Section 13-6 - cbafaculty.org

EQUATIONS OF MOTION: CYLINDRICAL COORDINATES Todays Objectives: Students will be able to: 1. Analyze the kinetics of a particle using cylindrical coordinates. Dynamics, Fourteenth Edition R.C. Hibbeler In-Class Activities: Check Homework Reading Quiz Applications Equations of Motion using Cylindrical Coordinates Angle between Radial and Tangential Directions Concept Quiz Group Problem Solving Attention Quiz

Copyright 2016 by Pearson Education, Inc. All rights reserved. READINGQUIZ QUIZ READING 1. The normal force which the path exerts on a particle is always perpendicular to the _________ A) radial line. B) transverse direction. C) tangent to the path. D) None of the above. 2. When the forces acting on a particle are resolved into cylindrical components, friction forces always act in the __________ direction. A) radial C) transverse Dynamics, Fourteenth Edition

R.C. Hibbeler B) tangential D) None of the above. Copyright 2016 by Pearson Education, Inc. All rights reserved. APPLICATIONS The forces acting on the 100-lb boy can be analyzed using the cylindrical coordinate system. How would you write the equation describing the frictional force on the boy as he slides down this helical slide? Dynamics, Fourteenth Edition R.C. Hibbeler Copyright 2016 by Pearson Education, Inc.

All rights reserved. APPLICATIONS (continued) When an airplane executes the vertical loop shown above, the centrifugal force causes the normal force (apparent weight) on the pilot to be smaller than her actual weight. How would you calculate the velocity necessary for the pilot to experience weightlessness at A? Dynamics, Fourteenth Edition R.C. Hibbeler Copyright 2016 by Pearson Education, Inc. All rights reserved. CYLINDRICAL COORDINATES (Section 13.6) This approach to solving problems has some external similarity to the normal & tangential method just studied. However, the path may be more complex or the problem may have other attributes that

make it desirable to use cylindrical coordinates. Equilibrium equations or Equations of Motion in cylindrical coordinates (using r, q , and z coordinates) may be expressed in scalar form as: . .. Fr = mar = m (r ..r q 2 ) . . Fq = maq = m .. (r q 2 r q ) Fz = maz = m z Dynamics, Fourteenth Edition R.C. Hibbeler Copyright 2016 by Pearson Education, Inc. All rights reserved. CYLINDRICAL COORDINATES (continued) If the particle is constrained to move only in the r q plane (i.e., the z coordinate is constant), then only the first

two equations are used (as shown below). The coordinate system in such a case becomes a polar coordinate system. In this case, the path is only a function of q. . .. Fr = mar = m(r rq 2 ). .. . Fq = maq = m(rq 2rq ) Note that a fixed coordinate system is used, not a bodycentered system as used in the n t approach. Dynamics, Fourteenth Edition R.C. Hibbeler Copyright 2016 by Pearson Education, Inc. All rights reserved. TANGENTIAL AND NORMAL FORCES If a force P causes the particle to move along a path defined by r = f (q ), the normal force N exerted by the path on the particle is always perpendicular to the paths tangent. The frictional force F always acts along the tangent in the opposite direction of motion. The directions of N and F can

be specified relative to the radial coordinate by using angle y . Dynamics, Fourteenth Edition R.C. Hibbeler Copyright 2016 by Pearson Education, Inc. All rights reserved. DETERMINATION OF ANGLE y The angle y, defined as the angle between the extended radial line and the tangent to the curve, can be required to solve some problems. It can be determined from the following relationship. If y is positive, it is measured counterclockwise from the radial line to the tangent. If it is negative, it is measured clockwise. Dynamics, Fourteenth Edition R.C. Hibbeler Copyright 2016 by Pearson Education, Inc.

All rights reserved. EXAMPLE Given: The 0.2 kg pin (P) is constrained to move in the smooth curved slot, defined by r = (0.6 cos 2q ) m. The slotted arm OA has a constant angular velocity of = 3 rad/s. Motion is in the vertical plane. Find: Force of the arm OA on the pin P when q = 0. Plan: Dynamics, Fourteenth Edition R.C. Hibbeler Copyright 2016 by Pearson Education, Inc.

All rights reserved. n: EXAMPLE Given: The 0.2 kg pin (P) is constrained to move in the smooth curved slot, defined by r = (0.6 cos 2q) m. The slotted arm OA has a constant angular velocity of = 3 rad/s. Motion is in the vertical plane. Find: Force of the arm OA on the pin P when q = 0. 1) Draw the FBD and kinetic diagrams. 2) Develop the kinematic equations using cylindrical coordinates. 3) Apply the equation of motion to find the force.

Dynamics, Fourteenth Edition R.C. Hibbeler Copyright 2016 by Pearson Education, Inc. All rights reserved. EXAMPLE (continued) Solution : 1) Free Body and Kinetic Diagrams: Establish the r, q coordinate system when q = 0, and draw the free body and kinetic diagrams. Free-body diagram q W r Kinetic diagram =

maq mar N Dynamics, Fourteenth Edition R.C. Hibbeler Copyright 2016 by Pearson Education, Inc. All rights reserved. EXAMPLE (continued) 2) Notice that , therefore: Kinematics: at q = 0, = 3 rad/s, = 0 rad/s2. Acceleration components are ar = = - 21.6 (0.6)(-3)2 = 27 m/s2 aq = = (0.6)(0) + 2(0)(-3) = 0 m/s2 Dynamics, Fourteenth Edition R.C. Hibbeler Copyright 2016 by Pearson Education, Inc.

All rights reserved. EXAMPLE (continued) Equation of motion: q direction (+) Fq = maq N 0.2 (9.81) = 0.2 (0) N = 1.96 N Free-body diagram q W r N Dynamics, Fourteenth Edition R.C. Hibbeler Kinetic diagram = maq mar

ar = 27 m/s2 aq = 0 m/s2 Copyright 2016 by Pearson Education, Inc. All rights reserved. CONCEPT QUIZ 1. When a pilot flies an airplane in a vertical loop of constant radius r at constant speed v, his apparent weight is maximum at A) Point A C) Point C B C r B) Point B (top of the loop) D) Point D (bottom of the loop)

A D 2. If needing to solve a problem involving the pilots weight at Point C, select the approach that would be best. A) Equations of Motion: Cylindrical Coordinates B) Equations of Motion: Normal & Tangential Coordinates C) Equations of Motion: Polar Coordinates D) No real difference all are bad. E) Toss up between B and C. Dynamics, Fourteenth Edition R.C. Hibbeler Copyright 2016 by Pearson Education, Inc. All rights reserved. lan: GROUP PROBLEM SOLVING I Given: The smooth can C is lifted from A to B by a rotating rod.

The mass of can is 3 kg. Neglect the effects of friction in the calculation and the size of the can so that r = (1.2 cos q) m. Find: Forces of the rod on the can when q = 30 and = 0.5 rad/s, which is constant. 1) Find the acceleration components using the kinematic equations. 2) Draw free body diagram & kinetic diagram. 3) Apply the equation of motion to find the forces. Dynamics, Fourteenth Edition R.C. Hibbeler Copyright 2016 by Pearson Education, Inc. All rights reserved. GROUP PROBLEM SOLVING (continued) Solution: 1) Kinematics: When q = 30, = 0.5 rad/s and = 0 rad/s2.

= 1.039 m = 0.3 m/s = 0.2598 m/s2 Accelerations: ar = = 0.2598 (1.039) 0.52 = 0.5196 m/s2 aq = + 2 = (1.039) 0 + 2 (0.3) 0.5 = 0.3 m/s2 Dynamics, Fourteenth Edition R.C. Hibbeler Copyright 2016 by Pearson Education, Inc. All rights reserved. GROUP PROBLEM SOLVING (continued) 2) Free Body Diagram q 30 Kinetic Diagram 3(9.81) N maq

r mar = 30 N F Apply equation of motion: Fr = mar -3(9.81) sin30 + N cos30 = 3 (-0.5196) Fq = maq F + N sin30 3(9.81) cos30 = 3 (-0.3) N = 15.2 N, F = 17.0 N Dynamics, Fourteenth Edition R.C. Hibbeler Copyright 2016 by Pearson Education, Inc. All rights reserved.

ATTENTION QUIZ 1. For the path defined by r = q rad is A) 10 C) 26 2 , the angle y at q = 0.5 B) 14 D) 75 2. If r = q 2 and q = 2t, find the magnitude of and when t = 2 seconds. A) 4 cm/sec, 2 rad/sec2 B) 4 cm/sec, 0 rad/sec2 C) 8 cm/sec, 16 rad/sec2 D) 16 cm/sec, 0 rad/sec2