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Newton’s law of gravitation

Newton’s law of gravitation can be expressed as

F 12 = G m 1 m 2 r 2 r ^ 12

where F 12 is the force on object 1 exerted by object 2 and r ^ 12 is a unit vector that points from object 1 toward object 2.

As shown in [link] , the F 12 vector points from object 1 toward object 2, and hence represents an attractive force between the objects. The equal but opposite force F 21 is the force on object 2 exerted by object 1.

The figure shows two circular objects, one smaller, labeled as mass m1 on the lower left, and the larger one labeled m2 on the upper right. The center of each object is labeled C M. A line is drawn joining the center of the objects and is labeled as r. Two red arrows and two black arrows, one each from the center of each object, are drawn toward each other. The black arrow from the center of mass 1 is labeled r hat 1 2 and the red arrow from mass 1 is labeled F 1 2. The black arrow from the center of mass 2 is labeled r hat 2 1 and the red arrow from mass 2 is labeled F 2 1.
Gravitational force acts along a line joining the centers of mass of two objects.

These equal but opposite forces reflect Newton’s third law, which we discussed earlier. Note that strictly speaking, [link] applies to point masses—all the mass is located at one point. But it applies equally to any spherically symmetric objects, where r is the distance between the centers of mass of those objects. In many cases, it works reasonably well for nonsymmetrical objects, if their separation is large compared to their size, and we take r to be the distance between the center of mass of each body.

The cavendish experiment

A century after Newton published his law of universal gravitation, Henry Cavendish determined the proportionality constant G by performing a painstaking experiment. He constructed a device similar to that shown in [link] , in which small masses are suspended from a wire. Once in equilibrium, two fixed, larger masses are placed symmetrically near the smaller ones. The gravitational attraction creates a torsion (twisting) in the supporting wire that can be measured.

The constant G is called the universal gravitational constant    and Cavendish determined it to be G = 6.67 × 10 −11 N · m 2 /kg 2 . The word ‘universal’ indicates that scientists think that this constant applies to masses of any composition and that it is the same throughout the Universe. The value of G is an incredibly small number, showing that the force of gravity is very weak. The attraction between masses as small as our bodies, or even objects the size of skyscrapers, is incredibly small. For example, two 1.0-kg masses located 1.0 meter apart exert a force of 6.7 × 10 −11 N on each other. This is the weight of a typical grain of pollen.

The figure illustrates the Cavendish experiment. A horizontal bar, centered on a stand, supports two spheres of mass M, one at either of its ends. The point where the bar sits on the stand is labeled
Cavendish used an apparatus similar to this to measure the gravitational attraction between two spheres ( m ) suspended from a wire and two stationary spheres ( M ). This is a common experiment performed in undergraduate laboratories, but it is quite challenging. Passing trucks outside the laboratory can create vibrations that overwhelm the gravitational forces.

Although gravity is the weakest of the four fundamental forces of nature, its attractive nature is what holds us to Earth, causes the planets to orbit the Sun and the Sun to orbit our galaxy, and binds galaxies into clusters, ranging from a few to millions. Gravity is the force that forms the Universe.

Problem-solving strategy: newton’s law of gravitation

To determine the motion caused by the gravitational force, follow these steps:

  1. Identify the two masses, one or both, for which you wish to find the gravitational force.
  2. Draw a free-body diagram, sketching the force acting on each mass and indicating the distance between their centers of mass.
  3. Apply Newton’s second law of motion to each mass to determine how it will move.

Questions & Answers

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Practice Key Terms 2

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