1. Introduction

Mechanics is the branch of the physical sciences that deals with the mechanical motion of bodies, i.e. changing of relative position of bodies in space in the course of time.

Our course is subdivided into four parts: statics, kinematics, dynamics, and analytical mechanics.

Statics is the branch of mechanics which treats of bodies that are at rest or in the state of uniform motion. Statics studies the laws of composition of forces and the conditions of equilibrium of engineering structures under the action of forces.

Kinematics is study of the geometry of motion without regard to the forces the cause that motion.

Dynamics deals with the action of forces in producing or modifying the motion of bodies.

2. Basic conceptions: space, time, frame of reference

As you know motion is changing with time of the particle (or body) position with respect to the position of some other particle (or body). So to observe the motion we have to use two base notions: space and time.

In Newtonian mechanics we use ideal model of space that can be visualized by nonlimited rigid body. It is supposed that the space affects the other physical phenomena, but the space itself is not affected by those phenomena. Such space is called absolute space. The absolute space is Euclid space, it means that Euclid’s geometry is valid in the space.

To determine particle (body) position in space we choose at least one frame of reference consisting of two components: a datum or origin and a system of three linear independent directions (coordinate axes). In the Newtonian mechanics it is postulated that there is at least one fixed frame of reference in which all Newton’s laws are valid. Such frame of reference is called absolute or inertial. For our purposes we can choose as the inertial frame of reference the heliocentric reference (or geocentric reference). Frames of reference where objects violate Newton's first law are called noninertial.

The second ideal model of Newtonian mechanics is the absolute time. It is supposed that time runs at the same rate for all the observers in the absolute space.

3. Axioms of dynamics

The first three axioms known as Newton’s lows of motion:

1. A particle isolated from other bodies remains at rest or continues to move in straight line with a constant velocity if there is no unbalanced force acting on it. (A particle isolated from other bodies and acted on by no forces, or by a system whose resultant is nil, is either at rest or in uniform rectilinear motion).

In other words, an object initially at rest is predicted to remain at rest if the total force acted on it is zero, and an object in motion remains in motion with the same velocity in the same direction.

The converse of Newton's first law is also true: if we observe an object moving with constant velocity along a straight line, then the total force on it must be zero.

2. A particle acted on by a single force is accelerated; the acceleration is in the direction of the force and is directly proportional to the force and inversely proportional to the mass of the particle.

. (1.1)

The property, by virtue of which a particle tends to remain at rest or in uniform rectilinear motion, and to resist being accelerated, is called inertia. Inertial mass is a measure of inertia of the particle which is its resistance to rate of change of velocity when a force is applied. An object with small inertial mass changes its motion more readily, and an object with large inertial mass does so less readily.

In the solution of problems Eq. (1.1) is usually expressed in scalar component form using one of the coordinate systems developed in kinematics (Cartesian, natural, polar, cylindrical). Equation 1.1, or any one of the component forms of the force-mass-acceleration equation, is usually referred to as the equation of motion. The equation of motion gives the instantaneous value of the acceleration corresponding to the instantaneous values of the forces which are acting.

3. To every action there is always an equal and contrary reaction: or, the mutual actions of any two bodies are always equal and oppositely directed.

4. The parallelogram law. Two forces applied at one point of a body have as their resultant a force applied at the same point and represented by the diagonal of a parallelogram constructed with the two given forces as its sides, i.e. a force system is equivalent to its resultant.

Fig. 1.1

5. Principle of superposition.

The resulting acceleration caused by two or more Forces is the geometrical sum of the accelerations which would have been caused by each force individually

. (1.2)

4. Different forms of free particle equation of motion in inertial frame of reference:

1. In vector form:

; (2)

. (3)

2. in coordinate form:

a. In a Cartesian coordinate system:

(4)

b. In a natural coordinate system:

(5)

c. In a polar coordinate system (for plane motion of a particle)

(6)

The choice of the appropriate coordinate system is dictated by the type of motion involved and is a vital step in the formulation of any problem.

5. The two problems of dynamics

We encounter two types of problems when applying Eq. 1.1. In the first type the acceleration is either specified or can be determined directly from known kinematic conditions. The corresponding forces which act on the particle whose motion is specified are then determined by direct substitution into Eq. 1.1. This problem is generally quite straightforward.

If motion is given in coordinate form

(7)

then force produced this motion has components

(8)

In the second type of problem the forces are specified and the resulting motion is to be determined. If the forces are constant, the acceleration is constant and is easily found from Eq. 1.1 When the forces are functions of time, position, velocity, or acceleration, Eq. 1.1 becomes a differential equation which must be integrated to determine the velocity and displacement. The Eq. 1.1 have to supplement the proper quantity of initial conditions to obtain single-valued solution (6 initial conditions for 3D case). This problem is called inverse.

Case 1. Force is constant (force of gravity) or function of time (force of interaction between core and magnetizing coil driven by alternating current):

(9)

To determine motion we rewrite equations (4) and integrate twice

(10)

and integrate twice

(11)

(12)

The equations (11) and (12) can be satisfied by the substituting the initial conditions into equations and solving for the constants

Case 2. Force is function of particle velocity.

Aerodynamics drag force acting on particle in rectilinear motion is

(13)

For a particle under the action of aerodynamics drag force directly proportional to the speed

(14)

For a particle under the action of aerodynamics drag force directly proportional to the speed squired

. (15)

The example of problem about particle motion under the action of aerodynamics drag force will be considered at the end of the lecture.

Lorentz force (Axis Ox is parallel to magnetic field

) is

. (16)

Case 3. Force is function of particle coordinates

Force of elasticity

(17)

(18)

Gravitational force

(19)

(20)

Summarizing

In general case force is function of time, particle position and velocity

(21)

Equations (4) can be rewriting with help of equations (21) in the following form

(22)

The system (22) is system of second order differential equations with respect to unknown functions

. This system is termed main differential equations of particle motion in noninertial frame of reference.

The first integral of the system (22) is

(23)

The second integral of the system (22) is

(24)

The equations (23) and (24) can be satisfied by the substituting the six initial conditions into equations and solving for the constants

when

Final equations of the particle motion are

(25)

So the second problem of dynamics is problem Cauchy problem or initial value problem.

6. Constrained and unconstrained motion

There are two physically distinct types of motion, both described by Eq. 1.1. The first type is unconstrained motion where the particle is free of mechanical guides and follows a path determined by its initial motion and by the forces which are applied to it from external sources. An airplane or rocket in flight and an electron moving in a charged field are examples of unconstrained motion.

The second type is constrained motion where the path of the particle is partially or totally determined by restraining guides. An ice hockey puck moves with the partial constraint of the ice. A train moving along its track and a collar sliding along a fixed shaft are examples of more fully constrained motion. Some of the forces acting on a particle during constrained motion may be applied from outside sources and others may be the reactions on the particle from the constraining guides. All forces, both applied and reactive, which act on the particle must be accounted for in applying Eq. 1.1.

The choice of a coordinate system is frequently indicated by the number and geometry of the constraints. Thus, if a particle is free to move in space, as is the center of mass of the airplane or rocket in free flight, the particle is said to have three degrees of freedom since three independent coordinates are required to specify the position of the particle at any instant. All three of the scalar components of the equation of motion would have to be applied and integrated to obtain the space coordinates as a function of time. If a particle is constrained to move along a surface, as is the hockey puck or a marble sliding on the curved surface of a bowl, only two coordinates are needed to specify its position, and in this case it is said to have two degrees of freedom. If a particle is constrained to move along a fixed linear path, as is the collar sliding along a fixed shaft, its position may be specified by the coordinate measured along the shaft. In this case the particle would have only one degree of freedom.

Example 1: High-speed land racer of mass m moves horizontal. At initial moment of time its velocity was

. If the air drag force is

where

is constant, determine the time t required for it to reduce its speed twice and the distance traveled.

Solution

(26)

for v=v/2 we get

So time of deceleration is function of the initial speed.

Determine the distance

To integrate, we perform a change of variable. Thus

We then have as a replacement for our equation

Now replacing z, we get

When t=0 we take x=0 and so

We then have

And on combining the logarithmic terms, we obtain

There is another way of the equation (26) solution:

Integrating we get

When

we take x=0 and so

, and

Substitute

we get x=13680 m.

So distance traveled during the velocity decreasing is not function of the initial speed value.

Example 2. Missile of mass m moves verticaly. At initial moment of time its velocity was

. If the air drag force is given by where k is constant, derive the maximum height and duration of missile lift.

Solution.

1. Choose the x direction as direction of missile motion, so that x=0 if h=0, axis x positive direction is down. Draw free-body diagram: show force of gravity and force of air resistance.

2. Apply equation of motion in the x-direction to get