8 Particle Kinetics
8.1 The Balance of Linear Momentum
A particle is endowed with an inertial measure \(m\), called its mass.
The linear momentum of a particle is defined to be \[\begin{align} {\bf G} = m{\bf v}. \end{align}\] Taking the time derivative: \[\begin{align} \dot{\bf G} = \dot{m}{\bf v}+m\dot{\bf v}. \end{align}\]
Question: Can you think of systems whose mass is not conserved?
In this class, we consider cases where mass is conserved, i.e. \(\dot{m} = 0\). This is not the case, for example, for a launching rocket that is losing mass. The \(\dot{m}{\bf v}\) term is useful in control volume analysis.
Then, in our case of interest: \[\begin{align} \dot{\bf G} = m\dot{\bf v} = m{\bf a}. \end{align}\] We postulate the following balance law: \[\begin{align} {\bf F} = \dot{\bf G}, \end{align}\] where \({\bf F}\) is the resultant external force acting on the particle. This is known as the Balance of Linear Momentum, Newton’s Second Law, or Euler’s First Law.
Important: \({\bf a}\) here is the absolute acceleration vector of the particle.
Question: Under what conditions is the linear momentum \({\bf G}\) conserved?
An unbalanced net force \({\bf F}\neq {\bf 0}\) results in a change of \({\bf G}\). If \({\bf F}={\bf 0}\), then \({\bf G}={\bf c}\) where \({\bf c}\) is a constant vector. This is Newton’s First Law.
If \({\bf c}={\bf 0}\), we recover statics.
8.1.1 Representations and Projections
Vectors are represented on a basis. For example: \[\begin{align} {\bf F} &= F_x{\bf E}_x+F_y{\bf E}_y+ F_z{\bf E}_z,\\ {\bf a} &= a_x{\bf E}_x+a_y{\bf E}_y+a_z{\bf E}_z. \end{align}\]
Question: How many independent scalar equations can we obtain from \({\bf F}=m{\bf a}\)? How do we obtain them?
We get useful scalar equations by projecting \({\bf F} = \dot{\bf G}\) along unit directions: \[\begin{align} \lp{\bf F}=\dot{\bf G}\rp\cdot{\bf E}_x \implies F_x = ma_x. \end{align}\]
8.2 Newton’s Third Law
When dealing with the forces of interaction between particles, a particle and a rigid body, or rigid bodies, we also invoke Newton’s third law: “For every action there is an equal and opposite reaction.” For example, the force exerted by a surface on a particle is equal in magnitude and opposite in direction to the force exerted by the particle on the surface.