eng1040 lec06
DESCRIPTION
TRANSCRIPT
![Page 1: ENG1040 Lec06](https://reader036.vdocument.in/reader036/viewer/2022062418/553bc37955034632548b4620/html5/thumbnails/1.jpg)
Faculty of Engineering
ENG1040 – Engineering Dynamics
ENG1040Engineering Dynamics
Pulley Systems,Free Body Diagrams : Example Questions
Dr Lau Ee Von – Sunway
Lecture 6
![Page 2: ENG1040 Lec06](https://reader036.vdocument.in/reader036/viewer/2022062418/553bc37955034632548b4620/html5/thumbnails/2.jpg)
Past exam question
• Question 2, Sem 2, 2007• Draw free body diagrams for Blocks A and B
when Block B is translating and accelerating downwards.
• How is acceleration of Block B related to the acceleration of block A?
2
![Page 3: ENG1040 Lec06](https://reader036.vdocument.in/reader036/viewer/2022062418/553bc37955034632548b4620/html5/thumbnails/3.jpg)
Past exam question
• How do you approach a system with several pulleys (pulley system)?
3
![Page 4: ENG1040 Lec06](https://reader036.vdocument.in/reader036/viewer/2022062418/553bc37955034632548b4620/html5/thumbnails/4.jpg)
Lecture Outline
• Pulley systems:• How to gain mechanical leverage• Example Questions: FBD
4
![Page 5: ENG1040 Lec06](https://reader036.vdocument.in/reader036/viewer/2022062418/553bc37955034632548b4620/html5/thumbnails/5.jpg)
Pulley systems
• Pulley systems have been used for Millennia to reduce the force required to lift weights.
• Employed largely in sailing, they are believed to have been invented by Archimedes (200BC).
5
![Page 6: ENG1040 Lec06](https://reader036.vdocument.in/reader036/viewer/2022062418/553bc37955034632548b4620/html5/thumbnails/6.jpg)
Pulley systems
• Pulleys in everyday life:
6
![Page 7: ENG1040 Lec06](https://reader036.vdocument.in/reader036/viewer/2022062418/553bc37955034632548b4620/html5/thumbnails/7.jpg)
The Simplest Pulley system
7
The simplest type of pulley system is shown here.
A free-body diagram of this pulley system shows that the total load is split into half on either pulley rope to maintain equilibrium.
But there must be a trade-off...
... The amount of work applied does not change.
dFEnergy
![Page 8: ENG1040 Lec06](https://reader036.vdocument.in/reader036/viewer/2022062418/553bc37955034632548b4620/html5/thumbnails/8.jpg)
Kinematics of Pulleys
8
Therefore to raise the mass a distance d, the rope must be hoisted a distance 2d.
This also implies that if the rope is pulled with a velocity v, then the mass will move with a velocity v/2.
![Page 9: ENG1040 Lec06](https://reader036.vdocument.in/reader036/viewer/2022062418/553bc37955034632548b4620/html5/thumbnails/9.jpg)
Further improvements
9
Often we want to pull down to pull a weight up – Gun tackle system
In this case, the beam has to support 1½ times the weight just to maintain equilibrium.
![Page 10: ENG1040 Lec06](https://reader036.vdocument.in/reader036/viewer/2022062418/553bc37955034632548b4620/html5/thumbnails/10.jpg)
10
We can make further improvements!
The Luff Tackle (shown here) has a mechanical advantage of 3.
Note, to maintain equilibrium, the tension in the rope is the same at all locations.
Pulley systems
![Page 11: ENG1040 Lec06](https://reader036.vdocument.in/reader036/viewer/2022062418/553bc37955034632548b4620/html5/thumbnails/11.jpg)
11
Once again, we can change the system so that we are pulling downwards to lift the weight.
Pulley systems
The more pulleys, the greater the mechanical advantage.Why stop at 4:1?
The greater the mechanical advantage, the further you have to pull the rope in order to shift the mass.
![Page 12: ENG1040 Lec06](https://reader036.vdocument.in/reader036/viewer/2022062418/553bc37955034632548b4620/html5/thumbnails/12.jpg)
Pulley systems
12
• Note: the analysis described on the previous slides assumes that the pulleys are massless...
![Page 13: ENG1040 Lec06](https://reader036.vdocument.in/reader036/viewer/2022062418/553bc37955034632548b4620/html5/thumbnails/13.jpg)
Analysis procedure
1. Establish a coordinate system
2. Draw Free Body Diagram(s)• Graphical representation of all forces
acting on the system.
3. Establish known & unknown quantities
4. Apply Equation(s) of Motion in each direction
5. Evaluate kinematics to solve problem
Kinetics/Kinematics problems...
![Page 14: ENG1040 Lec06](https://reader036.vdocument.in/reader036/viewer/2022062418/553bc37955034632548b4620/html5/thumbnails/14.jpg)
14
B
A
Free body diagrams – Pulley system
Draw the FBD for the following pulley systems, assuming the pulleys and ropes are massless
![Page 15: ENG1040 Lec06](https://reader036.vdocument.in/reader036/viewer/2022062418/553bc37955034632548b4620/html5/thumbnails/15.jpg)
Free body diagrams – Pulley system
Draw the FBD for the following pulley systems, assuming the pulleys and ropes are massless
15
![Page 16: ENG1040 Lec06](https://reader036.vdocument.in/reader036/viewer/2022062418/553bc37955034632548b4620/html5/thumbnails/16.jpg)
Free body diagrams – Pulley system
16
A
F
BC
Rope 1Rope 2
Question 2, Sem 1, 2012
![Page 17: ENG1040 Lec06](https://reader036.vdocument.in/reader036/viewer/2022062418/553bc37955034632548b4620/html5/thumbnails/17.jpg)
Question 3, Sem 2, 2011
Free body diagrams – Pulley system
![Page 18: ENG1040 Lec06](https://reader036.vdocument.in/reader036/viewer/2022062418/553bc37955034632548b4620/html5/thumbnails/18.jpg)
Kinematics
18
𝑥𝐴+𝑥𝐵+𝑥𝐶=𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡
∆ 𝑠𝐴+∆ 𝑠𝐵+∆𝑠𝐶=0
𝑑𝑣𝐴
𝑑𝑡+𝑑𝑣𝐵
𝑑𝑡+𝑑𝑣𝐶
𝑑𝑡=0
Position vector from origin (fixed point)
Displacement = xfinal - xinitial
𝑎𝐴+𝑎𝐵+𝑎𝐶=0Equation for the acceleration relationship between masses
![Page 19: ENG1040 Lec06](https://reader036.vdocument.in/reader036/viewer/2022062418/553bc37955034632548b4620/html5/thumbnails/19.jpg)
Example Question
• Question 12.12 [Kinetics] (MECHANICS FOR ENGINEERS: DYNAMICS by Ferdinand P. Beer)
19
The two blocks shown are originally at rest. Neglecting the masses of the pulleys and the effect of friction in the pulleys and between the blocks and the incline, determine
a) The acceleration of each blockb) The tension in the cable
x
y
![Page 20: ENG1040 Lec06](https://reader036.vdocument.in/reader036/viewer/2022062418/553bc37955034632548b4620/html5/thumbnails/20.jpg)
Example Question
• Question 12.12 [Kinetics]
20
The two blocks shown are originally at rest. Neglecting the masses of the pulleys and the effect of friction in the pulleys and between the blocks and the incline, determine
a) The acceleration of each blockb) The tension in the cable
Ax
y
![Page 21: ENG1040 Lec06](https://reader036.vdocument.in/reader036/viewer/2022062418/553bc37955034632548b4620/html5/thumbnails/21.jpg)
Example Question
• Question 12.12 [Kinetics]
21
The two blocks shown are originally at rest. Neglecting the masses of the pulleys and the effect of friction in the pulleys and between the blocks and the incline, determine
a) The acceleration of each blockb) The tension in the cable
A
)(30sin xAAo
Ax amgmTF
T
NFgmA
x
y
![Page 22: ENG1040 Lec06](https://reader036.vdocument.in/reader036/viewer/2022062418/553bc37955034632548b4620/html5/thumbnails/22.jpg)
Example Question
• Question 12.12 [Kinetics]
22
The two blocks shown are originally at rest. Neglecting the masses of the pulleys and the effect of friction in the pulleys and between the blocks and the incline, determine
a) The acceleration of each blockb) The tension in the cable
Bx
y
![Page 23: ENG1040 Lec06](https://reader036.vdocument.in/reader036/viewer/2022062418/553bc37955034632548b4620/html5/thumbnails/23.jpg)
Example Question
• Question 12.12 [Kinetics]
23
The two blocks shown are originally at rest. Neglecting the masses of the pulleys and the effect of friction in the pulleys and between the blocks and the incline, determine
a) The acceleration of each blockb) The tension in the cable
Bx
y
)(30sin3 xBBo
Bx amgmTF
T3
NFgmB
![Page 24: ENG1040 Lec06](https://reader036.vdocument.in/reader036/viewer/2022062418/553bc37955034632548b4620/html5/thumbnails/24.jpg)
Example Question
• Question 12.12 [Kinetics]
24
The two blocks shown are originally at rest. Neglecting the masses of the pulleys and the effect of friction in the pulleys and between the blocks and the incline, determine
a) The acceleration of each blockb) The tension in the cable
)(30sin3 xBBo
B amgmT
)(30sin xAAo
A amgmT
How many unknowns do I have?
Do I have enough equations?
![Page 25: ENG1040 Lec06](https://reader036.vdocument.in/reader036/viewer/2022062418/553bc37955034632548b4620/html5/thumbnails/25.jpg)
Example Question
• Question 12.12 [Kinetics]
25
The two blocks shown are originally at rest. Neglecting the masses of the pulleys and the effect of friction in the pulleys and between the blocks and the incline, determine
a) The acceleration of each blockb) The tension in the cable
If we consider the kinematics of the problem we can relate the acceleration of block A with the acceleration of block B:
03 )()( xBxA aa
constant3 BA xx
0
![Page 26: ENG1040 Lec06](https://reader036.vdocument.in/reader036/viewer/2022062418/553bc37955034632548b4620/html5/thumbnails/26.jpg)
Example Question
• Question 12.12 [Kinetics]
26
The two blocks shown are originally at rest. Neglecting the masses of the pulleys and the effect of friction in the pulleys and between the blocks and the incline, determine
a) The acceleration of each blockb) The tension in the cable
)(30sin3 xBBo
B amgmT
)(30sin xAAo
A amgmT
)()( 3 xBxA aa aA = -3.30 m/s2
aB = 1.10 m/s2
T = 16 N
![Page 27: ENG1040 Lec06](https://reader036.vdocument.in/reader036/viewer/2022062418/553bc37955034632548b4620/html5/thumbnails/27.jpg)
Example Question
• Question 12.32 [kinetics]
27
The weight of blocks A, B, and C are wa = wc = 20 lb, and wb=10 lb. Knowing that P = 50 lb and neglecting the masses of the pulleys and the effect of friction, determinea) The acceleration of each blockb) The tension in the cable.
1st step: Convert to SI units (see back of text book)
1 lb of force = 4.448 N of force
![Page 28: ENG1040 Lec06](https://reader036.vdocument.in/reader036/viewer/2022062418/553bc37955034632548b4620/html5/thumbnails/28.jpg)
Example Question
• Question 12.32 [kinetics]
28
The weight of blocks A, B, and C are wa = wc = 20 lb, and wb=10 lb. Knowing that P = 50 lb and neglecting the masses of the pulleys and the effect of friction, determinea) The acceleration of each blockb) The tension in the cable.
mAg = mCg = 88.96 N mA = mC = 9.07 kg
mBg = 44.48 N mB = 4.54 kg
P = 222.4 N
![Page 29: ENG1040 Lec06](https://reader036.vdocument.in/reader036/viewer/2022062418/553bc37955034632548b4620/html5/thumbnails/29.jpg)
Example Question
• Question 12.32 [kinetics]
29
The weight of blocks A, B, and C are wa = wc = 20 lb, and wb=10 lb. Knowing that P = 50 lb and neglecting the masses of the pulleys and the effect of friction, determinea) The acceleration of each blockb) The tension in the cable.
xy
A
![Page 30: ENG1040 Lec06](https://reader036.vdocument.in/reader036/viewer/2022062418/553bc37955034632548b4620/html5/thumbnails/30.jpg)
Example Question
• Question 12.32 [kinetics]
30
The weight of blocks A, B, and C are wa = wc = 20 lb, and wb=10 lb. Knowing that P = 50 lb and neglecting the masses of the pulleys and the effect of friction, determinea) The acceleration of each blockb) The tension in the cable.
A
)(3 xAAx amTF x
![Page 31: ENG1040 Lec06](https://reader036.vdocument.in/reader036/viewer/2022062418/553bc37955034632548b4620/html5/thumbnails/31.jpg)
Example Question
• Question 12.32 [kinetics]
31
The weight of blocks A, B, and C are wa = wc = 20 lb, and wb=10 lb. Knowing that P = 50 lb and neglecting the masses of the pulleys and the effect of friction, determinea) The acceleration of each blockb) The tension in the cable.
B
)(2 xBBx amTF x
![Page 32: ENG1040 Lec06](https://reader036.vdocument.in/reader036/viewer/2022062418/553bc37955034632548b4620/html5/thumbnails/32.jpg)
Example Question
• Question 12.32 [kinetics]
32
The weight of blocks A, B, and C are wa = wc = 20 lb, and wb=10 lb. Knowing that P = 50 lb and neglecting the masses of the pulleys and the effect of friction, determinea) The acceleration of each blockb) The tension in the cable.
)(4 xCCx amTPF xC
![Page 33: ENG1040 Lec06](https://reader036.vdocument.in/reader036/viewer/2022062418/553bc37955034632548b4620/html5/thumbnails/33.jpg)
Example Question
• Question 12.32 [kinetics]
33
The weight of blocks A, B, and C are wa = wc = 20 lb, and wb=10 lb. Knowing that P = 50 lb and neglecting the masses of the pulleys and the effect of friction, determinea) The acceleration of each blockb) The tension in the cable.
)(4 xCCamTP
)(2 xBBamT )(3 xAAamT
How many unknowns do I have?
Do I have enough equations?
![Page 34: ENG1040 Lec06](https://reader036.vdocument.in/reader036/viewer/2022062418/553bc37955034632548b4620/html5/thumbnails/34.jpg)
Example Question
• Question 12.32 [kinetics]
34
The weight of blocks A, B, and C are wa = wc = 20 lb, and wb=10 lb. Knowing that P = 50 lb and neglecting the masses of the pulleys and the effect of friction, determinea) The acceleration of each blockb) The tension in the cable.
If we consider the kinematics of the problem we can relate the three acceleration terms.
First, we note that the pulley system is attached to the ground at this point.
We will measure the length of rope from this point.
0
![Page 35: ENG1040 Lec06](https://reader036.vdocument.in/reader036/viewer/2022062418/553bc37955034632548b4620/html5/thumbnails/35.jpg)
• Question 12.32 [kinetics]
We notice that two lengths of rope connect mass B to the fixed point.
Therefore, part of the rope’s length is defined as:
This is two times the distance from mass B to the fixed point.
Example Question
35
The weight of blocks A, B, and C are wa = wc = 20 lb, and wb=10 lb. Knowing that P = 50 lb and neglecting the masses of the pulleys and the effect of friction, determinea) The acceleration of each blockb) The tension in the cable.
If we consider the kinematics of the problem we can relate the three acceleration terms.
bx20
![Page 36: ENG1040 Lec06](https://reader036.vdocument.in/reader036/viewer/2022062418/553bc37955034632548b4620/html5/thumbnails/36.jpg)
Example Question
• Question 12.32 [kinetics]
36
The weight of blocks A, B, and C are wa = wc = 20 lb, and wb=10 lb. Knowing that P = 50 lb and neglecting the masses of the pulleys and the effect of friction, determinea) The acceleration of each blockb) The tension in the cable.
If we consider the kinematics of the problem we can relate the three acceleration terms.
We notice that three lengths of rope connect mass A to the fixed point.
Therefore, part of the rope’s length is defined as:
This is three times the distance from mass A to the fixed point.
ax30
![Page 37: ENG1040 Lec06](https://reader036.vdocument.in/reader036/viewer/2022062418/553bc37955034632548b4620/html5/thumbnails/37.jpg)
Example Question
• Question 12.32 [kinetics]
37
The weight of blocks A, B, and C are wa = wc = 20 lb, and wb=10 lb. Knowing that P = 50 lb and neglecting the masses of the pulleys and the effect of friction, determinea) The acceleration of each blockb) The tension in the cable.
If we consider the kinematics of the problem we can relate the three acceleration terms. Finally, we notice that four
lengths of rope connect mass C to the fixed point.
Therefore, part of the rope’s length is defined as:
This is four times the distance from mass C to the fixed point.
cx4
0
![Page 38: ENG1040 Lec06](https://reader036.vdocument.in/reader036/viewer/2022062418/553bc37955034632548b4620/html5/thumbnails/38.jpg)
Example Question
• Question 12.32 [kinetics]
38
The weight of blocks A, B, and C are wa = wc = 20 lb, and wb=10 lb. Knowing that P = 50 lb and neglecting the masses of the pulleys and the effect of friction, determinea) The acceleration of each blockb) The tension in the cable.I can then sum all these lengths of rope together to form:
From this equation, I can determine an equation for velocity and acceleration...
constant423 cba xxx
![Page 39: ENG1040 Lec06](https://reader036.vdocument.in/reader036/viewer/2022062418/553bc37955034632548b4620/html5/thumbnails/39.jpg)
Example Question
• Question 12.32 [kinetics]
39
The weight of blocks A, B, and C are wa = wc = 20 lb, and wb=10 lb. Knowing that P = 50 lb and neglecting the masses of the pulleys and the effect of friction, determinea) The acceleration of each blockb) The tension in the cable.First the velocity:
0423 dt
dx
dt
dx
dt
dx
dt
dx cba
0423 cba vvvv
![Page 40: ENG1040 Lec06](https://reader036.vdocument.in/reader036/viewer/2022062418/553bc37955034632548b4620/html5/thumbnails/40.jpg)
Example Question
• Question 12.32 [kinetics]
40
The weight of blocks A, B, and C are wa = wc = 20 lb, and wb=10 lb. Knowing that P = 50 lb and neglecting the masses of the pulleys and the effect of friction, determinea) The acceleration of each blockb) The tension in the cable.Then the acceleration:
0423 dt
dv
dt
dv
dt
dv
dt
dv cba
0423 cba aaaa
I now have an equation relating the acceleration of the three weights.
![Page 41: ENG1040 Lec06](https://reader036.vdocument.in/reader036/viewer/2022062418/553bc37955034632548b4620/html5/thumbnails/41.jpg)
Example Question
• Question 12.32 [kinetics]
41
The weight of blocks A, B, and C are wa = wc = 20 lb, and wb=10 lb. Knowing that P = 50 lb and neglecting the masses of the pulleys and the effect of friction, determinea) The acceleration of each blockb) The tension in the cable.
)(4 xCCamTP
)(2 xBBamT )(3 xAAamT
0423 cba aaa
Using these equations, I can solve the problem. Note that I have four equations and four unknowns.
aA = 8.9 m/s2
aB = 11.9 m/s2
aC = 12.6 m/s2
T = 27 N