Curvilinear Motion

  • Introduction

Curvilinear stands for along a curve. This is the most general specification of motion. It can either be two-dimensional or three-dimensional.

Examples of 2D motions are motion of a projectile and circular motion. These involve motions along planar curves.

Examples of 3D motions are a roller coaster, satellite launching etc. They follow a space curve. These are slightly complicated to analyze as 3 space variables are involved.

  • Description

We defined the quantities distance s, displacement \vec r, position vector \vec {r_p}, average speed v_{avg}, instantaneous speed v_{inst}, average velocity \vec v_{avg}, instantaneous velocity \vec v_{inst} and instantaneous acceleration \vec a_{inst} and jerk \vec j ” for rectilinear motion. Their definitions are valid for curvilinear motion as well. We will use them extensively.

  • Specifying the Quantities – Coordinate Systems

The quantities are specified by considering certain reference (the frame). There are 3 general systems of specifications:

  • Rectangular Coordinate System

cartesian
Cartesian (Rectangular) Coordinate System

It involves 3 mutually perpendicular axes, X, Y and Z. The axes intersect at a point, known as origin. These axes are fixed. Thus, the unit vectors \hat i, \hat j and \hat k are fixed. (THIS IS IMPORTANT)

Any vector can be resolved into its components by projecting it on respective axes. Thus,

{\vec r = x \hat i + y \hat j + z \hat k}

{\vec v = \frac {dx}{dt} \hat i + \frac {dy}{dt} \hat j + \frac {dz}{dt} \hat k}

{\vec a = \frac {d^2x}{dt^2} \hat i + \frac {d^2y}{dt^2} \hat j + \frac {d^2z}{dt^2} \hat k}

When motion is 2-dimensional, \hat k component is absent.

  • Specifying Radial and Transverse Components (Polar Coordinates)

rad_trans
Polar Coordinates (Radial and Transverse Components)

The radial direction is that direction to which the radius vector (or position vector) \vec r is directed. The unit vector along radial direction is denoted by \hat r. The transverse direction is obtained, when \hat r is rotated through 90^o. Thus,

{\vec r = r \hat r}

{\vec v = \frac {d}{dt} (r \hat r) = \frac {dr}{dt} \hat r + r \frac {d}{dt} \hat r = (\dot r) \hat r + (r \dot \theta) \hat s}

\hat s is the unit vector in transverse direction. Similarly,

{\vec a = \frac {d \vec v}{dt} = [\ddot r - r(\dot \theta)^2]\hat r + [2 \dot r \dot \theta + r \ddot \theta] \hat s}

\theta denotes the angular position of the particle. Hence, \dot \theta is the rate of rotation and \ddot \theta is the rate of change of rate of rotation.

Thus, both velocity and accelerations have radial and transverse components.

Note: Radial component of velocity describes the rate of movement of an object from an observer. In astronomy, the radial velocity describes how quick is the star receding the earth. Transverse component of velocity describes the rate of movement of an object perpendicular to the observer. For an observer observing a star, this is perpendicular to his line of sight.

stars
Applying Radial and Transverse Components in Astronomy

These 2 apparent components tell us the actual relative velocity of star w.r.t earth.

  • Specifying Normal and Tangential Components

tang_norm.jpg
Tangential and Normal Components

In this type, a vector is resolved along 2 components, one along the tangent to the path and the other normal to the path. The unit vector along tangent is \hat e_T and along normal is \hat e_N. Unlike \hat i , \hat j and \hat k, the unit vectors \hat e_T, \hat e_N may change their directions. (THIS IS IMPORTANT).

The velocity vector \vec v is always tangential to the path. Hence,

{\vec v = v \hat e_T + 0 \hat e_N}

{\vec a = \frac {d \vec v}{dt} = \frac {d}{dt} (v \hat e_T) = \frac {dv}{dt} \hat e_T + v \frac {d \hat e_T}{dt} .... by \ Chain \ Rule}

The time derivative of \hat e_T is not equal to zero, as it changes its direction. It is equal to \frac {v}{\rho} e_N. Therefore,

{\vec a = \frac {dv}{dt} \hat e_T + \frac {v^2}{\rho} \hat e_N}

\rho is the radius of curvature of the curve at that point.

Note: Since the velocity is always tangential to the path, it has only 1 component in this representation. Normal component is zero.

  • Concept of Radius of Curvature

The term radius is well defined for circles. We also say that for a straight line, the radius is infinite. For any other curve, radius of curvature at a point is the radius of that circle, which best fits the curve or, which shares common tangent with the curve.

Mathematically, if y=f(x) is the equation of curve, then radius of curvature is given by

{\rho = \frac {\Big[1+ \big( \frac {dy}{dx} \big)^2 \Big]^{3/2}} {\big( \frac {d^2y}{dx^2}\big) }}

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