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 # 3

12. 3 Rectangular Kinematics: Erratic Motion Omitted

Objective • To investigate particle motion along a curved path using three coordinate systems – Rectangular Components – Normal and Tangential Components – Polar & Cylindrical Components

• Section 12. 4 in your text • Path is described in three dimensions • Position, velocity, and acceleration are vectors

Position * S is a path function * The position of the particle measured from a fixed point O is given by the position vector r = r(t) Example : r = {sin (2 t) i + cos (2 t) j – 0. 5 t k}

Displacement • The displacement Dr represents the change in the particle’s position • Dr = r’ - r

Velocity • Average velocity • Instantaneous velocity • As Dt = 0 then Dr = Ds • Speed • Since D r is tangent to the curve at P, then the velocity is tangent to the curve

Acceleration • Average acceleration: • Hodograph curve “velocity arrowhead points” • Instantaneous acceleration: Hodograph

Acceleration (con. ) • a acts tangent to the hodograph • a is not tangent to the path of motion • a directed toward the inside or concave side

12. 5 Curvilinear Motion: Rectangular Components • Rectangular : x, y, z frame

Position • Position vector r • r=xi+yj+zk • The magnitude of r is always positive and defined as • Unit vector • The direction cosines are

Velocity • Velocity is the first time derivative of r • Where • Magnitude of velocity • Direction is always tangent to the path

Problem • The position of a particle is described by r. A= {2 t i +(t 2 -1) j} ft. where t is in seconds. Determine the position of the point and the speed at 2 second. •

Acceleration • Acceleration is the first time derivative of v • Where • Magnitude of acceleration • Direction is not tangent to the path

Example 12. 9 • The distance of the balloon from A at 2 sec • The magnitude and direction of velocity at 2 sec • The magnitude and direction of acceleration at 2 sec Position Velocity Acceleration A X=8 t

Example 12. 10 t in second arguments in radians At t = 0. 75 s find location, velocity, and acceleration Note: Put your calculator in Rad Mode