Potential functions Potential functions arise frequently in physics and engineering. A potential function has the property that a field of interest (for example, an electric field, a gravitational field, or a velocity field) is the gradient of the potential (or sometimes the negative of the gradient of the potential). (Potential functions are considered in depth in Chapter 17.) 82. Gravitational potential The gravitational potential associated with two objects of mass M and m is φ = − GMm / r, where G is the gravitational constant. If one of the objects is at the origin and the other object is at P ( x, y, z), then r 2 = x 2 + y 2 + z 2 is the square of the distance between the objects. The gravitational field at P is given by F = − ∇ φ where ∇ φ is the gradient in three dimensions. Show that the force has a magnitude. | F | = GMm/ r 2. Explain why this relationship is called an inverse square law.
Potential functions Potential functions arise frequently in physics and engineering. A potential function has the property that a field of interest (for example, an electric field, a gravitational field, or a velocity field) is the gradient of the potential (or sometimes the negative of the gradient of the potential). (Potential functions are considered in depth in Chapter 17.) 82. Gravitational potential The gravitational potential associated with two objects of mass M and m is φ = − GMm / r, where G is the gravitational constant. If one of the objects is at the origin and the other object is at P ( x, y, z), then r 2 = x 2 + y 2 + z 2 is the square of the distance between the objects. The gravitational field at P is given by F = − ∇ φ where ∇ φ is the gradient in three dimensions. Show that the force has a magnitude. | F | = GMm/ r 2. Explain why this relationship is called an inverse square law.
Potential functions Potential functions arise frequently in physics and engineering. A potential function has the property that a field of interest (for example, an electric field, a gravitational field, or a velocity field) is the gradient of the potential (or sometimes the negative of the gradient of the potential). (Potential functions are considered in depth in Chapter 17.)
82. Gravitational potential The gravitational potential associated with two objects of mass M and m is φ = −GMm/r, where G is the gravitational constant. If one of the objects is at the origin and the other object is at P(x, y, z), then r2= x2 + y2 + z2 is the square of the distance between the objects. The gravitational field at P is given by F = − ∇φ where ∇φ is the gradient in three dimensions. Show that the force has a magnitude. |F| = GMm/r2.Explain why this relationship is called an inverse square law.
Kepler's Third Law of Orbital Motion states that you can approximate the period P (in Earth years) it takes a planet to complete one orbit of the sun
d
where d is the distance (in astronomical units, AU) from the planet to the sun. How many Earth years would it take for a
using the function p =
planet that is 0.76 AU from the sun?
A. 0.83
ов. 0.19
ос. 0.66
O D. 0.22
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