Translational Dynamics
SI units & Physics constants
Translational dynamics investigates translational motion of objects and deals with effects that forces have on motion
Here (all units see here):
m is mass of object
is velocity vector
is linear acceleration vector
is linear momentum vector
is force vector
General formulas and definition
Definition of mass: mass, m, of a substance or a body is scalar quantity, which defines its inertial and gravitational properties. The unit of mass is kilogram, 1 kg, defined by etalon of mass as specific platinumiridium cylinder
Linear momentum is vector fefined by
Force is vector fefined by
Resultant force or net force of n forces applied to object is defined by vector sum
The polygon rule for net force for case n = 3
Some forces in Mechanics
Gravitational force
where g is freefall acceleration
Frictional force
where:
is coefficient of friction
N is normal component of total force acting on object from the surface
Spring force
where:
k is spring constant
x_{o} is length of unstretched spring
x lengths of spring stretched by force F
Newton's laws of motion in inertial reference frames
Inertial reference frame is a frame of reference which is at rest or moving at a constant velocity, which obeys the Newton's First Law
Newton's First Law
If then , that is the object is in uniform motion, or at rest
Newton's Second Law:
 for general case
 for constant mass
Newton's Third Law
where: is force on mass m_{1} applied by mass m_{2}, and is force on mass m_{2} applied by mass m_{1}, shown in the below diagram
Law of conservation of linear momentum
Total linear momentum of system of n particles is defined by vector sum
where is linear momentum of ith particle with mass and velocity
The law of conservation of linear momentum
If then
where is resultant force of all external forces applied to the system of particles
Totally inelastic collision is the collision after which the objects are moving together as one body
The velocity after inelastic collision
Totally elastic collision is the collision when total machanical energy conserves in the collision
Collision for unidirectional motion:
The velocities of objects after collision:
where is coefficient of restitution
For totally elastic collision
For totally inelastic collision
ImpulseMomentum Theorem
Impulse of force
where is average force acting on object during time interval given by integral
ImpulseMomentum theorem
where and are initial and final linear momentums of the object respectively
Motion of center of mass of system of particles
The position vector of center of mass of n particles:
where:
and are mass and radius vector of position of ith particle
is total mass of the system of particles
The velocity vector of center of mass
The Newton's Second Law for motion of center of mass
where is resultant force of all external forces applied to the system of particles
Propulsion motion of object with variable mass
Where:
m is current mass of object
is current velocity of object
is trust (or reactive force)
is velocity of ejected mass with respect to the object
The thrust force
where is rate of change of mass m
The equation for motion of object with variable mass
Current speed of object with mass m starting from rest with original mass
