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PNG University of Technology Department of Applied Physics
AP 274 Classical Mechanics:
Syllabus: Review of kinematics and particle dynamics. Conservation theorems. Potential energy and conservative forces. Motion on a curve.
Translation of coordinate systems. Inertial and on-inertial reference systems. Rotation of coordinate system. Effects of Earth’s rotation. Coriolis force and centrifugal force. The Foucault pendulum. Gravitation and central forces. Gravitational potential. Motion in an inverse-square repulsive force field. Equations of motion, potential energy and differential equations describing the motion of a particle in a central force field. Centre of mass, kinetic energy, linear momentum and angular momentum of a system of particles.
Motion of two interacting bodies. The reduced mass. Collisions. The laboratory and centre of mass coordinate systems. Impulsive force. Motion of a body of variable mass. Rocket motion.
Rotation of a rigid body about a fixed axis. Moment of inertia. The physical pendulum. General theorem concerning angular momentum. Laminar motion of a rigid body. Rigid body rolling down an inclined plane.
Rotation of a rigid body about an arbitrary axis. Principal moments and product of inertia. Rotational kinetic energy of a rigid body. Principal axes and their directions. Euler’s equations. Motion of a rigid body under no torques. Free rotation of a rigid body with an axis of symmetry. Gyroscopic precession. Motion of a top-Gyroscopes. Generalized coordinates. Degrees of freedom. Constraints. D’Alembert’s principle. Lagrange’s equations. Calculus of variations. Hamilton’s equations. Hamilton-Jacobi equations.
Lecture Notes:
Lecture Week 1:
Kinematics and Particle Dynamics
Dynamics is a branch of applied mathematics (specifically classical mechanics) concerned with the study of forces and torques and their effect on motion (as opposed to kinematics , which studies the motion of objects without reference to its causes).
Kinematics is the study of only the motion of particles without taking into consideration the causes of said motion. It doesn't ask "how did the velocity of the body change?" Only, "by how much did it change?"
Dynamics on the other hand is the study of motion of the particles along with their cause ( id est , forces and torques). It asks why did the velocity change.
Before the mid-20th century, Dynamics was called Kinetics.
In classical mechanics "kinematics" generally refers to the study of properties of motion-- position, velocity, acceleration, etc.-- without any consideration of why those quantities have the values they do. "Dynamics" means a study of the rules governing the interactions of these particles, which allow you to determine why the quantities have the values they do.
Thus, for example, problems involving motion with constant acceleration ("A car starts from rest and accelerates at 4m/s^2. How long does it take to cover 100m?") are classified as “ kinematics” , while problems involving forces ("A 100g mass is attached to a spring with a spring constant of 10 N/m and hangs vertically from a support. How much does the spring stretch?") are classified as "dynamics".
An example in mechanics: swing a pendulum in a vertical plane, swing sufficiently fast so that the trajectory is a circle. What is the tension in the pendulum when it passes in the lowest point of the circle. The tension is a dynamical quantity, because it is a force. Now, when you solve the problem, you don't write down the full equation of Newton and solve them. You use the kinematic information you have about the trajectory: it's a circle, in the lowest part of the trajectory there is no tangential acceleration, so the acceleration is directed radially inwards and is v^2 /r. From this you can find the tension by using purely kinematic considerations and never solving F =ma → as a differential equation.
What is kinematics and kinetics?
Kinetics : The study of forces that cause motion (ex. torque, gravity, friction, etc.) and can be classified into two groups; Linear and angular motion. Kinematics : The study of describing movement (ex. displacement, time, velocity, etc.)
Kinematics Vs Dynamics:
Physics is the study of matter, their energy and their interaction. Physics is also the study of motion of objects. This study is also known as dynamics, a word that is derived from a Greek word dunamis that means power. Now study of motion is not possible without knowing the causes of motion which are various forces operating on bodies. It is these forces and their knowledge that helps us to know all about motion and also be able to predict motions. However, if someone is interested purely in motion without getting into forces that cause motion, it is possible through kinematics which is a branch of physics that deals solely in terms of different kinds of motions. There are obvious differences between dynamics and kinematics.
In Brief:
- Both dynamics and kinematics are study of motion but whereas causal forces are taken into account in dynamics there is no regard to such forces of interaction in kinematics.
Conservation Theorems:
Where the limits of integration on the right side are from x 0 to x. U(x 0 ) is an arbitrary constant of integration of potential energy.
Example:
Potential Energy Derivative:
Gravitational Potential energy:
Fy = - dU/dy = - d(mgy)/dy
Fy = - mg
Elastic Potential Energy:
Fx = - dU/dx = - d(1/2kx^2 )/dx
Fx = - kx
Potential Energy Integral: Gravitational Potential Energy:
U = - ʃ- mg dy = mgh
Where the range of integration is from y = 0 to h.
Elastic Potential Energy:
U = ʃ- (-kx)dx = 1/2(kx^2 )
Where the range of integration is from x = 0 to x.
Conservative Forces:
In a conservative force field the work done in moving between any two points A and B are path independent. The word “conservative” means if we move from A to B by one path and return by the other there is no net loss of energy or any closed path to A takes net zero work. So conservative forces are position dependent forces and independent of path.
Lecture Week 2 – 3:
Translation of Coordinate Systems:
Consider a coordinate system undergoing pure translation as shown below:
Oxyz are primary coordinate axis (assumed fixed) and Ox'y'z' are moving axis. The position vector of a particle P is denoted by r in the fixed system and r ' in the moving system. The displacement OOof the moving origin is denoted by **R** 0. Thus from the triangle OOP
r = R 0 + r '
Therefore
v = V 0 + v ' a = A 0 + a '
If the moving system is not accelerating A 0 = 0, then
a = a '
If the moving system is accelerating
F = m A 0 + m a ' or
F - m A 0 = m a '
F** = m **a** ' Where **F= F + (-mA 0 ).
Inertial and Non-Inertial Systems :
The first law of motion defines a particular type of reference frame called the inertial system. Therefore, inertial system is the one in which the Newton’s first law holds good. Suppose there are two observers A and B in two frames of reference i.e. S and S', where S is an inertial system and S' a non- inertial system. Both the observers are observing a common object C moving with acceleration. S being an inertial system means that the observer A is moving with a uniform velocity, while the system S' being non-inertial means that observer B has acceleration.
Now to find the time derivative d i' /dt, d j' /dt and d k' /dt consider the change in the unit vector Δi' in the unit vector i' due to a small rotation Δθ about the axis of rotation.
Now
|Δ i'| = (sin Φ)Δθ
Where Φ is the angle between i' and ω.
d i' /dt = ω x i'
Similarly
d j' /dt = ω x j' and d k' /dt = ω x k'
Therefore
x'd i' /dt + y'd j' /dt + z'd k' /dt = x'( ω x i') + y' (ω x j') + z' (ω x k')
= ω x (i' x' + j' y' + k' z')
= ω x r' This is the velocity of particle P due to rotation of the primed coordinate system. Therefore,
v = v' + ω x r' or, more explicitly
(d r/ dt)fixed = (d r'/ dt)rot + ω x r' = [(d / dt)rot + ω x ] r'
This implies, that the operation of differentiating the position vector w.r.t. time in the fixed system is equivalent to the operation of taking the time derivative in the rotating system plus the operation ω x. The same applies to any vector. If that vector is the velocity then we have
(dv / dt)fixed = (dv / dt)rot + ω x v
But v = v' + ω x r', so
(dv / dt)fixed = (d / dt)rot(v' + ω x r') + ω x (v’ + ω x r’)
= (dv' / dt)rot + [d(ω x r')/dt]rot + ω x v' + ω x (ω x r’)
= (d v'/ dt)rot + (d ω /dt)rot x r' + ω x (d r'/ dt)rot
Since, (d ω /dt)fixed = (d ω /dt)rot + ω x ω
this implies
(d ω /dt)fixed = (d ω /dt)rot = d ω/ dt
Since, v' = ( d r'/ dt)rot and a' = ( d v'/ dt)rot
Therefore,
a = a' + d ω/ dt x r' + 2 ω x v' + ω x ( ω x r' )
This gives the acceleration in the fixed system in terms of the position, velocity and acceleration in the rotating system.
If the primed system is undergoing both translation and rotation, we must add the velocity and translation V 0 to the right hand side of the expression for the velocity in the fixed system. And
a = a' + d ω/ dt x r' + 2 ω x v' + ω x ( ω x r' ) + A 0
The term 2 ω x v' is known as is known as Coriolis acceleration and the term ω x ( ω x r' ) is called the centripetal acceleration. Coriolis acceleration appears whenever a particle moves in a rotating coordinate system (except when velocity v' is parallel to the axis of rotation). The term d ω/ dt x r' is called the transverse acceleration since it is perpendicular to the position vector r'.
