3rd Class - Edition 2.0
Solutions to Self Test Problems
A1 - CHAPTER 7
1. A 380 kg object requires an effort of 450 N to li it 14 m, using a simple machine that is 92% efficient.
How far must the effort move and what is the mechanical advantage?

Load = 380 kg x 9.81 N/kg = 3727.8 N
3727.8 N
load ________
   =  
 
   
= 8.28 (Ans.)
MA = _____
 
450 N
effort
MA
Efficiency =  ____  x 100
VR
8.284 x 100
MA x 100 __________
  
= 
 
 
VR = ________
 
 
92
efficiency
= 90
distance of effort
  
 
 
VR = _______________
 
distance of load
 Distance effort moves = VR x distance load moves
= 9.0 x 14 m
= 126 m (Ans.)
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2. If a simple machine opera on has a velocity ra o of 7.5 and an actual mechanical advantage of 6.5,
how much load can be li ed by an effort of 80 kN and how far does the load move if the effort moves
65 m?
Load = MA x effort
= 6.5 x 80 kN
= 520 kN (Ans.)
effort distance
  
 
 
Load distance = ____________
 
VR
65
m
= _____
    
7.5
= 8.67 m (Ans.)
3. How efficient is a wheel-and-axle where the wheel diameter is 0.95 m, the axle diameter is 80 mm,
and a load of 700 kN is li ed 21 m by an effort of 65 kN?
D 0.95 m
VR = __
    = ______
 
  
= 11.875
d 0.08 m
load 700 kN
 
  
= 10.77
MA =  _____   = ______
effort 65 kN
MA
% Efficiency =  ____  x 100
VR
10.77
   
= ______
 
x 100
11.875
= 90.7% (Ans.)
4. What load can be li ed by an effort of 150 kN applied to a wheel-and-axle, if the wheel diameter is 2.0 m,
the axle radius is 50 mm, and the efficiency of the device is only 78%?
2m
D
 
   
= 20
VR = __
    = ______
d 0.10 m
MA = VR x efficiency
= 20 x 0.78
= 15.6

Load = MA x effort
= 15.6 x 150 kN
= 2340 kN (Ans.)
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5. Which of the following wheel-and-axle machines is most effficient and what is the difference in the
two efficiencies?
Machine A: has a wheel radius of 1.4 m and an axle diameter of 31 cm, but requires an effort of
3.5 kN to move a 3 tonne load.
Machine B: li a load that is exactly 13 mes the effort, while the wheel rope moves 14.5 mes
further than the load rope.
Machine A:
2.8
D
 
    
VR = __
    = ______
= 9.03
d 0.31 m
load 3000 kg x 9.81 N/kg
  
 
 
MA =  _____   =  ________________
 
3500 N
effort
= 8.41
MA
% Efficiency = ____
    x 100%
VR
8.41
  x 100% = 9.31%
 
= ____
 
9.03
Machine B:
MA = 13
VR = 14.5
MA
% Efficiency = ____
    x 100%
VR
13
= ____
     x 100 = 89.6%
14.5
 Machine A is more efficient by: 93.1 − 89.6
= 3.5% (Ans.)
6. A rope block with 4 pulleys in the top block and 3 pulleys in the bo om block li s a piece of machinery, with mass of 1220 kg, through a height of 5 m. A downward effort of 2740 N is required.
a) What length of rope is pulled through the pulleys?
b) What is the mechanical advantage?
c) What is the efficiency?
Since effort is downwards, VR = number of pulleys = 7
a)
Length of rope pulled = distance effort moves
= VR x load distance
=7x5m
= 35 m (Ans.)
b)
c)
1220 kg x 9.81 N/kg
load _________________
   =  
  
 
 
 
Mechanical Advantage (MA) = _____
 
2740 N
effort
= 4.37 (Ans.)
MA
% Efficiency = ____
    x 100%
VR
4.37
= ____
      x 100%
7
= 62.4% (Ans.)
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7. A 20 tonne load is li ed by an overhead crane, which has a total of nine pulleys between the top
and bo om blocks. If the actual mechanical advantage of the li is 9.0, find the effort applied by the
crane motor and the efficiency of the li .
load 20 000 kg x 9.81 N/kg
 
 
 
 
Effort = ____
    =  __________________
MA
9
= 21.8 kN (Ans.)
Since the effort is applied upwards, the
VR = number of pulleys + 1
= 10
MA
% Efficiency = ____
    x 100%
VR
9
= ___
     x 100% = 90% (Ans.)
10
8. What effort must be applied to a screw jack to li a 5.0 tonne load if the jack is 62% efficient, the
handle is 90 cm long, and the pitch is 15 mm?
2 x 3.1416 x 0.9 m
2  L _______________
  =  
   
 
 
VR =  _____ 
0.015 m
pitch
= 377
MA = efficiency x VR
= 0.62 x 377
= 233.7
load
Effort = ____
    
MA
5000 kg x 9.81 N/kg
  
 
 
=  _________________
233.7
= 210 N = 0.21 kN (Ans.)
9. For each revolu on of a screw jack, the load rises 12 mm. The velocity ra o of the jack is 220. If the
efficiency is 58%, how much load, in tonnes, can be li ed with an effort of 4.0 kN, and how long is
the jack handle?
MA
Efficiency =  ____  
VR
MA = efficiency x VR
= 0.58 x 220 = 127.6
Load = effort x MA
= 4.0 kN x 127.6
= 510.4 kN
510.4
  tonnes = 52 tonnes (Ans.)
= _____
 
9.81
2L
  
 
VR = _____
 
pitch
VR x pitch

L =  _________
 
  
2
220 x 0.012 m
     
 
= ____________
 
2 x 3.1416
= 0.420 m = 420 mm (Ans.)
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10. A solid sphere has a radius of 20 cm and it “weighs” 2800 N. What is the rela ve density of the
sphere’s material?
4
Volume of sphere =  __   r3
3
4
= __
    x 3.1416 x 0.2 x 0.2 x 0.2 m3
3
= 0.03351 m3
2800 N
Mass of sphere = _________
 
   
9.81 N/kg
= 285.42 kg


mass
Density of sphere = _______
 
   
volume
285.42
=  ________ 
  
= 8517 kg/m3
0.03351
density of sphere
  
Rela ve density = _______________
 
 
 
density of water
8517 kg/m3
= __________
 
  
= 8.52 (Ans.)
1000 kg/m3
11. How many tonnes of oil, with specific gravity of 0.86, can be held in a storage tank that is 5 m diameter and 6 m high?
Volume of storage tank = 0.7854 d2 x h
= 0.7854 x 5 x 5 x 6
= 117.81 m3
Specific Gravity of oil = 0.86
 
Density of oil = 0.86 x density of water
= 0.86 x 1000 kg/m3
= 860 kg/m3
  Mass of oil in storage tank = 117.81 m3 x 860 kg/m3
= 101 316 kg
1 tonne = 1000 kg
101 316
 
  
= 101.3 tonnes (Ans.)

Oil in tank =  _______
1000
12. What is the density of a material if it has a rela ve density of 8.5 and a volume of 5 m3?
Density = rela ve density x density of water
= 8.5 x 1000 kg/m3
= 8500 kg/m3 (Ans.)
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13. What pressure exists at the bo om of a 1.5 metre high column of a liquid that has a rela ve density
of 6.0?
Density of liquid () = rela ve density x 100 kg/m3
 = 6.0 x 1000
= 6000 kg/m3
Pressure = h x  x g
= 1.5 m x 6000 kg/m3 x 9.81
= 8829 pa
= 88.29 kPa (Ans.)
14. How many millimetres of mercury (mm Hg) would be equivalent to the pressure caused by a column
of oil (rela ve density = 0.94) that is 2.0 m high?
Pressure caused by oil = h x 0.94 x 1000 x 9.81
= 2.0 x 0.94 x 1000 x 9.81 pa
= 18 442.8 pa
= 18.443 kPa
1 mm Hg = 0.133 kPa
18.443 kPa
  
 
 
Equivalent height of Hg = _____________
 
0.133 kPa/mm
= 138.67 mm (Ans.)

15. A pressure gauge at the inlet to a boiler exhaust stack indicates a pressure of 3.3 kPa. If a dra gauge,
calibrated for mm H2O, is connected at the same loca on to verify the gauge pressure, what should
the reading be?



1 mm H2O = 0.00981 kPa
3.3
3.3 kPa = _______
 
    
mm H2O
0.00981
= 336.4 mm H2O (Ans.)
16. If atmospheric pressure at the me of the test in ques on 15 is 99.8 kPa, what is the absolute pressure at the inlet to the boiler stack?
Absolute pressure = gauge pressure + atmospheric pressure
= 3.3 kPa + 99.8 kPa
= 103.1 kPa absolute (Ans.)
17. A pressure gauge, located 1 metre from the bo om of a tank, reads 68.0 kPa when the tank is completely full. If the tank is 4 m diameter and 6 m high, what is the rela ve density of the liquid in the
tank and what will the pressure gauge reading be when the tank is half full?

Pressure on gauge = h x  x g
68 000 Pa = 5 x  x 9.81
68 000
 
  
= 1386 kg/m3

 = _______
 
5 x 9.81
1386

Rela ve density = _____
 
  = 1.386 (Ans.)
1000
When tank is half full, h = 2 m

Pressure = 2 x 1386 x 9.81 Pa
= 27 193 Pa
= 27.2 kPa (Ans.)
• 3rd Class - Edition 2.0
18. A closed, horizontal, rectangular storage tank is 5 m long, 2 m wide, and 1.5 m high. A vent pipe
from the top of the tank rises through the roof of the building. The tank is full of liquid, with rela ve
density of 1.2, plus the vent pipe has a level that is 3 m above the top of the vessel. What is the force
on the ends and the sides of the vessel?
Area of tank sides = 5 m x 1.5 m = 7.5 m2
Area of tank ends = 2 m x 1.5 m = 3 m2
Consider pressure at top of tank (i.e., bo om of vent):
P = h x  x 9.81
= 3 m x 1.2 x 1000 kg/m3 x 9.81
= 35 316 pa = 35.316 kPa
Consider pressure at bo om of tank:
P = h x  x 9.81
= 4.5 m x 1.2 x 1000 x 9.81
= 52 974 pa
= 52.974 kPa
 Average pressure on sides and ends:
35.316 + 52.974
= ______________
 
 
 
 
 
2
= 44.145 kPa
Force = average pressure x area
Force on ends = 44.145 kPa x 3 m2
= 132.4 kN (Ans.)
Force on sides = 44.145 kPa x 75 m2
= 331.1 kN (Ans.)
19. How many kilograms of water per minute are passing through a pipeline valve if the velocity is
11.5 m/s in the pipe, which has an ID of 254 mm?

Density of water = 1000 kg/m3
Cross-sec onal area of pipe = 0.7854 x 0.2542 m2
= 0.0507 m2
Volume flowing per second = velocity x area
= 11.5 m/s x 0.0507 m2
= 0.583 m3
Volume per minute = 0.583 m3 x 60
= 34.98 m3/min
kg of water per minute = 34.98 m3/min x 1000 kg/m3
= 34 980 kg/min (Ans.)
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20. An orifice in a water pipe restricts the flow diameter to 2/3 of the pipe diameter. If the flow in the
pipe is 1.2 m3/s and the velocity is 2.7 m/s, what is the velocity through the orifice?
Use:

a1 v1 = a2 v2
a
v2 = __
 a12  x v1
a1 = 0.7854 d12
but

a2 = 0.7854 d22
2
d2 = __
    d1
3
a
d2 9
0.7854 d12
__
 a12  = ___________
 
  
= ____
 _4 1  2  = __
  
 
2
_
2
0.7854 ( 3  d1)  9  d1 4
9
    x v1
v2 = __
4
9
= __
    x 2.7 m/s = 6.08 m/s (Ans.)
4
21. Oil, with rela ve density = 0.85, flows through a 20 km sec on of pipeline in one hour. If the pipeline
ID is 300 mm, find Q and the mass flow per minute of the oil.
distance
Velocity = _______
 
   
 
me
20 000 m
= ________
 
  
= 5.556 m/sec
3600 sec
Cross-sec onal area = 0.7854 x 0.32 m2
= 0.0707 m2

Q= v x a
= 5.556 m/sec x 0.0707 m2
= 0.392 m3/s (Ans.)
Density of oil = 0.85 x 1000 kg/m3
= 850 kg/m3
 Mass flow/minute = density x Q x 60 sec/min
= 850 x 0.392 x 60
= 19 992 kg/min (Ans.)