DYNAMICS - Lecture Notes / Mehmet Zor

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23 Agust, 2024

1.2.5 Curvilinear Motion in Polar Coordinates :

we will determine the position, velocity and acceleration of an object moving curvilinearly on a plane according to the polar coordinates r - θ, at a time t

At the examined time t, (at point P):

• r and θ axes are placed at point P.

1.2.5 Kinematics of Particle / Curvilinear Motion / Polar Components

(tvid 1.2.4 a)

• + r axis is in direction of OP.

• The +θ axis is obtained by rotating the r axis 90⁰ counterclockwise.

• According to the right hand rule, we rotate the +x axis to the +y axis with the 4 fingers of our right hand and close it. This rotation direction is the +q direction and in this case our thumb points to the +z direction perpendicular to the plane.

Figure 1.2.23

Figure 1.2.22

DYNAMICS - Lecture Notes / Mehmet Zor

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23 Agust, 2024

1.2.5 Kinematics of Particle / Curvilinear Motion / Polar Components

dθ

θ

First, we will derive the relationships between the unit vectors on the r and θ axes:

dθ

When we move from position P by the angle dq, we come to position P’. Unit vectors at position P’:

magnitudes of unit vectors are 1:

Magnitude of diff. unit vectors:

(We liken it to the equation radius x central angle = arc length.)

//

//

Position at time t :

instantaneous velocity vector :

Since

instantaneous acceleration vector :

instantaneous

acceleration components:

instantaneous velocity components :

(1.27)

(1.28)

(1.29)

(1.30)

Figure 1.2.24

Figure 1.2.25

DYNAMICS - Lecture Notes / Mehmet Zor

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23 Agust, 2024

Example 1.2.6

On a sufficiently long arm AC, the ring B slides freely. The position of the ring B at time t is described by the equations

r = 2t² -2t +20, θ = 0.2t²

Accordingly, calculate the velocity and acceleration of the ring when θ = 120^o

Solution:

r = 2t² -2t +20

θ = 0.2t²

1.2.5 Kinematics of Particle / Curvilinear Motion / Polar Components

(from equations 1.27-1.30)

r

Figure 1.2.26

Figure 1.2.27

DYNAMICS - Lecture Notes / Mehmet Zor

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23 Agust, 2024

Question 1.2.3 (*)

1.2.5 Kinematics of Particle / Curvilinear Motion / Polar Components

Figure 1.2.28

DYNAMICS - Lecture Notes / Mehmet Zor

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23 Agust, 2024

Example 1.2.7

x

y

8.66km

θ

r

r

Valid for rectilinear motion with constant acceleration.

Timeless Velocity Equation (1.9):

Solution:

1.2.5 Kinematics of Particle / Curvilinear Motion / Polar Components

8.66km

Figure 1.2.29

Figure 1.2.30

angular accelaration

DYNAMICS - Lecture Notes / Mehmet Zor

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23 Agust, 2024

Example 1.2.8

1.2.5 Kinematics of Particle / Curvilinear Motion / Polar Components

Solution:…>>>

Figure 1.2.31

DYNAMICS - Lecture Notes / Mehmet Zor

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23 Agust, 2024

Velocity components of the bullet at time t= 0 (oblique shot)

The bullet starts moving with the speed of the car + its own speed:

the bullet is in the rise zone.

Figure 1.2.32

1.2.5 Kinematics of Particle / Curvilinear Motion / Polar Components

DYNAMICS - Lecture Notes / Mehmet Zor

8

23 Agust, 2024

Question 1.2.4 (*)

1.2.5 Kinematics of Particle / Curvilinear Motion / Polar Components

a-)

Answers:

b-) 6369.8m

Figure 1.2.33

DYNAMICS - Lecture Notes / Mehmet Zor

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23 Agust, 2024

Question1.2.5 (*)

A

1.2.5 Kinematics of Particle / Curvilinear Motion / Polar Components

Hasan

Figure 1.2.34