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 platinum-iridium 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 free-fall 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 m₁ applied by mass m₂, and  is force on mass m₂ applied by mass m₁, 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 i-th 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

Impulse-Momentum Theorem

Impulse of force

where is average force acting on object during time interval given by integral

Impulse-Momentum 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 i-th 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