1 two of newton’s law of motions 1) in the absence of any forces applied, an object at rest will...

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Two of Newton’s Law of Motions

1) In the absence of any forces applied, an object at rest will stay at rest, and a body moving at a constant velocity in straight line continues doing so indefinitely.

2) When a force is applied to an object, it accelerates. The acceleration is in the direction of the force and proportional to its strength, and is also inversely proportional to the mass (m) of the object. In suitable units:

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Newton’s Law of Gravitation

Gravitational force: an attractive force that exists between all objects with mass; an object with mass attracts another object with mass; the magnitude of the force is directly proportional to the masses of the two objects and inversely proportional to the square of the distance between the two objects.

Where:m1 & m2 are the masses of the two objectsr is the distance between the objectsG is the universal gravitation constant

G = 6.67384 x 10-20 km3/(kg*seconds2 )

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Consider a single object

t = timem = mass of objectF = force on object (vector)a = acceleration object (vector)x = location of object (vector)v = velocity of object (vector)

The following equationsapply:

Note: F, a, v and x are all vectors and are functions of t.

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Two Gravitating Particle Masses

m1

m2

Each particle has a scalar mass quantity

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Particle Positions

x1

x2

(0,0)

Each particle has a vector positionrelative to the origin:

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Particle Velocities

v1

v2

Each particle has a vector velocity

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Particle Accelerations

a1 a2

Each particle has a vector acceleration

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• In two dimensions, vectors can be represented by two signed floats (or doubles):

Vector Operations

• The length or magnitude of a vector is a scalar quantity and can be computed as:

• A unit length vector U with the same direction as a non-unit length vector A can be computed as:

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More Vector Ops• A unit vector U can be given a known magnitude (length) as

follows:

Where U is a unit vector and force is a scalarquantity which is to be the magnitude of F.

• Vector addition and subtraction consist of adding and subtracting the vectors’ x and y components:

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Two-body Newtonian Gravitation• Two objects of mass m1 and m2 with position vectors X1 and X2 exert a

gravitational force on each other. The magnitude of the force is given by:

• The vector F21, representing the gravitational force exerted by m2 on m1 is then:

• The net resulting force vector on any mass is the vector sum of all the force vectors generated by masses other than itself.

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Newtonian Gravitation• Newton’s second law (F = mA) can be applied to the two body

scenario:

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Newtonian Gravitation• Using velocity:

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N-Body Gravitation• Generalizing to N bodies from two bodies we must sum gravitational

force vectors induced by all N bodies except the one we are computing the acceleration for:

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N-Body Newtonian Gravitation Simulation

• Problem: Plot the position of the bodies as a function of time.

• We need to specify the initial velocity and positions of the objects.

• Next we need a numerical scheme to advance the equations in time.

• Can use Euler’s Method …. as a first approach.

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Numerical Approach

For m=1 to FinalTime/dt For n=1 to number of objects

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