EN4: Dynamics and Vibrations
Division of Engineering
Brown University
1.2 Fundamental Postulates of Classical Mechanics
Newton’s Laws of Motion
All the ideas that will be discussed in EN4 can (and will) be deduced from Newton’s laws of motion:
1. When the sum of the forces acting on a particle is zero, its velocity is constant
2. The sum of forces acting on a particle of constant mass is equal to the product of the mass of the particle and its acceleration
3. The forces exerted by two particles on each other are equal in magnitude and opposite in direction.
To understand Newton’s laws of motion, then, we must discuss what we mean by a `particle;’ a `force;’ and we must find a way to describe accelerations. Finally, we need to understand how to use the laws of motion to solve problems.
Definition: A particle is a concentrated mass at some position in space. The inertial mass of a particle is defined through Newton’s second law.
A particle has two properties:
(i) its position (a vector) r
(ii) its mass (a scalar) m
There is actually no such thing as a particle, but many mechanical systems can be idealized as systems of particles to predict approximately how they behave. Some examples of systems that can be idealized as particles are:
(i) Orbiting planets
(ii) Vehicles
(iii) Infinitesimal volume elements inside a deformable body
As you see, an object need not necessarily be small to be idealized as a particle. As a general rule, we idealize a body as a particle if
(i) it doesn’t change its shape significantly, and
(ii) we are only interested in the position of the body, and not its orientation.
Hint: until Section 5 of this course, we will idealize everything as a particle, so you don’t need to decide for yourself!
Definition: A force is something that causes a particle to accelerate. Forces
are defined through Newton’s second law.
A force has two properties:
(i) Magnitude
(ii) Direction
It is therefore a vector.
When solving problems in dynamics, we start by idealizing our system (either as one or more particles, or for more complex problems, as rigid bodies). We then work out the forces acting on each particle. Forces may arise from a number of different effects, including
(i) Gravity
(ii) Electromagnetism or electrostatics
(iii) Buoyancy
(iv) Wind or fluid induced drag or lift forces
(v) Forces induced by deformable elements, such as springs
(vi) Contact forces or constraint forces, which force the particle to move in along a prescribed path
(vii) Friction
We will assume that you are comfortable with identifying the forces acting in a system as a result of having survived EN3, and that you are also happy with expressing forces as vectors.
