This page provides key dynamics formulae used in engineering and physics, including equations for linear motion, constant acceleration, simple harmonic motion and Newton’s laws of motion.
The information supports engineers and students in analysing motion, forces and energy relationships in mechanical systems and real-world applications.
The following equations describe motion under different conditions and are widely used in mechanical engineering analysis and design.
For constant velocity. Distance moved at constant velocity
s = v . t
v = v_o + a . t
s = v_o . t + 1/2 . a . t2
v2 = v_o2 + 2 . a . s
Simple Harmonic Motion is characterised by the relationship that the rate of
acceleration of a body towards a central position is directly related to the distance
of the body from the central position.
The result is a cyclic motion with a frequency (Cycles/s) = ω / 2 . p Cycles/s.
Note: ω = angular velocity in rads/s
f ''(x) = - ω2 . x
The solution for the displacement from and the velocity towards the central position is below;
x = a . cos ( ω. t + ε )
x ' = - ω . a . sin ( ω . t + ε )
Newtons First Law;
Every body continues in a state of rest or of uniform rectilinear
motion unless acted upon by a force.
The time rate of change of linear momentum of a body is proportional to the unbalanced force acting on the body and occurs in the direction in which the force acts.
To each action (or force) there is an equal and opposite reaction. The mutual force of the two bodies acting upon each other are equal in magnitude, opposite in direction and are colinear.
Momentum
Momentum = m . v
Force
F = m . a
The mass moment of inertia of a body about an axis has been defined as the sum of the products of mass-elements and the square of their distance from the axis
For a mass rotating about a centre. The force tending to accelarate the mass
towards the centre and restraining the mass to move around the centre is the centripetral force.
The reaction to this force tending the accelarate the mass away is the centrifugal force.
(If the string breaks the mass would fly away under the effect of the centrifugal force.
Angular acceleration α = (ω2 - ω1) / t
Acceleration Torque T = I . α
Related topics: Vectors and Forces, Kinematics, and Atomic Physics.