Written by tutor Maria W.
Gravitation is a universal phenomenon that describes the tendency of physical bodies to attract, or move toward each other. The gravitational force is responsible for keeping the Moon in orbit about the Earth, holding together the billions of stars in our galaxy, and most importantly, keeping you attracted to the Earth, safe from floating off into space.
Newton's Universal Law of Gravitation
In 1687, Sir Isaac Newton published a three volume book called Principia in which he introduced mathematical laws for motion and gravitation. Within this book, Newton introduced his universal law of gravitation, which states that an object attracts any other object with a force that is inversely proportional to the square of the distances between them, with the magnitude of this force given by:
where G=6.673x10-11 m3/kg*s2 (or N*m2/kg2) is called the gravitational constant, m1 and m2 are the masses (in kg) of the two objects attracted to each other, and r is the distance (in m) between the centers of the two objects. Gravitational forces are vectors, denoted here by a bolded letter, such as F, and have a magnitude given by the above equation, as well as a direction.
The force exerted on m1 by m2, written as F21, is directed along a straight line toward the center of m2. Force F12, the force exerted on m2 by m1, is equal in magnitude, but opposite in direction to F21. Therefore, F21 = -F12.
Given a group of objects with masses, the total gravitational force acting on a chosen object is the sum of the gravitational forces exerted on that object by all the others. This is known as the principle of superposition. Suppose we wish to calculate the magnitude and direction of the gravitational force on m2 due to m1 and m3, all of which are located along the x-axis:
Note that F12 is a negative value. This is because F12 is pointing in the negative x direction, while F32 is pointing in the positive x direction. Depending on the values of the masses, F2,net could be either positive or negative. If negative, F2,net is directed toward the left, and if positive, F2,net is directed toward the right. To calculate gravitational forces in more than one dimension, the force vectors must be broken up in to their x, y, and z components, then the principle of superposition can be applied.
Acceleration Due to Gravity
The acceleration caused by the gravitational force between the Earth and objects near the Earth’s surface is denoted as g = -9.81 m/s2. When the acceleration due to gravity g is written as a vector, it has a negative value because it is directed downward toward the center of the Earth. The magnitude of g for any planetary body can be calculated from first principles by starting with Newton’s Second Law, F=ma, and substituting the gravitational force for F:
where M is the mass of the large planetary body, m is the mass of the smaller object, and r is the distance between the centers of the two objects. For objects very near the surface of the planetary body, we assume that the distance r is approximately the radius of the planetary body. This allows us to assume the magnitude of g is constant near the surface, even though the equation shows that g varies inversely with the square of r. The weight of any object can be calculated by multiplying its mass and g: Wg = mg. On smaller and less massive astronomical objects than the Earth, such as the Moon, the acceleration due to gravity is g ~ 1.6 m/s2. Although an object’s mass remains the same wherever it might be in the universe, its weight would be significantly less on the Moon than on the Earth.
Gravitational Potential Energy
Potential energy can be thought of as energy a body attains or stores as a result of its position in space. For an object near the Earth’s surface, this energy is U=mgh, where m is the mass of the object, g is the acceleration due to gravity, and h is the height of an object above a reference point of h0=0. We often use this simplified equation for potential energy because we assume g remains constant near the Earth’s surface. However, for objects significantly beyond the Earth’s surface, such as satellites, the Moon, planets, other galaxies, and so on, the gravitational potential energy of a two object system is given by:
which is a scalar, not a vector as is the gravitational force. If the system contains more than two objects, the potential energy of the system is the sum of the potential energies for the object pairs (superposition principle). Suppose we wish to calculate the potential energy of the following three mass system:
The work required to arrange the three mass configurations is related to the change in gravitational potential energy of the system.
Suppose all three of the masses are initially infinitely far away. At infinity, r is very large, so the initial potential energy of the system is zero. Therefore, the work required to arrange the above configuration by bringing the masses in from infinity is:
This will give a positive value for work, since Uf is also negative. A negative value for work in this case indicates that work is being done against the attractive gravitational force. This would happen if we wanted calculate the work required to take the masses in Figure 4 out to infinity.
If a projectile is fired upward from a planetary body with a large enough speed, it can escape the gravitational pull of the planetary body, thus continuing off in to space. The minimum required speed of the projectile for this to occur can be calculated by considering the conservation of energy:
Assuming the final velocity of the object at infinity is zero, as well as the gravitational potential energy at a distance of infinity, we derive below the equation for the escape velocity of a projectile:
where M is the mass of the large planetary body, m is the mass of the projectile, and r is the radius of planetary body. We use the radius of the planetary body for r in this case because we assume the projectile is initially sitting on or very near the surface of the Earth. The escape speed for Earth is about 11.2 km/s.
Uniform Circular Motion: Orbits in Space
In order for an object to remain in a stable circular orbit about a larger body, the gravitational force must balance the centripetal force. Starting from Newton’s Second Law, F=ma, and inserting the gravitational force for F and centripetal acceleration for a, we can derive an equation for the tangential velocity vt, also known as the orbital speed, of the orbiting body, m2. Note that the gravitational force on m2 is directed inward toward the center of the orbit. The centripetal acceleration of m2 is also always directed inward, therefore the gravitational force and the centripetal acceleration of m2 are always of the same sign.
In fact, from this equation, we see that the mass of the orbiting object does not even need to be known. Additionally, the period of orbit can also be related to the distance between the two objects if we substitute the following equation in for vt:
Then the previous equation we derived for the tangential speed is now:
And solving for the period squared, we arrive at the following equation, which relates the square of the period (in seconds) to the cube of the distance between m1 and m2:
In the early 1600’s, Johannes Kepler published a series of three laws describing planetary motion. Nearly 80 years later, Newton recognized that Kepler’s Laws arise from his own laws of motion and the law of universal gravitation. Following is a summary of Kepler’s Three Laws as they pertain to our solar system:
1. All planets move in elliptical orbits, with the Sun at one focus of the ellipse.
The eccentricity of an ellipse is a value between zero and one, which describes how much the shape of an ellipse deviates from that of a circle. A circle has an eccentricity of zero, and the two foci merge to a point in the exact center. The eccentricity of the Earth’s orbit is only 0.0167, so the Earth orbits in very nearly a circular path about the Sun.
Note: The semi-major axis of the ellipse is denoted by a.
2. A line drawn from the Sun to any planet sweeps out equal areas in equal time intervals.
As a result, a planet will move slowly when it is farthest away from the Sun, and more rapidly when it is closest to the Sun.
3. The square of the orbital period (T) of any planet is proportional to the cube of the semi-major axis (a) of its orbit.
If we approximate the elliptical orbit as a circular orbit, and replace the semi-major axis a with r, this equation is the same as derived at the end of the previous section, where M is the mass of the body at one foci of the ellipse.
Note: Period (T) must be in seconds, semi-major axis (a) in meters, and mass (M) in kilograms.
Einstein’s View of Gravitation
No review of gravitation would be complete without mentioning Einstein and the curvature of spacetime. Thus far, we have discussed gravitation as being an attractive force between objects. However, Einstein introduced a whole new way of thinking about gravitation by positing that gravitation is due to a curvature of space created by objects with mass. Spacetime is a way of describing the universe as existing in four dimensions: three spatial dimensions (x,y,z) and one additional dimension devoted to time.
Imagine spacetime as a flat rubber sheet in empty space. Now visualize what will happen when a massive object is placed on the rubber sheet. The rubber sheet, or spacetime, stretches and deforms due to the presence of the massive body, becoming curved in the vicinity of a massive object. Suppose now that a body with a smaller mass and zero initial velocity begins to move toward the massive object from infinity. In Einstein’s view, this smaller mass moves toward the massive body not because of the gravitational force, but because of the curvature of spacetime near the massive body. If Einstein’s theory is correct, light would also be deflected by the curvature of spacetime near a massive object. This effect, known as gravitational lensing, was proven in May 1919 during a total solar eclipse. While Einstein’s theory of gravitation is more accurate than Newton’s, Newton’s theory is much simpler, and is a good approximation in most situations.
The magnitude of the gravitational force between two masses, m1 and m2, is F. If m2 is now doubled and the distance between the masses is also doubled, by what factor does the gravitational force increase/decrease?
In Figure 2, calculate the magnitude and direction of the gravitational force exerted by m1 and m3 on m2 . m1 =1.0 kg, m2 =2.0 kg, m3 =3.0 kg, and d =2.0 m.
Calculate the acceleration due to gravity on the surface of Mars. The mass of Mars is 6.419x1023 kg, and the diameter of Mars is 6794 km.
What is the work required to move the masses in Figure 4 from their current position out to infinity? m1=1.0 kg, m2=2.0 kg, m3=3.0 kg, and d=2.0 m.
Calculate the escape speed of a projectile on Mars. The mass of Mars is 6.419x1023 kg, and the diameter of Mars is 6794 km.
Calculate the period of orbit and tangential speed (orbital speed) of Venus around the Sun, assuming that Venus orbits the Sun in a circle with a radius of 108.2x106 km. In fact, this is a very good assumption since the eccentricity of Venus’ orbit is only 0.007. The mass of the Sun is 1.989x1030 kg.
In Figure 8, between which two positions will the planet have the fastest orbital speed about the Sun:
Choose the FALSE statement below about Newton’s and Einstein’s view of gravitation: