EQUATIONS OF MOTION: CYLINDRICAL COORDINATES

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Stuart Potter
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1 Today s Objectives: Students will be able to: 1. Analyze the kinetics of a particle using cylindrical coordinates. EQUATIONS OF MOTION: CYLINDRICAL COORDINATES 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

2 READING QUIZ 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 B) tangential C) transverse D) None of the above.

3 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?

4 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?

5 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:... F r = ma r = m (r r q 2 ).... F q = ma q = m (r q 2 r q ).. F z = ma z = m z

6 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.... F r = ma r = m(r rq 2 )... F q = ma q = m(rq 2rq. ) Note that a fixed coordinate system is used, not a bodycentered system as used in the n t approach.

7 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 path s 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.

8 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. tan y = r dq dr = r dr/dq If y is positive, it is measured counterclockwise from the radial line to the tangent. If it is negative, it is measured clockwise.

9 EXAMPLE Given: The ball (P) is guided along the vertical circular. path. W.. = 0.5 lb, q = 0.4 rad/s, q = 0.8 rad/s 2, r c = 0.4 ft Find: Plan: Force of the arm OA on the ball when q = 30.

10 EXAMPLE Given: The ball (P) is guided along the vertical circular. path. W.. = 0.5 lb, q = 0.4 rad/s, q = 0.8 rad/s 2, r c = 0.4 ft Find: Force of the arm OA on the ball when q = 30. Plan: Draw a FBD. Then develop the kinematic equations and finally solve the kinetics problem using cylindrical coordinates. Solution: Notice that r = 2r c cos q, therefore:.. r = -2r c sin q q..... r = -2r c cos q q 2 2r c sin q q

11 EXAMPLE (continued) Free Body Diagram and Kinetic Diagram : Establish the r, q inertial coordinate system and draw the particle s free body diagram. mg = ma q ma r N s 2q N OA

12 EXAMPLE (continued) Kinematics: at q = 30 r = 2(0.4) cos(30 ) = ft. r = - 2(0.4) sin(30 )(0.4) = ft/s.. r = - 2(0.4) cos(30 )(0.4) 2 2(0.4) sin(30 )(0.8) = ft/s 2 Acceleration components are... a r = r rq 2 = (0.693)(0.4) 2 = ft/s a q = rq + 2rq = (0.693)(0.8) + 2(-0.16)(0.4) = ft/s 2

13 F r = ma r 0.5 N s cos(30 ) 0.5 sin(30 ) = (-0.542) 32.2 N s = lb EXAMPLE (continued) Equation of motion: r direction ma q ma r =

F q = ma q 0.5 N OA + 0.279 sin(30 ) 0.5 cos(30 ) = (0.426) 32.

14 F q = ma q 0.5 N OA sin(30 ) 0.5 cos(30 ) = (0.426) 32.2 N OA = 0.3 lb EXAMPLE (continued) Equation of motion: q direction ma q ma r =

15 CONCEPT QUIZ 1. When a pilot flies an airplane in a vertical loop of constant radius r at C r constant speed v, his apparent weight is maximum at D A) Point A B) Point B (top of the loop) C) Point C D) Point D (bottom of the loop) 2. If needing to solve a problem involving the pilot s 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. B A

GROUP PROBLEM SOLVING Given: The smooth particle is attached to an elastic cord extending from O to P and due to the slotted arm guide moves along the horizontal circular path.

16 GROUP PROBLEM SOLVING Given: The smooth particle is attached to an elastic cord extending from O to P and due to the slotted arm guide moves along the horizontal circular path. The cord s stiffness is k=30 N/m unstretched length = 0.25 m m=0.08 kg, r = (0.8 sin q) m Find: Forces of the guide on the particle when q = 60 and q = 5 rad/s, which is constant. Plan:... Determine r and r by differentiating r. Draw Free Body Diagram & Kinetic Diagram. Solve for the accelerations, and apply the equation of motion to find the forces.

17 Solution: GROUP PROBLEM SOLVING (continued) Kinematics: r = 0.8 (sin q ) r = 0.8 (cos q ) q r = -0.8 (sin q ) q (cos q ) q When q = 60 ; q = 5 rad/s, q = 0 rad/s 2. r = m r = 2 m/s r = m/s 2 Accelerations : a r = r r q 2 = (0.6928) 5 2 = m/s 2 a q = r q + 2 r q = (0.6928) (2) 5 = 20 m/s 2

18 GROUP PROBLEM SOLVING (continued) Free Body Diagram & Kinetic Diagram = ma q ma r where the spring force F will be F s = k s = 30 ( ) = N Kinetics: F r = ma r => N cos 30 = 0.08 (-34.64) N = 12.1 N

19 GROUP PROBLEM SOLVING (continued) Kinetics: F q = ma q => F - N sin 30 = 0.08 (20) F = 7.67 N = ma q ma r

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

EQUATIONS OF MOTION: CYLINDRICAL COORDINATES (Section 13.6)

EQUATIONS OF MOTION: CYLINDRICAL COORDINATES (Section 13.6) Today s Objectives: Students will be able to analyze the kinetics of a particle using cylindrical coordinates. APPLICATIONS The forces acting