Plane curvilinear motion is the motion of · Plane curvilinear motion is the motion of a particle...
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Transcript of Plane curvilinear motion is the motion of · Plane curvilinear motion is the motion of a particle...
Plane curvilinear motion is the motion of a particle along a curved path which lies in a single plane. Before the description of plane curvilinear motion in any specific set of coordinates, we will use vector analysis to describe the motion, since the results will be independent of any particular coordinate system.
At time t the particle is at position A, which is located by the position vector measured from the fixed origin O. Both the magnitude and direction of are known at time t. At time t+Dt, the particle is at A' , located by the position vector .
rr
rr
D
O
A
A'
t
t+Dt
Ds
A
A'
A
Path of particle
rr
D
r
r
D
V
V
VV
V
D
a
The displacement of the particle during Dt is the vector which represents the vector change of position and is independent of the choice of origin. If another point was selected as the origin the position vectors would have changed but would remain the same.
r
D
O
A
A'
t
t+Dt
Ds
A
A'
A
Path of particle
rr
D
r
r
D
V
V
VV
V
D
a
r
D
The distance actually travelled by the particle as it moves along the path from A to A' is the scalar length Ds measured along the path. It is important to distinguish between Ds and .
r
D
O
A
A'
t
t+Dt
Ds
A
A'
A
Path of particle
rr
D
r
r
D
V
V
VV
V
D
a
r
D
The average velocity of the particle between A and A' is defined as
which is a vector whose direction is that of . The magnitude of is .
The average speed of the particle between A and A' is
Clearly, the magnitude of the average velocity and the speed approach one another as the interval Dt decreases and A and A' become closer together.
t
rvav
D
D
r
D
t
s
D
D
Velocity
t
s
D
D
t
r
D
D
avv
t
r
D
D
The instantaneous velocity of the particle is defined as the limiting value of the average velocity as the time Dt approaches zero.
We observe that the direction of approaches that of the tangent to the path as Dt approaches zero and, thus, the velocity is always a vector tangent to the path.
rdt
rd
t
rv
t
D
D
D 0
lim
v
sdt
dsvv
v
The magnitude of is called the speed and is the scalar
v
The change in velocities, which are tangent to the path and are at A and at A‘ during time Dt is a vector . Here indicates both change in magnitude and direction of . Therefore, when the differential of a vector is to be taken, the changes both in magnitude and direction must be taken into account.
v
v
v
D
v
Dv
vv
v
D
which is a vector whose direction is that of . Its magnitude is
The instantaneous acceleration of the particle is defined as the limiting value of the average acceleration as the time interval approaches zero.
t
vaav
D
D
v
D
rvdt
vd
t
vlima
0t
D
D
D
The average acceleration of the particle between A and A' is defined as
Acceleration
t
v
D
D
The acceleration includes the effects of both the changes in magnitude and direction of . In general, the direction of the acceleration of a particle in curvilinear motion is neither tangent to the path nor normal to the path.
As Dt becomes smaller and approaches zero, the direction of approaches . v
D vd
v
If the acceleration was divided into two components one tangent and the other normal to the path, it would be seen that the normal component would always be directed towards the center of curvature.
If velocity vectors are plotted from some arbitrary point C, a curve, called the hodograph, is formed. Acceleration vectors are tangent to the hodograph.
Three different coordinate systems are commonly used in describing the vector relationships for plane curvilinear motion of a particle. These are: • Rectangular (Cartesian) Coordinates (Kartezyen veya Dik Koordinatlar) • Normal and Tangential Coordinates (Doğal veya Normal-Teğetsel Koordinatlar) • Polar Coordinates (Polar veya Kutupsal Koordinatlar) The selection of the appropriate reference system is a prerequisite for the solution of a problem. This selection is carried out by considering the description of the problem and the manner the data are given.
Cartesian Coordinate system is useful for describing motions where the x- and y-components of acceleration are independently generated or determined. Position, velocity and acceleration vectors of the curvilinear motion are indicated by their x and y components.
O
A
Path of particle
q
x
y
A
a
r
ix
i
jy
j
yv
xv
v
xa
ya
a
Let us assume that at time t the particle is at point A. With the aid of the unit vectors and , we can write the position, velocity and acceleration vectors in terms of x- and y-components.
As we differentiate with respect to time, we observe that the time derivatives of the unit vectors are zero because their magnitudes and directions remain constant.
jyixjvivjaiaa
jyixjvivv
jyixr
yxyx
yx
i
O
A
Path of particle
q
x
y
A
a
r
ix
i
jy
j
yv
xv
v
xa
ya
a
j
The magnitudes of the components of and are:
v
a
yvaxva
yvxv
yyxx
yx
In the figure it is seen that the direction of ax is in –x direction. Therefore when writing in vector form a “-” sign must be added in front of ax.
O
A
Path of particle
q
x
y
A
a
r
ix
i
jy
j
yv
xv
v
xa
ya
a
x
y
yx
yx
x
y
yx
yx
a
a
aaa
aaa
v
v
vvv
vvv
a
q
tan
tan
22
222
22
222
The direction of the velocity is always tangent to the path. No such thing can be said for acceleration.
If the coordinates x and y are known independently as functions of time, x=f1(t) and y=f2(t), then for any value of the time we can obtain . Similarly, we combine their first derivatives to obtain and their second derivatives to obtain . Inversely, if ax and ay are known, then we must take integrals in order to obtain the components of velocity and position. If time t is removed between x and y, the equation of the path can be obtained as
y=f(x).
avr
yandx yandx
Projectile Motion (Eğik Atış Hareketi) An important application of two-dimensional kinematic theory is the problem of projectile motion. For a first treatment, we neglect aerodynamic drag and the curvature and rotation of the earth, and we assume that the altitude change is small enough so that the acceleration due to the gravity can be considered constant. With these assumptions, rectangular coordinates are useful to employ for projectile motion.
Acceleration components;
ax=0
ay= -g
vox= vocosq
voy= vosinq vo
q
g
v
vx
vy
v'
vx
v'y
x
y Apex; vy=0
vx
Horizontal Vertical
)(2
2
1
0
02
02
00
2000
0
yygvvtvxx
gttvyyvv
gtvvconstantv
gaa
yyx
yxx
yyx
yx
--
-
-
-
We can see that the x- and y-motions are independent of each other. Elimination of the time t between x- and y-displacement equations shows the path to be parabolic.
If motion is examined separately in horizontal and vertical directions,
1. The particle P moves along the curved slot, a portion of which is shown. Its distance in meters measured along the slot is given by s=t2/4, where t is in seconds. The particle is at A when t = 2.00 s and at B when t =2.20 s. Determine the magnitude aav of the average acceleration of P between A and B. Also express the acceleration as a vector using unit vectors and .
ava
i
j
2. With what minimum horizontal velocity u can a boy throw a rock at A and have it just clear the obstruction at B?
3. For a certain interval of motion, the pin P is forced to move in the fixed parabolic slot by the vertical slotted guide, which moves in the x direction at the constant rate of 20 mm/s. All measurements are in mm and s. Calculate the magnitudes of and of pin P when x = 60 mm.
v
a
4. A projectile is fired with a velocity u at right angles to the slope, which is inclined at an angle q with the horizontal. Derive an expression for the distance R to the point of impact.
5. Pins A and B must always remain in the vertical slot of yoke C, which moves to the right at a constant speed of 6 cm/s. Furthermore, the pins cannot leave the elliptic slot. What is the speed at which the pins approach each other when the yoke slot is at x = 50 cm? What is the rate of change of speed toward each other when the yoke slot is again at x = 50 cm?
100 cm 60 cm
x
6 cm/s
yoke
6. A 7 m long conveyor band makes an angle a with the horizontal surface. Sand, thrown at point B freely falls to the point C on the surface. If the band is moving with a constant velocity v0 = 3 m/s, calculate the maximum distance d between A and C and also find a.
A
d
7 m B
C
