rectangular plate) suspended by the fiber, all enclosed within a vacuum vessel. Precisely weighed and positioned massive spheres are placed on a turntable that surrounds the vacuum vessel. Rotating the turntable with the outer spheres about the fiber axis modulates the gravitational torque that the spheres exert on the pendulum and changes the fiber's twist angle.

This type of experiment accounts in large part for the currently accepted value of (6.674280.00067) x 10-11 N-m2/kg2 for the universal gravitational constant. It is the least precisely known of the fundamental constants because the weakness of gravity requires the use of relatively large masses, whose homogeneities and positioning are challenging to determine with high precision. Dividing GME found from satellite and lunar orbits by the laboratory value for G allows us to deduce the mass of the Earth: 5.98 X 1024 kilograms.

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