Chapter 12 Kinematics

ME 242 Chapter 12 Question 1 We obtain the acceleration fastest (A)By taking the derivative of x(t) (B)By Integrating x(t) twice (C)By integrating the accel as function of displacement (D)By computing the time to liftoff, then choosing the accel such that the velocity is 160 mph

Question 2 The acceleration is approximately (A)92 ft/s2 (B)66 ft/s 2 (C)85.3 ft/s2 (D)182 ft/s2 (E)18.75 ft/s 2 ME 242 Chapter 12 Ya pili 160 mi/h = 235 ft/s

Question 2 The acceleration is approximately (A)92 ft/s 2 (B)66 ft/s2 (C)85.3 ft/s2 (D)182 ft/s2 (E) ft/s 2 ME 242 Chapter 12 Ya kwanza Solution: 160 mi/h = 235 ft/s We use v*dv = a*dx Integrate: 1 / 2 v 2 = a*d, where d is the length of the runway, and the start velocity = 0

Question 3 Road map: We obtain the velocity fastest (A)By Taking the derivative of a(t) (B)By Integrating a(t) (C)By integrating the accel as function of displacement (D)By computing the time to bottom, then computing the velocity.

ME242 Tutoring Graduate Assistant Ms. Yang Liu will be available to assist with homework preparation and answer questions. Coordination through the Academic Success Center in TBE-A 207 Tuesday and Friday mornings. Contact hours: MW after class

ME242 Reading Assignments Look up the next Homework assignment on Mastering Example: your second assignment covers sections 12.5 and 12.6 Study the text and practice the examples in the book An I-Clicker reading test on each chapter section will be given at the start of each lecture More time for discussion and examples

Supplemental Instruction ME 242 Questions –Yang Liu – PhD student in ME –Lab: SEB 4261

Chapter 12-5 Curvilinear Motion X-Y Coordinates

Here is the solution in Mathcad

Example: Hit target at Position (360’, -80’)

Example: Hit target at Position (360, -80)

NORMAL AND TANGENTIAL COMPONENTS (Section 12.7) When a particle moves along a curved path, it is sometimes convenient to describe its motion using coordinates other than Cartesian. When the path of motion is known, normal (n) and tangential (t) coordinates are often used. In the n-t coordinate system, the origin is located on the particle (the origin moves with the particle). The t-axis is tangent to the path (curve) at the instant considered, positive in the direction of the particle’s motion. The n-axis is perpendicular to the t-axis with the positive direction toward the center of curvature of the curve.

NORMAL AND TANGENTIAL COMPONENTS (continued) The position of the particle at any instant is defined by the distance, s, along the curve from a fixed reference point. The positive n and t directions are defined by the unit vectors u n and u t, respectively. The center of curvature, O’, always lies on the concave side of the curve. The radius of curvature, , is defined as the perpendicular distance from the curve to the center of curvature at that point.

ACCELERATION IN THE n-t COORDINATE SYSTEM Acceleration is the time rate of change of velocity: a = dv/dt = d(vu t )/dt = vu t + vu t.. Here v represents the change in the magnitude of velocity and u t represents the rate of change in the direction of u t.... a = v u t + (v 2 /  ) u n = a t u t + a n u n. After mathematical manipulation, the acceleration vector can be expressed as:

ACCELERATION IN THE n-t COORDINATE SYSTEM (continued) So, there are two components to the acceleration vector: a = a t u t + a n u n The normal or centripetal component is always directed toward the center of curvature of the curve. a n = v 2 /  The tangential component is tangent to the curve and in the direction of increasing or decreasing velocity. a t = v or a t ds = v dv. The magnitude of the acceleration vector is a = [(a t ) 2 + (a n ) 2 ] 0.5

NORMAL AND TANGENTIAL COMPONENTS (Section 12.7) When a particle moves along a curved path, it is sometimes convenient to describe its motion using coordinates other than Cartesian. When the path of motion is known, normal (n) and tangential (t) coordinates are often used. In the n-t coordinate system, the origin is located on the particle (the origin moves with the particle). The t-axis is tangent to the path (curve) at the instant considered, positive in the direction of the particle’s motion. The n-axis is perpendicular to the t-axis with the positive direction toward the center of curvature of the curve.

Normal and Tangential Coordinates Velocity Page 53

Normal and Tangential Coordinates ‘e’ denotes unit vector (‘u’ in Hibbeler)

‘e’ denotes unit vector (‘u’ in Hibbeler)

Learning Techniques Complete Every Homework Team with fellow students Study the Examples Ask: Ms. Yang, peers, me Mathcad provides structure and numerically correct results

Course Concepts Math Think Conceptually Map your approach BEFORE starting work

12.8 Polar coordinates

Polar coordinates ‘e’ denotes unit vector (‘u’ in Hibbeler)

Polar coordinates ‘e’ denotes unit vector (‘u’ in Hibbeler)

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12.10 Relative (Constrained) Motion v A is given as shown. Find v B Approach: Use rel. Velocity: v B = v A +v B/A (transl. + rot.)

Vectors and Geometry x y  t  r(t) 

Make a sketch: A V_rel v_Truck B The rel. velocity is: V_Car/Truck = v_Car -vTruck Relative (Constrained) Motion V_truck = 60 V_car = 65

Example Vector equation: Sailboat tacking at 50 deg. against Northern Wind (blue vector) We solve Graphically (Vector Addition)

Example Vector equation: Sailboat tacking at 50 deg. against Northern Wind An observer on land (fixed Cartesian Reference) sees V wind and v Boat. Land

Plane Vector Addition is two-dimensional Relative (Constrained) Motion vBvB vAvA v B/A

Example cont’d: Sailboat tacking against Northern Wind 2. Vector equation (1 scalar eqn. each in i- and j- direction). Solve using the given data (Vector Lengths and orientations) and Trigonometry i

Chapter Relative Motion

Vector Addition

Differentiating gives:

Exam 1 We will focus on Conceptual Solutions. Numbers are secondary. Train the General Method Topics: All covered sections of Chapter 12 Practice: Train yourself to solve all Problems in Chapter 12

Exam 1 Preparation: Start now! Cramming won’t work. Questions: Discuss with your peers. Ask me. The exam will MEASURE your knowledge and give you objective feedback.

Exam 1 Preparation: Practice: Step 1: Describe Problem Mathematically Step2: Calculus and Algebraic Equation Solving

And here a few visual observations about contemporary forms of socializing, sent to me by a colleague at the Air Force Academy. Enjoy!

Having coffee with friends.

A day at the beach.

Cheering on your team.

Out on an intimate date.

Having a conversation with your BFF

A visit to the museum

Enjoying the sights