Curvilinear motion is defined as motion that occurs when a particle travels along a curved path. The curved path can be in two dimensions (in a plane), or in three dimensions. This type of motion is more complex than rectilinear (straight-line) motion.

Three-dimensional curvilinear motion describes the most general case of motion for a particle.

To find the velocity and acceleration of a particle experiencing curvilinear motion one only needs to know the position of the particle as a function of time.

Let’s say we are given the position of a particle

The velocity of the particle

The acceleration of the particle

As you can see, if we know the position of a particle as a function of time, it is a fairly simple exercise to find the velocity and acceleration. You simply take the first derivative to find the velocity and the second derivative to find the acceleration.

The magnitude of the velocity of particle

The magnitude of the acceleration of particle

Note that the direction of velocity of the particle

However, the acceleration component tangent to the curve is equal to the time derivative of the magnitude of velocity of the particle

In addition, the acceleration component normal to the curve (

where

The figure below illustrates the acceleration components

For the specific case where the path of the blue curve is given by

where |

However, it is usually not necessary to know the radius of curvature

It is sometimes convenient to express the planar (two-dimensional) motion of a particle in terms of polar coordinates (

For a particle

Note that the circumferential direction is perpendicular to the radial direction.

The position of the particle

To find the velocity, take the first derivative of

To find the acceleration, take the second derivative of

Without loss of generality we can evaluate the velocities and accelerations at angle

Setting

Equations (1), (2), (3), and (4) fully describe the curvilinear motion of a particle

The term

The term

Since

Since

A slotted link is rotating about fixed pivot

The angle

The radial velocity of the rod is given by equation (1):

(The radial velocity is in the direction of increasing

The circumferential velocity of the rod is given by equation (3):

(The circumferential velocity is in the direction of increasing

The radial acceleration of the rod is given by equation (2):

(The radial acceleration is in the direction of decreasing

The circumferential acceleration of the rod is given by equation (4):

(The circumferential acceleration is in the direction of increasing

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