Particle Dynamics:
Represents objects in motion as point masses, with no identifiable dimensions.
This “particle dynamics” model need not necessarily only apply to objects whose dimensions are negligibly small on an engineering scale.
For example, useful analyses of the orbit of the Moon and Earth around the Sun can be done by assuming that they are all point masses.
Rigid Body Motion:
Even if it is necessary to consider the dimensions of objects in motion it may be acceptable to neglect any deformations (e.g., flexure, extension) in the objects themselves.
In other words, the objects themselves may be treated as rigid bodies, which can translate and rotate but cannot change shape.
Such “rigid body dynamics” models are far simpler than the alternative.
Cartesian (right-handed) reference frame:
A point in space can also be defined by a “position vector” r (as shown) which, for the present, can be thought of as the vector from the origin of the reference frame to the point in space.
r = x i + y j + z k
In this course, we focus on moving particles, which means that their positions (or position vectors) depend on time t. A simple example might be:
r = (5 + t) i + t^2 j + 3 k
velocity:
we define the velocity v, which is a vector, as the rate of change of position with time, as follows:
v = 𝑑/𝑑𝑡𝐫 ≡ 𝐝𝐫/dt
Infinitesimal change in the position vector:
The vector r + dr.
According to the law of vector addition, the
infinitesimal vector dr has a simple physical meaning: it is simply the infinitesimal vector directed along the path of motion.
The velocity vector is always directed along the path of motion. Note that the direction of this vector is not directly related to the direction of the position vector.
The magnitude of the velocity |v| depends of course on how quickly the position vector changes with time.
For convenience, we employ the term “speed” to signify the magnitude of the velocity.
We obtain the velocity by differentiating each term of the position vector with respect to time.
Acceleration:
By definition, the acceleration is also the first derivative of velocity: it is the rate of
change of velocity with time.
Thus, acceleration is:
a= (𝑑/𝑑𝑡)𝐯 = (𝑑^2/𝑑𝑡^2)𝐫
The direction of the acceleration must always be specified.
Kinematics of Rectilinear Motion:
The two differential equations of rectilinear motion (namely):
𝑣= 𝑑𝑥/𝑑𝑡 and 𝑎=𝑑𝑣/𝑑𝑡
can be combined, by eliminating the time variable dt to form a third differential equation:
𝑣 𝑑𝑣=𝑎 𝑑𝑥
(a) Constant Acceleration:
Rewrite, a = dv/dt, equation as: dv = a dt.
Integrate both sides of this equation and put in the limits of integration as follows:
{u-v}∫ 𝑑𝑣=𝑎 {0-T}∫ 𝑑𝑡
v = u + at
