Particles kinematics
Kinematics is a branch of mechanics that focuses on describing the motion of objects without considering the forces that cause it. It examines parameters like position, velocity, and acceleration to analyze and predict the trajectories of particles and rigid bodies. By using concepts such as displacement, speed, and relative motion, this section provides tools to understand and solve problems involving motion of particles.
Point particle
We begin consider the point particle model. We agree that a relevant description of an object’s motion can be achieved by tracking a single point associated with that object. We will consider one point of the object and describe its movement. Furthermore, we attribute the object’s entire mass to this point, hence the term “point particle”.
The discrepancy between the predictions of the point particle model and experimental observations can be of two kinds:
- Quantitative difference: Consider a pendulum made from a rigid rod attached to a pivot. If we approximate this rod as having all its mass concentrated at its center of mass, measurements will reveal an error, but this error is merely quantitative (e.g., in the calculated period).
- Qualitative difference: If you model billiard balls as point masses, the description is adequate for many translational motions. However, if you spin a billiard ball rapidly, its behavior has nothing in common with what the point particle model can predict. The model fundamentally cannot account for rotation.
Coordinate systems
When discussing velocity and acceleration, it is essential to specify what we are measuring these quantities with respect to. This entity is called the frame of reference.
If we perform a free fall experiment in a laboratory, the laboratory is a suitable frame of reference. Conversely, if we want to discuss the Earth’s orbit around the Sun, we must consider a much larger frame of reference, such as the Sun and distant stars.
In general, at least four non-coplanar points are needed to define a frame of reference. Three points define a plane; a fourth point outside this plane is sufficient to establish a frame.
Sometimes, a Cartesian coordinate system is chosen as a frame of reference. However, we shouldn’t confuse a coordinate system with the frame of reference itself. The coordinate system is a tool within a frame of reference and is not the frame itself.
The choice of frame of reference is very important. For instance, centrifugal and Coriolis forces arise when we make certain choices of frame of reference. Furthermore, Einstein’s famous theory of relativity originates from general considerations regarding the choice of frames of reference.
The trajectory is the geometric locus of points in the frame of reference through which our point particle passes. The equation of motion is the function of time that gives us the position of our point particle as a function of time. This provides more information than just the trajectory. The trajectory is simply a geometric path, but the equation of motion tells us where the point particle is at any given time. You can consider the ultimate goal of mechanics to be determining the equation of motion for all point particles of the object under consideration.
Cartesian
There are several different coordinate systems, the most commonly used for curvilinear is the Cartesian for which the orthonormal vectors are \mathbf i, \mathbf j and \mathbf k. Since these are fixed, the position, velocity and acceleration are:
\begin{aligned} \mathbf r_P & = x \mathbf i + y \mathbf j + z \mathbf k \\ \mathbf v_P & = \dot x \mathbf i + \dot y \mathbf j + \dot z \mathbf k \\ \mathbf a_P & = \ddot x \mathbf i + \ddot y \mathbf j + \ddot z \mathbf k \\ \end{aligned}
Cylindrical
Another commonly used system for curvilinear motion is the cylindrical coordinate system described by three unitary vectors \mathbf e_\rho, \mathbf e_\phi, \mathbf e_z. The position, velocity, and acceleration of a point P are described by the coordinates (\rho, \phi, z), where \rho is the radial distance, \phi is the azimuthal angle, and z is the height along the z-axis. The unit vectors in this system are \mathbf{e}_\rho, \mathbf{e}_\phi, and \mathbf{e}_z, which vary with position due to the curvilinear nature of the system.
The expressions for position, velocity, and acceleration in cylindrical coordinates are:
\begin{aligned} \mathbf{r}_P &= \rho \mathbf{e}_\rho + z \mathbf{e}_z \\ \mathbf{v}_P &= \dot{\rho} \mathbf{e}_\rho + \rho \dot{\phi} \mathbf{e}_\phi + \dot{z} \mathbf{e}_z \\ \mathbf{a}_P &= \left( \ddot{\rho} - \rho \dot{\phi}^2 \right) \mathbf{e}_\rho + \left( 2 \dot{\rho} \dot{\phi} + \rho \ddot{\phi} \right) \mathbf{e}_\phi + \ddot{z} \mathbf{e}_z \\ \end{aligned}
Spherical
In the spherical coordinates system, the position, velocity, and acceleration are described by the coordinates (r, \theta, \phi), where r is the radial distance, \theta is the polar angle, and \phi is the azimuthal angle. The corresponding unit vectors are \mathbf{e}_r, \mathbf{e}_\theta, and \mathbf{e}_\phi, which also vary with position.
The expressions for position, velocity, and acceleration in spherical coordinates are:
\begin{aligned} \mathbf{r}_P &= r \mathbf{e}_r \\ \mathbf{v}_P &= \dot{r} \mathbf{e}_r + r \dot{\theta} \mathbf{e}_\theta + r \sin \theta \, \dot{\phi} \mathbf{e}_\phi \\ \mathbf{a}_P &= \left( \ddot{r} - r \left( \dot{\theta}^2 + \sin^2 \theta \, \dot{\phi}^2 \right) \right) \mathbf{e}_r + \left( 2 \dot{r} \dot{\theta} + r \ddot{\theta} - r \sin \theta \cos \theta \, \dot{\phi}^2 \right) \mathbf{e}_\theta + \left( 2 \dot{r} \sin \theta \, \dot{\phi} + 2 r \dot{\theta} \cos \theta \, \dot{\phi} + r \sin \theta \, \ddot{\phi} \right) \mathbf{e}_\phi \\ \end{aligned}
These formulations provide a complete description of motion in Cartesian, cylindrical, and spherical coordinate systems, with each system suited for specific types of curvilinear motion.
The Cartesian, cylindrical, and spherical coordinate systems described above are extrinsic coordinate systems, meaning they are defined with respect to fixed axes in space. In contrast, intrinsic coordinate systems, such as the tangential and normal coordinate systems, are defined based on the motion of the particle itself. This makes intrinsic systems particularly useful for describing curvilinear motion along a path or trajectory.
Tangential and normal
In the tangential and normal (or Frenet-Serret) coordinate system, we define the motion of a point P along a path using the following unit vectors:
- \mathbf{e}_t: the unit tangent vector, pointing in the direction of the particle’s instantaneous velocity.
- \mathbf{e}_n: the unit normal vector, pointing towards the center of curvature of the path.
- \mathbf{e}_b: the unit binormal vector, perpendicular to both \mathbf{e}_t and \mathbf{e}_n, completing a right-handed coordinate system.
The position vector is generally described by the path or trajectory itself, while the velocity and acceleration can be expressed in terms of the intrinsic coordinates:
\begin{aligned} \mathbf{v}_P &= v \, \mathbf{e}_t \\ \mathbf{a}_P &= \frac{d v}{d t} \, \mathbf{e}_t + \frac{v^2}{\rho} \, \mathbf{e}_n \\ \end{aligned}
where:
- v = \frac{ds}{dt} is the magnitude of the velocity, where s is the arc length along the path.
- \frac{d v}{d t} represents the tangential acceleration, indicating how the speed of the particle changes along the path.
- \frac{v^2}{\rho} represents the normal (or centripetal) acceleration, which is directed towards the center of curvature of the path and is responsible for changing the direction of the velocity vector.
- \rho is the radius of curvature of the path at the particle’s current position.
The binormal component of the acceleration is zero if the motion is confined to a plane. However, in three-dimensional motion, the binormal component of the acceleration vector would be governed by the torsion of the path and the derivative of \mathbf{e}_b.
Vector time derivatives
Considering a vector \mathbf r with a magnitude and a direction, the derivatives with respect of time is defined as:
\frac{\mathrm d \mathbf r}{\mathrm dt} \equiv \dot {\mathbf r}
so either or both can change as time passes.
When taking vector derivatives, it is necessary to specific a reference frame because the time derivative of a vector might differ in different reference frames: for example, considering a fixed vector on a rotating disc, when considering a position staying on the disc, the time derivatives is equal to zero, while considering a reference frame outside the disc, the same vector has a time derivatives that is not equal to zero.
A reference frame is a coordinate system or viewpoint used to measure and observe the position, orientation, and motion of objects. It provides a basis for describing physical quantities such as displacement, velocity, and acceleration, typically relative to a fixed or moving point in space.
Considering a generic vector \mathbf r = r_1 \mathbf e_1 + r_2 \mathbf e_2 + r_3 \mathbf e_3, and a reference frame RF, the time derivative of the vector with respect to that frame is:
\frac{\mathrm d \mathbf r}{\mathrm dt}\bigg|_{RF} = \sum_{i=1}^3 \left(\frac{\mathrm d r_i}{\mathrm dt}\mathbf e_i + r_i \frac{\mathrm d \mathbf e_i}{\mathrm dt}\bigg|_{RF} \right)
Equations of motion
Considering a point P, the position is defined as the vector which goes from the origin of a reference frame to P:
\overline {OP} = \mathbf r
If this position is varying with the time t we have what is called an equation of motion \mathbf r(t).
Velocity
To define the velocity, we first establish a frame of reference. Then we measure a displacement, and we divide the displacement by the time interval; this gives us a velocity. If we measure the displacement vectorially, when we divide by the time interval, we obtain a vector velocity.
Note first that from a physics point of view, velocity is fundamentally a vector quantity. A velocity describes a direction, the direction of the object’s motion; the magnitude of the velocity gives the speed in the common sense, what is called in physics the scalar velocity. So now we must find how to mathematically express this velocity in the physical sense.
We define a vector \mathbf{OP} that locates the position of my point particle and to determine how much the position changes during \mathrm d t, we divide by \mathrm d t:
\mathbf v(t) \equiv \lim_{\Delta t \to 0} \frac{\mathbf r (t + \Delta t) - \mathbf r (t)}{\Delta t} = \frac{\mathrm d \mathbf r}{\mathrm dt} = \dot{\mathbf r}
Here is a Cartesian coordinate system representing a frame of reference, we assume we know the trajectory, we locate the position of the point particle at time t by the vector \mathbf r(t).
At time t + \Delta t, the displacement \mathbf r(t + \Delta t) - \mathbf r(t), it is a vector, and in the diagram we have taken a finite \Delta t which gives \Delta r = \mathbf v \Delta t. So we can divide both side for \Delta t and take the limit as \Delta t approaches zero, and this will give us the velocity.
Acceleration
Let us now move on to the definition of vector acceleration. What is acceleration? It must be a change in velocity per unit of time. So our definition of acceleration will necessarily be given by the choice of a frame of reference, which is very important, trivial, quickly said but very important.
And we will mathematically express the fact that we want to calculate the change in velocity per unit of time. So here we go, we do it in the following way: we calculate \mathbf v at time t + \Delta t minus \mathbf v at time t, change in velocity divided by \Delta t. We take the limit as \Delta t approaches zero and we obtain the acceleration:
Acceleration is the derivative of velocity with respect to time. Which we can write like this:
\mathbf a(t) \equiv \lim_{\Delta t \to 0} \frac{\mathbf v (t + \Delta t) - \mathbf v (t)}{\Delta t} = \frac{\mathrm d \mathbf v}{\mathrm dt} = \dot{\mathbf v} = \frac{\mathrm d^2 \mathbf r}{\mathrm dt^2} = \ddot{\mathbf r}
We would like to express the acceleration in a diagram. We assume that the trajectory is known and we draw the position of the point particle at time t and at time t + \Delta t.
Now we have the velocity at time t, we compute the velocity at time t + \Delta t here, and we bring this velocity vector back to the position at time t to calculate the increment \mathbf v(t + \Delta t) - \mathbf v(t) which give us \mathbf a \Delta t. We can divide both side for \Delta t and take the limit as \Delta t approaches zero, and that will gives the acceleration.