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Provided that the r and θ axes are in the x-y plane, at point P (at time t):

 

 

 

 

 

 

 

velocity:

accelaration:

(1.27)

(1.28)

(1.29)

(1.30)

(1.31)

(1.32)

 

 

 

 

 

 

 

path

1.2.6 Examination of curvilinear motion according to Cylindrical Coordinates:

1.2.6 Kinematics of Particle / Curvilinear Motion / Cylindiral Components

(tvid1.2.4 b)

Figure 1.2.35

Figure 1.2.36

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DYNAMICS - Lecture Notes / Mehmet Zor

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2.6m

x

y

r

z

 

The position of a child who starts sliding from the top point of a curved slide in an aqua park at any time t can be expressed in cylindrical coordinates as

r = 0.2t2 -2t +6, θ = Sin(5t) + Cos(20t) + 2t, z = -2.3t + 20

It is known that the lowest point of the slide is 2.6m above the pool. Accordingly,a-) Calculate the speed and acceleration of the child at the 5th second,b-) calculate how far from the slide the point where the child falls into the pool will be.

Solution:

 

 

 

 

 

 

 

 

r = 0.2t2 -2t +6

θ = Sin(5t) + Cos(20t) + 2t

z = -2.3t + 20

Example 1.2.14

1.2.6 Kinematics of Particle / Curvilinear Motion / Cylindiral Components

Figure 1.2.37

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DYNAMICS - Lecture Notes / Mehmet Zor

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a-) at t=5s, v=?, a=?

 

 

 

 

 

 

 

 

y

r

2.6m

x

z

 

 

 

 

 

 

 

 

 

 

 

a-)

 

1.2.6 Kinematics of Particle / Curvilinear Motion / Cylindiral Components

(from eq. 1.27)

(from eq. 1.29)

(from eq. 1.28)

(from eq. 1.31)

(from eq. 1.30)

(from eq. 1.32)

 

 

 

Figure 1.2.38

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DYNAMICS - Lecture Notes / Mehmet Zor

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2.6m

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

First of all, we need to find the velocity components of the child at point A at the end of the slide:

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1.2.6 Kinematics of Particle / Curvilinear Motion / Cylindiral Components

b-) We will calculate how far away from the slide the point where the child falls into the pool will be.

After leaving point A, there is a projectile motion in the 𝑥′−𝑧′ plane...>>

Figure 1.2.40

Figure 1.2.39

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DYNAMICS - Lecture Notes / Mehmet Zor

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Projectile motion after exiting point A:

 

 

2.6m

 

 

 

 

 

 

 

 

 

 

 

1.2.6 Kinematics of Particle / Curvilinear Motion / Cylindiral Components

(The distance from the slide to the point where the child fell into the pool)

Figure 1.2.41

Figure 1.2.42

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DYNAMICS - Lecture Notes / Mehmet Zor

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1.2.6 Kinematics of Particle / Curvilinear Motion / Cylindiral Components

Figure 1.2.43

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DYNAMICS - Lecture Notes / Mehmet Zor

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1.2.6 Kinematics of Particle / Curvilinear Motion / Cylindiral Components

 

 

 

Answers:

;

;

Figure 1.2.44