ABSTRACT
EXAMPLE 2.1. Planar motion Two parameters are needed to specify the position of a particle in a two dimensional space. In Cartesian coordinates they are x and y. The time derivative of the position vector (2.1) can also be written as rr ererrv
GGGG +== (2.3) where the first term on the right is the component of the velocity directed radially outward from the origin. The second term gives the component of velocity in the direction perpendicular to
(Figure 2.1). If we allow a change ϕ∆ in the direction of the unit vector ,reG in a time interval ,t∆ we can write
lim limr rr t t de ee dt t t
∆ ∆= = = =∆ ∆ G GG G G (2.4)
Thus, for a planar motion, the position of a particle may be defined by two polar coordinates ,r ϕ and its velocity is given by r r rv re r e v e v eϕ ϕ ϕϕ= + = +G G G G G (2.5) where is known as the radial velocity, and is called the tangential velocity. We then rv ϕv differentiate to find the acceleration as vG
( )r r rdv da re r e r e r e r e r e r edt dt ϕ ϕ ϕ ϕϕ ϕ ϕ ϕ= = + = + + + + GG G G G G G G G
where the time derivative ϕe
G can be evaluated, using Figure 2.1, as
ef e
e ery
x
y v
v e v er r
r
er e∆e r
ϕ ∆ϕ
∆e
Figure 2.1. Cartesian and polar coordinates of a planar motion
The radial component includes a linear acceleration ra r in the radial direction, due to a change in radial speed and a centripetal acceleration Similarly, the tangential component .2ϕr− ϕa contains two terms: a linear acceleration ϕr in the tangential direction, due to a change in the magnitude of the angular velocity, and an acceleration 2 ,rϕ arising when r and ϕ both change with time, and this is called the Coriolis acceleration. The momentum of a particle is a vector parallel to its velocity, whose magnitude is zero when the particle is at rest and increases with the magnitude of velocity. Inertial mass is the scalar coefficient of velocity in the expression for the momentum vmp GG = (2.8) and is a measure of the ability of a particle to resist a change in its motion, called inertia. Experiments show that inertial mass is equivalent to the gravitational mass of a particle, which measures its response to gravitational fields, so that normally the term mass is used in both cases. Mass was defined by Newton as the quantity of matter present, given by the product of density and volume. It is additive, the mass of a system is the sum of the masses of the isolated parts. Mass is a secondary concept, derived from space, time and momentum. It is a convenient term that provides, in Newtonian mechanics, a quantitative measure of inertia. Momentum plays a central role in the formulation of the laws of motion. Newton's first law: An isolated particle, independent of any external
interactions, remains indefinitely in its state of rest or of uniform motion in a straight line.
