MEE 214 (Dynamics) Tuesday 8.30-11.20 Dr. Soratos Tantideeravit (สรทศ ตันติธีรวิทย์) soratos@oaep.go.th Lecture Notes, Course updates, Extra problems, etc No Homework Final Exam (Date & Time – TBD) 29/11/61 MEE214 – Dynamics
Course Overview Kinematics of a Particle Kinetics of a Particle Rectilinear and Curvilinear Motion Kinetics of a Particle Force and Acceleration Work and Energy Impulse and Momentum Kinetics of a System of Particles 29/11/61 MEE214 – Dynamics
Introduction to Dynamics Statics Engineering Mechanics Dynamics Kinematics Kinetics 29/11/61 MEE214 – Dynamics
Kinematics of a particle
Objectives Concepts of position, displacement, velocity, and acceleration. Particle motion along a straight line Particle motion along a curved path using different coordinate systems. Analysis of dependent motion of two particles. Principles of relative motion of two particles using translating axes. 29/11/61 MEE214 – Dynamics
Rectilinear Kinematics Origin Define a fixed point in space Position Defined by a position vector r or an algebraic scalar s 29/11/61 MEE214 – Dynamics
Rectilinear Kinematics Displacement Change in position Velocity 29/11/61 MEE214 – Dynamics
Rectilinear Kinematics Acceleration 29/11/61 MEE214 – Dynamics
Constant Acceleration 29/11/61 MEE214 – Dynamics
Problem 12-31 The acceleration of a particle along a straight line is defined by a=(2t-9) m/s2, where t is in seconds. At t=0, s=1 m and v=10 m/s. When t=9 s, determine (a) the particle’s position, (b) the total distance traveled and (c) the velocity. 29/11/61 MEE214 – Dynamics
General Curvilinear Motion Curvilinear motion occurs when the particle moves along a curved path Position. The position of the particle, measured from a fixed point O, is designated by the position vector r = r(t). 29/11/61 MEE214 – Dynamics
General Curvilinear Motion Displacement. Suppose during a small time interval Δt the particle moves a distance Δs along the curve to a new position P`, defined by r` = r + Δr. The displacement Δr represents the change in the particle’s position. 29/11/61 MEE214 – Dynamics
General Curvilinear Motion Velocity 29/11/61 MEE214 – Dynamics
General Curvilinear Motion Acceleration. 29/11/61 MEE214 – Dynamics
Curvilinear Motion: Rectangular Components Position. Position vector is defined by The magnitude of is always positive and defined as The direction of r is specified by the components of the unit vector 29/11/61 MEE214 – Dynamics
Curvilinear Motion: Rectangular Components Velocity. The velocity has a magnitude defined as the positive value of 29/11/61 MEE214 – Dynamics
Curvilinear Motion: Rectangular Components Acceleration. The acceleration has a magnitude defined as the positive value of 29/11/61 MEE214 – Dynamics
Curvilinear Motion: Rectangular Components The acceleration has a direction specified by the components of the unit vector . Since a represents the time rate of change in velocity, a will not be tangent to the path. 29/11/61 MEE214 – Dynamics
Motion of Projectile Constant downward acceleration, no air resistance Mathematical expressions, ↑ [=] +, → [=] + 29/11/61 MEE214 – Dynamics
Example 12.12 The chipping machine is designed to eject wood chips at vO = 25 ft/s. If the tube is oriented at 30° from the horizontal, determine how high, h, the chips strike the pile if they land on the pile 20 ft from the tube. 29/11/61 MEE214 – Dynamics
Curvilinear Motion: Normal and Tangential Components When the path of motion of a particle is known, describe the path using n and t coordinates which act normal and tangent to the path Consider origin located at the particle Tangential direction Normal direction 29/11/61 MEE214 – Dynamics
Curvilinear Motion: Normal and Tangential Components Velocity. Since the particle is moving, s is a function of time Particle’s velocity v has direction that is always tangent to the path and a magnitude that is determined by taking the time derivative of the path function s = s(t) 29/11/61 MEE214 – Dynamics
Curvilinear Motion: Normal and Tangential Components Acceleration Acceleration of the particle is the time rate of change of velocity 29/11/61 MEE214 – Dynamics
Curvilinear Motion: Normal and Tangential Components Acceleration Find 29/11/61 MEE214 – Dynamics
Curvilinear Motion: Normal and Tangential Components 29/11/61 MEE214 – Dynamics
Problem 12-120 The automobile is originally at rest at s=0. If it then starts to increase its speed at ft/s2 where t is in seconds, determine the magnitudes of its velocity and acceleration at s = 550 ft. 29/11/61 MEE214 – Dynamics
Problem 12-131 At a given instant the train engine at E has a speed of 20 m/s and an acceleration of 14 m/s2 acting in the direction shown. Determine the rate of increase in the train’s speed and the radius of curvature of the path. 29/11/61 MEE214 – Dynamics
Problem 12-152 If the speed of the box at point on the track is 30ft/s which is increasing at the rate of ft/s2 , determine the magnitude of the acceleration of the box at this instant. 29/11/61 MEE214 – Dynamics
Curvilinear Motion: Cylindrical Components Fixed origin Radial direction Transverse direction 29/11/61 MEE214 – Dynamics
Curvilinear Motion: Cylindrical Components Position 29/11/61 MEE214 – Dynamics
Curvilinear Motion: Cylindrical Components or Polar Velocity 29/11/61 MEE214 – Dynamics
Curvilinear Motion: Cylindrical Components Acceleration 29/11/61 MEE214 – Dynamics
Example 12-19 The searchlight casts a spot of light along the face of a wall that is located 100m from the searchlight. Determine the magnitudes of the velocity and acceleration at which the spot appears to travel across the wall at the instant θ = 45°. The searchlight is rotating at a constant rate of 4 rad/s 29/11/61 MEE214 – Dynamics
Problem 12-184 The slotted arm AB drives pin C through the spiral groove described by the equation r = (1.5Ө) ft, where Ө is in radians. If the arm starts from rest when Ө = 60˚ and is driven at an angular velocity of Ө̇ = (4t) rad/s, where t is in seconds, determine the radial and transverse components of velocity and acceleration of the pin C when t=1 s. 29/11/61 MEE214 – Dynamics
Absolute Dependent Motion Dependent motions of two particles are normally associated with systems of connected masses via inextensible cords and pulleys. 29/11/61 MEE214 – Dynamics
Absolute Dependent Motion 29/11/61 MEE214 – Dynamics
Problem 12-206 If the hydraulic cylinder at H draws in rod BC by 200 mm at 2ft/s, determine how far the slider A moves and the speed of the slider. 29/11/61 MEE214 – Dynamics
Example 3 A man at A is hoisting a safe S by walking to the right with a constant velocity vA = 0.5m/s. Determine the velocity and acceleration of the safe when it reaches the elevation at E. The rope is 30m long and passes over a small pulley at D. 29/11/61 MEE214 – Dynamics
Problem 12-208 If block A is moving downward with a speed of 6 ft/s while C is moving down at 18 m/s, determine the speed of block B. 29/11/61 MEE214 – Dynamics
Relative Motion Analysis The relative position of B with respect to A is given by The relative velocity and acceleration of B with respect to A are given by 29/11/61 MEE214 – Dynamics
Example 4 A train, traveling at a constant speed of 60 mi/h, crosses over a road. If automobile A is traveling t 45 mi/h along the road, determine the magnitude and direction of relative velocity of the train with respect to the automobile. 29/11/61 MEE214 – Dynamics
Problem 12-149 The two particles A and B start at the origin O and travel in opposite directions along the circular path at constant speeds vA=0.7 m/s and vB=1.5 m/s respectively. Determine at t= 2s, (a) the displacement along the path of each particle, (b) the position vector to each particle, and (c) the magnitude of the acceleration of particle B. 29/11/61 MEE214 – Dynamics