King Fahd University of Petroleum Minerals Mechanical Engineering

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 • Acceleration a = ax i + ay j +az k (tangent to path) (tangent to hodograph) – Normal and Tangential Components – Polar & Cylindrical Components

12. 7 Normal and Tangential Components • If the path is known i. e. – Circular track with given radius – Given function • Method of choice is normal and tangential components

Position • From the given geometry and/or given function • More emphasis on radius of curvature velocity and acceleration

Planer Motion • 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) • For the Circular motion : (r) = radius of the circle • For y = f(x):

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 • 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 = 5 s. Note: speed increasing at # this means the tangential acceleration

Problem • 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)

Centripetal acceleration • Recall that acceleration is defined as a change in velocity with respect to time. • Since velocity is a vector quantity, a change in the velocity’s direction , even though the speed is constant, represents an acceleration. • This type of acceleration is known as Centripetal acceleration

Problem • 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