EQUATIONS OF MOTION:

CYLINDRICAL COORDINATES

Today’s Objectives: In-Class Activities:

Students will be able to:
1. Analyze the kinetics of a • Check Homework
particle using cylindrical • Reading Quiz
coordinates. • Applications
• Equations of Motion using
Cylindrical Coordinates
• Angle between Radial and
Tangential Directions
• Example Problem
• Concept Quiz
• Group Problem Solving
• Attention Quiz
Dynamics, Fourteenth Edition in SI Units Copyright ©2017 by Pearson Education, Ltd.
R.C. Hibbeler All rights reserved.
 READINGQUIZ
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.

Dynamics, Fourteenth Edition in SI Units Copyright ©2017 by Pearson Education, Ltd.

R.C. Hibbeler All rights reserved.
 APPLICATIONS

The forces acting on the 50-kg

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 in SI Units Copyright ©2017 by Pearson Education, Ltd.

R.C. Hibbeler 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 in SI Units Copyright ©2017 by Pearson Education, Ltd.

R.C. Hibbeler 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 ) .
.
(r q – 2 r q )
 Fq = maq = m ..
 Fz = maz = m z
Dynamics, Fourteenth Edition in SI Units Copyright ©2017 by Pearson Education, Ltd.
R.C. Hibbeler 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 “body-
centered” system as used in the n – t approach.

Dynamics, Fourteenth Edition in SI Units Copyright ©2017 by Pearson Education, Ltd.

R.C. Hibbeler 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 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 .

Dynamics, Fourteenth Edition in SI Units Copyright ©2017 by Pearson Education, Ltd.

R.C. Hibbeler 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 in SI Units Copyright ©2017 by Pearson Education, Ltd.

R.C. Hibbeler 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 in SI Units Copyright ©2017 by Pearson Education, Ltd.

R.C. Hibbeler 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.
n: 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 in SI Units Copyright ©2017 by Pearson Education, Ltd.
R.C. Hibbeler All rights reserved.
 EXAMPLE (continued)

Solution :
1) Free Body and Kinetic Diagrams:
Establish the r,  coordinate
system when  = 0, and draw
the free body and kinetic
diagrams.

Free-body diagram Kinetic diagram


W ma
r mar
=
N
Dynamics, Fourteenth Edition in SI Units Copyright ©2017 by Pearson Education, Ltd.
R.C. Hibbeler 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 in SI Units Copyright ©2017 by Pearson Education, Ltd.

R.C. Hibbeler All rights reserved.
 EXAMPLE (continued)

Equation of motion:  direction

(+)  Fq = maq
N – 0.2 (9.81) = 0.2 (0)
N = 1.96 N 

Free-body diagram Kinetic diagram


W ma
r mar
= ar = –27 m/s2

N aq = 0 m/s2

Dynamics, Fourteenth Edition in SI Units Copyright ©2017 by Pearson Education, Ltd.

R.C. Hibbeler All rights reserved.
 CONCEPT QUIZ

B
1. When a pilot flies an airplane in a
vertical loop of constant radius r at
C r A
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.
Dynamics, Fourteenth Edition in SI Units Copyright ©2017 by Pearson Education, Ltd.
R.C. Hibbeler All rights reserved.
 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 ) m.

Find: Forces of the rod on the can when  = 30 and

= 0.5 rad/s, which is constant.
lan: 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 in SI Units Copyright ©2017 by Pearson Education, Ltd.
R.C. Hibbeler All rights reserved.
 GROUP PROBLEM SOLVING (continued)

Solution:
1) Kinematics:

When  = 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 in SI Units Copyright ©2017 by Pearson Education, Ltd.

R.C. Hibbeler All rights reserved.
 GROUP PROBLEM SOLVING (continued)

2) Free Body Diagram Kinetic Diagram


3(9.81) N ma
30
r mar
=
30 F
N
Apply equation of motion:
 Fr = mar  -3(9.81) sin30 + N cos30 = 3 (-0.5196)
 F = ma  F + N sin30  3(9.81) cos30 = 3 (-0.3)
N = 15.2 N, F = 17.0 N

Dynamics, Fourteenth Edition in SI Units Copyright ©2017 by Pearson Education, Ltd.

R.C. Hibbeler All rights reserved.
 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 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

Dynamics, Fourteenth Edition in SI Units Copyright ©2017 by Pearson Education, Ltd.

R.C. Hibbeler All rights reserved.
Dynamics, Fourteenth Edition in SI Units Copyright ©2017 by Pearson Education, Ltd.
R.C. Hibbeler All rights reserved.