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Direction Convention for Gravitational Motion
by Ron Kurtus (revised 15 May 2011)
Just as there is a convention as to which direction is positive and which is negative for gravity equations, it is also necessary to establish a similar convention for gravitational equations.
The direction of the gravitational attraction of the smaller object toward the larger is defined as positive, while the direction away from the object is negative. Also, the directions are with respect to the starting point of the smaller object.
Since gravitation also allows orbiting around the larger object, a form of radial coordinate system or r-t system is used with the zero-point at center of the smaller object.
Vectors and scalars used are similar to those used in the gravity convention.
Questions you may have include:
- What is the gravity coordinate system?
- What is the gravitation coordinate system?
- What are the vectors and scalars used?
This lesson will answer those questions.
Useful tool: Metric-English Conversion
Gravity coordinate system
The gravity convention for direction considers the motion toward the Earth—in the direction of gravity—as positive and upward or away from Earth as negative. The directions are with respect to the starting point of an object held above the ground.
When studying gravity motion equations, a modified x-y coordinate system is used. We defined a direction convention that set the starting point of an object as the zero-point and designated that downward—toward the ground—was the positive direction and upward was negative. This meant essentially inverting or flipping the y-axis to change the common directions. Motion in the x-direction remained the same.
Gravity coordinate system
(See Convention for Direction in Gravity Equations for more information.)
Gravitation coordinate system
Our considerations of gravitation concern two objects in space and the interaction between them. Although both objects move with respect to the center of mass (CM) between them, it is often more convenient to consider the motion of a smaller object with respect to a larger one. This reduces the need for two motion equations, such that only a single equation is needed.
Since the gravitational force on the smaller object (m) is toward the larger object (M), that direction is designated as the positive direction. Thus, when object m moves away from object M, the velocity is in a negative direction.
Because gravitation also allows orbiting around the larger object, a form of radial-tangential coordinate system—or r-t system—is used instead of the x-y coordinate system of gravity. The result is that the coordinate system is inverted, similar to that in gravity.
Gravitation coordinate system
Gravitational vectors and scalars
Vectors are divided into those along the axis and those that are perpendicular to the axis of rotation. Scalar quantities are similar to those used in the gravity convention.
Radial vectors
Vectors used in gravitation equations along the axis between the two objects include:
- F: Gravitational force
- R: Radial displacement
- vR: Radial velocity
Velocity and displacement vectors pointing away from the CM are negative vectors.
Tangential vectors
Linear tangential velocity (vT) is the primary tangential vector used for gravitational orbit equations. This is a straight-line velocity that is perpendicular to the axis of rotation and tangent to the curved path of the object. There is no straight-line displacement tangential vector.
Pseudo-vectors that follow a curved path include:
- ω: Angular velocity
- θ: Angular displacement
Scalars
Scalar quantities related to the vectors in the gravitational equations include:
- s: Speed
- d: Distance
- l: Section of curved path
- m: Mass
All scalar magnitudes are positive numbers or quantities.
Magnitude of vectors
The magnitude or absolute value of a vector is a scalar quantity. For example, the absolute value of the velocity vector is the speed of the object:
|−vR| = s
where
- −vR is the radial velocity away from the larger object
- |−vR| is the absolute or positive value of the velocity, independent of the direction = speed
Summary
The gravitational direction convention is similar to the convention used for gravity equations. The direction of the gravitational attraction of the smaller object toward the larger is defined as positive, while the direction away from the object is negative. Directions are with respect to the starting point of the smaller object.
A form of radial coordinate system or r-t system is used with the zero-point at center of the smaller object. Vectors and scalars used are similar to those used in the gravity convention.
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Resources and references
Websites
Acceleration due to Gravity Calculations - from Western Washington University
Books
Top-rated books on Simple Gravity Science
Top-rated books on Advanced Gravity Physics
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Direction Convention for Gravitational Motion