MAÎTRISER LA CINÉMATIQUE : repères cartésien, cylindrique, sphérique et Frenet en action

Introduction to Kinematics

Overview of Kinematics

• The video introduces the topic of kinematics, focusing on key concepts such as the reference frame, position vector, velocity vector, and acceleration vector.

• It emphasizes expressing these vectors in Cartesian, cylindrical, or spherical coordinates as part of a broader series on Newtonian mechanics.

Engagement with Audience

• The presenter encourages viewers to subscribe and engage by commenting on topics they wish to learn more about.

Defining Reference Frames

Establishing a Reference Frame

• A solid reference frame is defined using the screen as a point of observation for studying the movement of point M.

• A temporal reference is also established based on the video's timeline, creating a complete reference frame.

Position Vector Definition

• The position of point M relative to a fixed origin (point 0) is described by its position vector, which varies over time and is measured in meters.

Understanding Velocity

Average vs. Instantaneous Velocity

• Velocity is defined as the ratio of distance traveled over time; average velocity applies over an interval Δt.

• To find instantaneous velocity, one must consider infinitesimally small changes in position and time leading to the relationship that velocity equals the derivative of the position vector.

Acceleration Concept

Deriving Acceleration from Velocity

• Acceleration arises from changes in velocity; it is mathematically represented as the derivative of the velocity vector or second derivative of the position vector.

Cartesian Coordinate System

Setting Up Cartesian Coordinates

• The Cartesian coordinate system involves three orthogonal axes (X, Y, Z), with unit vectors defined for each axis.

• Point M's coordinates are projected onto these axes; its position vector OM can be expressed using these coordinates.

Calculating Speed and Acceleration

• The speed is derived from differentiating the position vector while considering constant unit vectors whose derivatives are zero.

• The acceleration vector results from differentiating the speed expression again under similar conditions regarding unit vectors.

Cylindrical Coordinate System

Transitioning to Cylindrical Coordinates

• In cylindrical coordinates, an angle θ influences positioning; thus, point M's representation includes radial distance r and height z.

Deriving Velocity in Cylindrical Coordinates

Deriving the Velocity and Acceleration Vectors in Spherical Coordinates

Understanding the Derivative of the Position Vector

• The position vector mathbfr is expressed as r cdot mathbfe_r + r' cdot fracdmathbfe_rdt , where mathbfe_r varies with time due to its dependence on the angle theta(t) .

• A diagram is used to illustrate the axes (X, Y) and unit vectors ( e_x, e_y ), helping visualize how to project e_r .

• The projection of e_r = cos(theta)x + sin(theta)y; since there’s no Z component, it simplifies our calculations.

• The expression for e_theta = -sin(theta)x + cos(theta)y is derived using trigonometric identities.

• To find the time derivative of e_r, we apply the chain rule: multiplying by dtheta/dt = dottheta.

Deriving Velocity from Position

• The velocity vector is defined as the derivative of position; it consists of two terms: one related to Z motion and another involving radial motion.

• The complete expression for velocity combines contributions from both Z direction and radial components:

• v_z = z'ez + r'cdot e_r + rcdotdotthetacdot e_theta.

Analyzing Acceleration Components

• To derive acceleration, we differentiate each component of velocity. This involves applying product rules for derivatives.

• For the first term involving Z, since its derivative yields zero contribution from unit vector changes, we focus on other terms.

• The second term's differentiation leads us back to previously established expressions for derivatives involving angular motion.

Final Expression for Acceleration

• Each component contributes to a complex acceleration formula that includes:

• Radial acceleration,

• Angular acceleration,

• Contributions from both unit vectors involved in spherical coordinates.

• Notably, when differentiating e_theta, it introduces additional terms linked with angular rates due to its orientation relative to other vectors.

Transitioning to Spherical Coordinate Systems

• In spherical coordinates, new angles are introduced which affect how we define our position vector in relation to axes X, Y, and Z.

• As we express velocity in this system, it becomes crucial to account for both radial and angular components distinctly influenced by their respective angles ( theta, phi).

• We introduce a new unit vector based on projections onto different planes (like OXY), emphasizing how these relationships complicate deriving velocities further.

Projection and Derivation of Vectors in Spherical Coordinates

Projection of the Vector mathbfe_r

• The vector mathbfe_r is projected onto the basis formed by mathbfe_oh and mathbfe_z , using the angle theta .

• The derivative of the unit vector mathbfe_oh is expressed as a function of another vector, specifically involving Fcdotmathbfe_phi .

Time Derivative Calculation

• The time derivative of mathbfe_r involves terms like -dotthetasin(theta)mathbfe_z + cos(theta)dotthetamathbfe_oh + (sin(theta)textderivative of mathbfe_oh) .

• This results in an expression that can be factored to highlight the role of dottheta .

Understanding Vector Relationships

• A diagram is suggested to visualize how vectors relate within the projection framework, particularly focusing on how they align with axes.

• When projecting onto direction e_oh ,cos(theta)text and -sin(theta)text are derived from previous findings.

Final Expression for Velocity

• The time derivative leads to a final expression for velocity:

• v = r' + rdotthetamathbfe_theta + rsin(theta)dotphimathbfE_phi .

Acceleration in Spherical Coordinates

• The acceleration is defined as the time derivative of velocity, which will not be explored further in this context.

• Introduction to Frenet Frame: It consists of a mobile origin linked to point M's trajectory, defined by three components: a tangent unit vector, a normal unit vector directed towards the center.

Defining Velocity and Acceleration in Frenet Frame

• In this frame, velocity is simply given by multiplying speed with tangent direction.

• The norm indicates that angular motion relates directly to linear speed through radius.