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Lecture III Curvilinear Motion.

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Presentation on theme: "Lecture III Curvilinear Motion."— Presentation transcript:

1 Lecture III Curvilinear Motion

2 Topics Covered in Curvilinear Motion
Plane curvilinear motion Coordinates used for describing curvilinear motion Rectangular coords n-t coords Polar coords

3 Plane curvilinear Motion
Studying the motion of a particle along a curved path which lies in a single plane (2D). This is a special case of the more general 3D motion. 3D

4 Plane curvilinear Motion – (Cont.)
If the x-y plane is considered as the plane of motion; from the 3D case, z and j are both zero, and R becomes as same as r. The vast majority of the motion of particles encountered in engineering practice can be represented as plane motion.

5 Coordinates Used for Describing the Plane Curvilinear Motion
Normal-Tangential coordinates Polar coordinates Rectangular coordinates PC Path t y y q Path r Path n n PA P P n r t t PB q x x O O

6 Plane Curvilinear Motion – without Specifying any Coordinates (Displacement)
Actual distance traveled by the particle (it is s scalar) Note: Since, here, the particle motion is described by two coordinates components, both the magnitude and the direction of the position, the velocity, and the acceleration have to be specified. The vector displacement of the particle Ds (Dt) Note: If the origin (O) is changed to some different location, the position r(t) will be changed, but Dr(Dt) will not change. or r(t)+Dr(Dt)

7 Plane Curvilinear Motion – without Specifying any Coordinates (Velocity)
Note: vav has the direction of Dr and its magnitude equal to the magnitude of Dr divided by Dt. Average velocity (vav): Instantaneous velocity (v): as Dt approaches zero in the limit, Note: the average speed of the particle is the scalar Ds/Dt. The magnitude of the speed and vav approach one another as Dt approaches zero. Note: the magnitude of v is called the speed, i.e. v=|v|=ds/dt= s.. Note: the velocity vector v is always tangent to the path.

8 Plane Curvilinear Motion – without Specifying any Coordinates (Acceleration)
Average Acceleration (aav): Instantaneous Acceleration (a): as Dt approaches zero in the limit, Note: aav has the direction of Dv and its magnitude is the magnitude of Dv divided by Dt. Note: in general, the acceleration vector a is neither tangent nor normal to the path. However, a is tangent to the hodograph. P V1 C V1 Hodograph P V2 V2 a1 a2

9 The description of the Plane Curvilinear Motion in the Rectangular Coordinates (Cartesian Coordinates)

10 Plane Curvilinear Motion - Rectangular Coordinates
y v vy q vx P Path j r x O i a ay Note: the time derivatives of the unit vectors are zero because their magnitude and direction remain constant. ax P Note: if the angle q is measured counterclockwise from the x-axis to v for the configuration of the axes shown, then we can also observe that dy/dx = tanq = vy/vx.

11 Plane Curvilinear Motion - Rectangular Coordinates (Cont.)
The coordinates x and y are known independently as functions of time t; i.e. x = f1(t) and y = f2(t). Then for any value of time we can combine them to obtain r. Similarly, for the velocity v and for the acceleration a. If is a given, we integrate to get v and integrate again to get r. The equation of the curved path can obtained by eliminating the time between x = f1(t) and y = f2(t). Hence, the rectangular coordinate representation of curvilinear motion is merely the superposition of the components of two simultaneous rectilinear motions in x- and y- directions.

12 Plane Curvilinear Motion - Rectangular Coordinates (Cont
Plane Curvilinear Motion - Rectangular Coordinates (Cont.) – Projectile Motion y v vy vx vx vo vy g v Path (vy)o = vo sinq q x (vx)o = vo cosq

13 Exercises

14 Exercise # 1 Given the position vector , calculate the velocity, acceleration, and the magnitudes of position, velocity and acceleration.

15 Exercise # 2 2/62: A particle which moves with curvilinear motion has coordinates in millimeters which vary with the time t in seconds according to x = 2t2 - 4t and y = 3t2 – (1/3)t3. Determine the magnitudes of the velocity v and acceleration a and the angles which these vectors make with the x-axis when t = 2 s.

16 Exercise # 3 2/63: A roofer tosses a small tool toward a coworker on the ground. What is the minimum horizontal velocity vo necessary so that the tool clears point B? Locate the point of impact by specifying the distance s shown in the figure.

17 Exercise # 4 2/68: A rocket runs out of fuel in the position shown and continues in unpowered flight above the atmosphere. If its velocity in this position was 1000 km/h, calculate the maximum additional altitude h acquired and the corresponding time t to reach it. The gravitational acceleration during this phase of its flight is 9.39 m/s2.

18 Exercise # 5 2/80: The pilot of an airplane carrying a package of mail to a remote outpost wishes to release the package at the right moment to hit the recovery location A. What angle q with the horizontal should the pilot's line of sight to the target make at the instant of release? The airplane is flying horizontally at an altitude of 100 m with a velocity of 200 km/h.

19 Exercise # 6 2/92: The basketball player likes to release his foul shots at an angle q = 50° to the horizontal as shown. What initial speed vo will cause the ball to pass through the center of the rim?

20 Exercise # 7 2/73: A particle is ejected from the tube at A with a velocity v at angle q with the vertical y-axis. A strong horizontal wind gives the particle a constant horizontal acceleration a in the x-direction. If the particle strikes the ground at a point directly under its released position, determine the height h of point A. The downward y-acceleration may be taken as the constant g.


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