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King Fahd University of Petroleum Minerals

- Mechanical Engineering
- Dynamics ME 201
- BY
- Dr. Meyassar N. Al-Haddad
- Lecture 5

Objective

- To investigate particle motion along a curved

path Curvilinear Motion using three coordinate

systems - Rectangular Components
- Position vector r x i y j z k
- Velocity v vx i vy j vz k

(tangent to path) - Acceleration a ax i ay j az k

(tangent to hodograph) - Normal and Tangential Components
- Polar Cylindrical Components

12.6 Normal and Tangential Components

- If the path is known i.e.
- Circular track with given radius
- Given function
- Method of choice in normal and tangential

components - At any instant the origin is located at the

particle it self

Position

- Direct question
- From the geometry
- Less emphasis in n t
- More emphasis on radius of curvature velocity and

acceleration

Planer Motion

- The origin point is coincide with the location of

the particle. - At any instant the origin is located at the

particle it self - The t axis is tangent to the curve at P and in

the direction of increasing s. - The normal axis is perpendicular to t and

directed toward the center of curvature O. - un is the unit vector in normal direction
- ut is a unit vector in tangent direction

Radius of curvature (r)

- Circle (r) radius of the circle
- y f(x) is given by

Example

- Find the radius of curvature of the parabolic

path in the figure at x 150 ft.

Velocity

- The particle velocity is always tangent to the

path. - Magnitude of velocity is the time derivative of

path function s s(t) - From constant tangential acceleration
- From time function of tangential acceleration
- From acceleration as function of distance

Example 1

- A skier travel with a constant speed of 20 ft/s

along the parabolic path shown. Determine the

velocity at x 150 ft.

Problem 12.103

- A boat is traveling a long a circular curve. If

its speed at t 0 is 15 ft/s and is increasing

at , determine the

magnitude of its velocity at the instant t 5s.

Problem 12.106

- A truck is traveling a long a circular path

having a radius of 50 m at a speed of 4 m/s. For

a short distance from s 0, its speed is

increased by . Where s is

in meters. Determine its speed when it moved s

10 m.

Acceleration

- Acceleration is time derivative of velocity

Special case

- 1- Straight line motion
- 2- Constant speed curve motion (centripetal

acceleration)

Motion in a Circle

- Recall that acceleration is defined as a change

in velocity with respect to time. - Since velocity is a vector quantity, a change in

the velocitys direction , even though the speed

is constant, represents an acceleration. - This type of acceleration is known as
- Centripetal acceleration
- ac v2/r

Acceleration

- 3 types of acceleration
- linear
- radial (centripetal)
- angular

Acceleration

- Linear acceleration is a change in speed without

change in direction (increase in thrust in

straight-and-level flight) - Radial (or centripetal) acceleration when there

is a change in direction (turn, dive) - Angular acceleration when body speed and

direction are changed (tight spin)

Problem 12.106

- A truck is traveling a long a circular path

having a radius of 50 m at a speed of 4 m/s. For

a short distance from s 0, its speed is

increased by . Where s is

in meters. Determine its speed and the magnitude

of its acceleration when it moved s 10 m.

Review

- Example 12-14
- Example 12-15
- Example 12-16

Three-Dimensional Motion

- For spatial motion required three dimension.
- Binomial axis b which is perpendicular to ut and

un is used - ub ut x un

Thank You