Examples of gravitational constant in the following topics:
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- $P^2 \propto a^3$ , with a constant of proportionality of:
- In the case of a circular orbit, the proportionality constant is as follows:
- where $T$ is the period, $G$ is the gravitational constant, and $R$ is the distance between the center of mass of the two bodies.
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- The pressure is the scalar proportionality constant that relates the two normal vectors:
- For fluids near the surface of the earth, the formula may be written as $p = \rho g h$, where $p$ is the pressure, $\rho$ is the density of the fluid, $g$ is the gravitational acceleration, and $h$ is the depth of the liquid in meters.
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- Vector fields are often used to model the speed and direction of a moving fluid throughout space, for example, or the strength and direction of some force, such as the magnetic or gravitational force, as it changes from point to point.
- A gravitational field generated by any massive object is a vector field.
- For example, the gravitational field vectors for a spherically symmetric body would all point towards the sphere's center, with the magnitude of the vectors reducing as radial distance from the body increases.
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- The gravitational potential associated with a mass distribution given by a mass measure $dm$ on three-dimensional Euclidean space $R^3$ is:
- If there is a continuous function $\rho(x)$ representing the density of the distribution at $x$, so that $dm(x) = \rho (x)d^3x$, where $d^3x$ is the Euclidean volume element, then the gravitational potential is:
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- Conic sections are important in astronomy: the orbits of two massive objects that interact according to Newton's law of universal gravitation are conic sections if their common center of mass is considered to be at rest.
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- An indefinite integral is defined as $\int f(x)dx = F(x)+ C$, where $F$ satisfies $F'(x) = f(x)$ and where $C$ is any constant.
- We can add any constant $C$ to $F$ without changing the derivative.
- With this in mind, we define the indefinite integral as follows: $\int f(x)dx = F(x)+ C$ , where $F$ satisfies $F'(x) = f(x)$ and $C$ is any constant.
- Therefore, all the antiderivatives of $x^2$ can be obtained by changing the value of $C$ in $F(x) = \left ( \frac{x^3}{3} \right ) + C$, where $C$ is an arbitrary constant known as the constant of integration.
- Apply the basic properties of indefinite integrals, including the constant, sum, and difference rules
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- A complete solution contains the same number of arbitrary constants as the order of the original equation.
- Since our example above is a first-order equation, it will have just one arbitrary constant in the complete solution.
- Therefore, the general solution is $f(x) = Ce^{-x}$, where $C$ stands for an arbitrary constant.
- You can see that the differential equation still holds true with this constant.
- For a specific solution, replace the constants in the general solution with actual numeric values.
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- If $f(x)$ is a constant, then $f'(x) = 0$, since the rate of change of a constant is always zero.
- By extension, this means that the derivative of a constant times a function is the constant times the derivative of the function.
- The known derivatives of the elementary functions $x^2$, $x^4$, $\ln(x)$, and $e^x$, as well as that of the constant 7, were also used.
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- Vector calculus is used extensively throughout physics and engineering, mostly with regard to electromagnetic fields, gravitational fields, and fluid flow.
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- The line integral finds the work done on an object moving through an electric or gravitational field, for example.