Upload
malaika-azeem
View
222
Download
0
Embed Size (px)
Citation preview
7/29/2019 ENGR221 Lecture 25
1/43
Curvilinear Motion &
Equations of MotionENGR 221
April 16, 2003
7/29/2019 ENGR221 Lecture 25
2/43
Lecture Goals
Normal and Tangential Coordinates
Rectilinear Equations of Motion
7/29/2019 ENGR221 Lecture 25
3/43
Equations of motion
From Newtons Second Law of motion
where the force and acceleration are vectorquantities.
F ma
7/29/2019 ENGR221 Lecture 25
4/43
Equations of motion
The vectors can be broken into components similar to
equilibrium
x y z x y zx x y y z z& &
F i F j F k m a i a j a k
F ma F ma F ma
7/29/2019 ENGR221 Lecture 25
5/43
Equations of motion
From equilibrium equations
x
y
z
0
0
0
F
F
F
7/29/2019 ENGR221 Lecture 25
6/43
Equations of motion
The difference between the equilibrium equations and
the equations of motion is that the right hand side is
no longer zero.
x x
y y
z z
F ma
F maF ma
7/29/2019 ENGR221 Lecture 25
7/43
Procedure for solving the
Equations of Motion
1. Draw the Free-Body Diagram.
2. Write the equations of motion
3. Solve for the velocities and accelerations need
or obtain the equations for the components.
4. Solve for the forces
7/29/2019 ENGR221 Lecture 25
8/43
Example ProblemEquation of
MotionA 80-kg skydiver falls at 85 m/s
when she opens her parachute. If
her speed is reduced to 5 m/sduring the next 60 m of fall,
determine the average force exerted
on her body by the parachute
during this interval.
7/29/2019 ENGR221 Lecture 25
9/43
Example ProblemEquation of
MotionDraw the free-body diagram
of the skydiver.
y yF ma
7/29/2019 ENGR221 Lecture 25
10/43
Example ProblemEquation of
MotionRewrite the equation
The acceleration is going to be a
constant
y y
2
p y80 kg 9.81 m/s 80 kg
F ma
F a
p2
y9.81 m/s
80 kg
Fa
7/29/2019 ENGR221 Lecture 25
11/43
Example ProblemEquation of
MotionThe acceleration is defined as
60 m 5 m/s
0 85 m/s
dv dv dxadt dx dt
adx vdv
adx vdv
7/29/2019 ENGR221 Lecture 25
12/43
Example ProblemEquation of
MotionPlug in the acceleration equation
5 m/s60 m 2p2
0 85 m/s
60 m
p2 2 2
0
p
9.81 m/s80 kg 2
9.81 m/s 3600 m /s80 kg
5584.8 N
F vdx
Fx
F
7/29/2019 ENGR221 Lecture 25
13/43
Class ProblemRectilinear
MotionThe 2000-lb elevator cage is
brought to rest from an initial
speed of 25 ft/s in a distanceof 50 ft. Determine the
uniform deceleration and the
tension in the elevator cable
while the cage is coming to
rest.
7/29/2019 ENGR221 Lecture 25
14/43
Example ProblemRectilinear
Motion
a) Determine the acceleration of body
A and tension in the cable
connecting the bodies.
b) Determine the acceleration of body
B if body A is replaced with a
constant force of 245 N.
Two bodies A and B with masses of 25 kg
and 30 kg, respectively. During motion of
the bodies,
7/29/2019 ENGR221 Lecture 25
15/43
Example ProblemRectilinear
MotionDraw the free body diagram of the boxes A and B. The
normal weight, NB and R is resistance due to friction and
the tension T.
7/29/2019 ENGR221 Lecture 25
16/43
xampeProblem
Equation ofMotion
Find the equations of motion
for the block.
2
Ay A Ay Ay
Bx B Bx Bx
2
By B By B
B
25 kg 9.81 m/s 25 kg
30 kg
30 kg 9.81 m/s 0
294.3 N
F m a T a
F m a T R a
F m a N
N
7/29/2019 ENGR221 Lecture 25
17/43
xampeProblem
Equation ofMotion
If there is no friction
resistance thenaAy = aBx
2
Ay A Ay Ay
Bx B Bx Bx2
Bx Bx
2 2
Bx
25 kg 25 kg 9.81 m/s
30 kg
30 kg 25 kg 25 kg 9.81 m/s
4.46 m/s 30 kg 4.46 m/s 133.8 N
F m a T a
F m a T a
a a
a T
7/29/2019 ENGR221 Lecture 25
18/43
Example Problem
Equation of Motion
The force in the cable is T =245 N
Bx2
Bx
245 N 30 kg
8.167 m/s
a
a
7/29/2019 ENGR221 Lecture 25
19/43
Class Problem
Rectilinear Motion
a) Determine the acceleration of body A and tension
in the cable connecting the bodies if the fiction is
neglected.
Two bodies A and B with masses of 100 kg and
300 kg, respectively. During motion of the
bodies,
7/29/2019 ENGR221 Lecture 25
20/43
Example ProblemRectilinear
motion
a) The acceleration of A
b) The tension in the cable
c) The velocity of body B
after 5 sec.
Twp bodies A (40 kg) and B (30 kg) are connected by a flexible
cable. The kinetic coefficient of friction between body A and the
inclined surface is mk= 0.25 and the system is released from rest.
During the motion of the body determine.
7/29/2019 ENGR221 Lecture 25
21/43
Example ProblemRectilinear
motionDraw the free body diagrams of the bodies
xampe
7/29/2019 ENGR221 Lecture 25
22/43
xampeProblem
Rectilinearmotion
The equations are
2
Ax A Ax A Ax
2
Ay A Ay A
A
2
By B By By
32 40 kg 9.81 m/s 40 kg
5
440 kg 9.81 m/s 05
313.92 N
30 kg 9.81 m/s 30 kg
F m a T F a
F m a N
N
F m a T a
xampe
7/29/2019 ENGR221 Lecture 25
23/43
xampeProblem
Rectilinearmotion
Use the constraints of the two length of the cable
The direction so the block are moving the upward
A B
A B
A B
2 constant
2 0
2
L s s
s s
s s
xampe
7/29/2019 ENGR221 Lecture 25
24/43
xampeProblem
Rectilinearmotion
The frictional force is
Rearrange the equations in terms of T
A A A 0.25 313.92 N
78.48 N
F Nm
7/29/2019 ENGR221 Lecture 25
25/43
xampe
7/29/2019 ENGR221 Lecture 25
26/43
xampeProblem
Rectilinearmotion
The equations are
Ax Ax
Ax
2
Ax
2 30 kg 2 294.3 N 313.92 N 40 kg
274.68 N 160 kg
1.717 m/s 191.3 N
a a
a
a T
7/29/2019 ENGR221 Lecture 25
27/43
7/29/2019 ENGR221 Lecture 25
28/43
Equations of motion
The difference between the linear and curvilinear
equations of motion is the defined system.
n n
t t
F ma
F ma
7/29/2019 ENGR221 Lecture 25
29/43
Example ProblemCurvilinear
motionThe bob of a 2 m pendulum describes an arc of circle in
a vertical plane. If the tension in the cord is 2.5 times
the weight of the bob for the position show, find thevelocity and acceleration of the bob at the position. .
7/29/2019 ENGR221 Lecture 25
30/43
Example ProblemCurvilinear
motionDraw the free body diagram of the mass .
t t
o
t
n n
o
n
sin 30
2 cos 30
F ma
mg ma
F ma
mg mg ma
+
+
7/29/2019 ENGR221 Lecture 25
31/43
Example ProblemCurvilinear
motionSolve for the acceleration .
o 2 o
t t
2
t
o 2 o
n n
2
t
sin 30 9.81 m/s sin 304.90 m/s
2 cos 30 9.81 m/s 2 cos 30
16.30 m/s
mg ma aa
mg mg ma a
a
7/29/2019 ENGR221 Lecture 25
32/43
Example ProblemCurvilinear
motionCompute the tangential velocity .
2t
n t n
2
t 16.30 m/s 2 m
5.71 m/s
va v a
v
7/29/2019 ENGR221 Lecture 25
33/43
Example ProblemCurvilinear
motionDetermine the rated speed of a highway curve of radius
= 400 ft. banked through an angle, q = 18o. The rated
speed of the banked curve road is the speed at which acar should travel if no lateral friction force is to be
exerted on its wheel.
7/29/2019 ENGR221 Lecture 25
34/43
Example ProblemCurvilinear
motionDetermine the equation of motion
for the car with a radius = 400 ft.
banked through an angle, q = 18o
.
y
o
n n
o
n
0
cos 18 0
sin 18
F
R WF ma
R ma
7/29/2019 ENGR221 Lecture 25
35/43
Example ProblemCurvilinear
motionSolve for R and find an.
o
o o
n o
o 2 o
n
2
cos 18
sin 18 sin 18
cos 18
tan 18 32.2 ft/s tan 18
10.46 ft/s
mg
R
mgma R
a g
7/29/2019 ENGR221 Lecture 25
36/43
Example ProblemCurvilinear
motionCompute the tangential velocity .
2
t
n t n
2
t 10.46 ft/s 400 ft
64.7 ft/s
60 mph64.7 ft/s 44.1 mph
88 ft/s
v
a v a
v
7/29/2019 ENGR221 Lecture 25
37/43
Class ProblemCurvilinear
motionAn airplane is descending at anangle, q = 20o with respect to
the horizontal when it drops a
bomb. If the altitude at thetime of release is 5000 m and
speed of the plane 750 km/hr,
determine the range
(horizontal distance traveled)
of the bomb and the elapsed
time before it strikes the
ground. Neglect air friction.
7/29/2019 ENGR221 Lecture 25
38/43
Homework (Due 4/23/03)
Problems:
15-10, 15-21,15-23, 15-57, 15-60, 15-61,
15-62, 15-63, 15-64
7/29/2019 ENGR221 Lecture 25
39/43
Bonus ProblemCurvilinear
motionA block B of mass m may slidefreely on a frictionless arm OA,
which rotates in a horizontal
plane at a constant rate ,Knowing that B is released at a
distance r0 from O and express as
a function of r (a) the
components vrof the velocity of
B along AO, (b) the magnitude
of the horizontal force F exerted
on B by the arm OA.
q
7/29/2019 ENGR221 Lecture 25
40/43
Bonus ProblemCurvilinear
motionDraw the free body diagram ofthe block B.
r r
20
2
F ma
m r r
F ma
F m r r
q q
q
q q
+
+
7/29/2019 ENGR221 Lecture 25
41/43
Bonus ProblemCurvilinear
motionDetermine the velocity of the r component
Substitute into the equation for r double dot
r r
r rr r
v r r v
dv dvdrr v v
dr dt dr
2rr0
dvm v r
drq
7/29/2019 ENGR221 Lecture 25
42/43
Bonus ProblemCurvilinear
motionIntegrate
r
0
f
2 2rr r r
r
0.5
2 2r 0
v
o
dvv r v dv r dr dr
v r r
q q
q
7/29/2019 ENGR221 Lecture 25
43/43
Bonus ProblemCurvilinear
motionInsert the to find the
force.and 0r q q
0.52 2
0
0.5
2 2 20
0 2
2
F m r r r
m r r
q q
q