Читать книгу Foundations of Space Dynamics - Ashish Tewari - Страница 30

A particle is defined to have a finite mass but infinitesimal dimensions, and is therefore regarded to be a point mass. Since a particle has negligible dimensions, its position in space is completely determined by the radius vector, , measured from a fixed point, o, at any instant of time, . The components of are resolved in a right‐handed reference frame with origin at o, having three mutually perpendicular axes denoted by the unit vectors , and , and are defined to be the Cartesian position coordinates, , of the particle along the respective axes, , as shown in Fig. 2.1. The velocity, , of the particle is defined to be the time derivative of the radius vector, and given by

(2.18)

Figure 2.1 The position vector, , of a particle resolved in an inertial reference frame using Cartesian coordinates, ().

If the reference frame, , used to measure the velocity of the particle is at rest, then the components of the velocity, , resolved along the axes of the frame, , and , are simply the time derivatives of the position coordinates, , and , respectively. However, if the origin, , of the reference frame itself is moving with a velocity, , and the frame, , is rotating with an angular velocity, , with respect to an inertial frame3, then the velocity of the moving reference frame must be vectorially added to that of the particle in order to derive the net velocity of the particle in the stationary frame as follows:

(2.19)

The last term on the right‐hand side of Eq. (2.19) is the change caused by rotating axes, , each of which have the same angular velocity, . The relationship between the position and velocity described by a vector differential equation, either Eq. (2.18) or Eq. (2.19), is termed the kinematics of the particle.

Since the velocity of the particle could be varying with time, the acceleration, , of the particle is defined to be the time derivative of the velocity vector, and is given by

(2.20)

with the understanding that the derivatives are taken with respect to a stationary reference frame. If the reference frame in which the position and velocity of the particle are resolved is itself moving such that its origin, , has an instantaneous velocity, and an instantaneous acceleration, , and its axes are rotating with an instantaneous angular velocity, , all measured in a stationary frame, then the net acceleration of the particle is given by

(2.21)

Equation (2.21) is an alternative kinematical description of the particle's motion, and can be regarded as being equivalent to that given by Eq. (2.19), which has been differentiated in time according to the chain rule. Equation (2.21) is useful in finding the acceleration of the particle from the position and velocity measured in a moving reference frame. The first two terms on the right‐hand side of Eq. (2.21) represent the net acceleration due to the origin of the moving frame. The term is the Coriolis acceleration, and the centripetal acceleration of the particle in the moving reference frame. The term is the effect of the angular acceleration of the reference frame, whereas is the acceleration due to a changing magnitude of , and would be the only acceleration had the reference frame been stationary.

The application of Newton's second law to the motion of a particle of a fixed mass, , and acted upon by a force, , gives the following important relationship – called the kinetics – for the determination of the particle's acceleration:

(2.22)

The linear momentum, , of the particle is defined as the product of its mass, , and velocity, :

(2.23)

Since the particle's mass is constant, the second law of motion given by Eq. (2.22) can alternatively be expressed as follows:

(2.24)

which gives rise to the principle of linear momentum conservation if no force is applied to the particle.

The angular momentum, , of the particle about a point, o, is defined to be the vector product of the radius vector, , of the particle from o and its linear momentum, :

(2.25)

By virtue of Eq. (2.24), it is evident that the angular momentum of the particle about o can vary with time, if and only if a torque, defined by , acts on the particle about o:

(2.26)

This results in the principle of angular momentum conservation if no torque acts on the particle about o.

The work done on a particle by a force while moving from point A to point B is defined by the following integral of the scalar product of the force, , and the particle's displacement, :

(2.27)

The application of Newton's second law for the constant mass particle, Eq. (2.22), results in the following expression for the work done:

(2.28)

where and are the speeds of the particle at the points A and B, respectively. Thus the net work done on a particle equals the net change in its kinetic energy, .