Chapter Two

The Fundamental laws of Mechanics

Contents

• The concepts of displacement (位移), velocity, acceleration, angular velocity, angular acceleration and angular momentum (角动量)
• Newton’s laws of motion, rotational laws,
• Conservational (守恒的) laws of momentum, energy and angular momentum

§2.1 The motion of Point masses

2.1.1 Displacement and equation of motion

When a particle is moving in space, it changes its position with time. Generally, the motion could be described by a function of time. For example, a particle moves along x-direction in a constant speed v, its position can be written as x = vt + x0 ( x(t) (2.1)

x0 can be determined by the initial condition. If the particle moves in a three dimensional space, its path at time t can be described by the following equation: r(t) = x(t) i + y(t) j + z(t) k (2.2) r (r(t) (2.3) where r denotes vector [pic] and same for i, j, and k. Quite a few books used such bold form letters as vectors for simplicity. At the time of t+(t, r(t) becomes r(t+ (t). The change of its path during the interval (t is given by (r = r(t+ (t) - r(t). (r(t) is called the displacement of the particle in (t period. It also has three components in x, y and z directions. r(t) is called the equation of motion for the particle. 2.1.2 Velocity and Acceleration

1. Instantaneous velocity and acceleration Fig. 2.1 The displacements

The average velocity of the particle during (t with displacement of (r is (r/(t. It is also a vector, its direction is the same as the increment (r. The instantaneous velocity of the particle at time t is [pic] (2.4)

The instantaneous velocity is not necessarily a constant. It can be a function of time. At the moment t, [pic]will give the exact position of that particle. Using equation (2.4), we