APPENDIX C
VARIATION PROBLEMS
SECTION I
1.
2.
The safe uniformly distributed load P which a beam simply supported at each end can carry varies
directly as its width w, and the square of its depth d and inversely as the distance L between the
supports. If a 6 in width by 12 in depth wooden beam having a length of 18 ft between the
supports can carry a safe load of 2400 lb, determine:
(a)
a relationship between P,w,d
numerical constant of proportionality)
(b)
the safe load for a 2 in width by 8” depth beam of the same material, having 10 ft between
supports
The velocity of sound in air varies as the square root of its absolute temperature T. If
the velocity of sound V in air at 273 K is 331 m/s, determine:
(a)
a relationship between V and T in the units given (final answer must contain a
numerical constant of proportionality)
(b)
the temperature necessary for a velocity of 350 m/s
3.
The velocity of sound V, in a medium through which it travels is directly proportional to its
acoustical impedance Z and inversely proportional to its density ρ. If the velocity of sound is
6000 m/s when the density of a medium is 8.00 g/cm3 and the acoustical impedance is 4.8x107
Rayl, determine a relationship between V, Z and ρ. in the units given (your answer must contain a
numerical constant of proportionality)
4
The frequency f of vibration of a piano wire is directly proportional to the square root of its
tension T and inversely proportional to its length L. If a wire 50 cm long vibrates at 256 Hz when
under a tension of 245 N, determine:
5.
(a)
a relationship between f, T and L in the units given
(b)
the frequency of an identical wire 80 cm long and under a tension of 402 N
(c)
the amount by which the length of wire in (a) must be increased to produce a frequency of
200 Hz (the tension is maintained at 402 N)
(d)
if instead of increasing the length as in (b), what tension could be applied to achieve the
200 Hz with a wire length of 80 cm?
Rapid expansion of a gas follows a relationship in which the pressure P varies directly with the
1.4 power of volume V and inversely as the absolute temperature T.
MS273 Appendix C Solutions
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If the volume is 100 cm3 when the pressure is 200 kPa and the temperature is 300 K, determine:
(a)
a relationship between P, V and T in the units given
(b)
the pressure when the volume is 600 cm3 and the temperature is 500 K
(c)
the volume if the pressure is 750 kPa and the temperature if 475 K
Skip 6.The deflection D in a beam loaded at the centre and supported at the ends varies directly as the
load W at the centre, the cube of the distance L between the supports and inversely as the width b
of the beam and the cube of its depth h. If a 2.5 in. wide by 3 in. deep beam has its supports 12 ft.
apart and a load of 250 lb. at its centre, its deflection is 4 in., determine:
(a)
(b)
the deflection when W is 400 lb., L is 10 ft., h is 4 in. and b is 2 in.
INDIRECT OR INVERSE PROPORTION/VARIATION
SECTION II
1.
The time taken to do a certain job is inversely proportional to the number of people on the job. If
eight men do a certain job in twelve days, how many days will it take ten men to do the same job?
2.
The time taken to empty a storage tank is inversely proportional to the area of the drain pipe. If a
tank is emptied in ten minutes by a drain pipe whose area is four square inches, how long will it
take to empty the tank if the area of the pipe is one and a half square inches?
3.
A pump drains a storage tank in 25 minutes at a rate of 240 gallons per minute. How long will it
take to empty if the rate of pumping was increased to 330 gallons per minutes?
4.
A storage tank is filled with water in twelve hours when using four taps. How long will it take to
fill the same tank using nine taps?
5.
A certain job is completed in five working days by three plumbers. How long would it
take to do the job using two plumbers?
JOINT VARIATION
SECTION III
MS273 Appendix C Solutions
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1.
The lost of head due to friction increases as the length of the pipe increases and decreases as the
diameter increases. If thirty feet of head are lost in 350 feet of two-inch diameter pipe, how much
will be lost in 560 feet of three-inch pipe?
2.
The volume of a cylindrical tank increases jointly as the length of the tank and the area of the
circular ends. If the volume is 6,280 cubic feet when the area of the circular ends is 314 square
feet and the length of the tank is twenty feet, calculate the volume of the tank when the area of the
end is 600 square feet and the length of the tank is twenty-five feet.
3.
A certain job takes six men seven days at eight hours per day to complete. How long would it
take for ten men to complete the job working six hours per day?
4.
The pressure in a liquid varies jointly as the depth and the specific gravity. The pressure in
mercury at a depth of three feet is 17.67 p.s.i. and the specific gravity of mercury is 13.6. Find the
pressure in gasoline at a depth of four feet if the specific gravity of gasoline is 0.64.
5.
The time taken (days) to excavate a pine trench varies jointly as the length, width and depth of the
trench and inversely as the number of machines used to dig the trench. If six machines dig a
trench 100 m long, 1.4 m wide and 3 m deep in seven days, how long will it take ten machines to
dig a trench 320 m long, 1.1 m wide and 2.0 deep?
ANSWERS
Section I
1.
(b)
640 lbs
2.
(b)
305.24 k
3.
(a)
1.0 % 10 −3
4.
(b)
204.95 Hz
(c)
1.98 cm
5.
(b)
1473.3 Kpa
(c)
356.9 cm3
6.
(b)
1.953 in.
m$g
s $ RAYL $ cm 3
(d)
382.81
Section II
1.
4.
9.6 days
5.33 hr.
2.
5.
26.67 min.
7.5 days
3.
2.
5.
15,000 cu. ft.
3.
7.04 days
APPENDIX C
18.18 min.
Section III
1.
4.
32 ft.
1.11 p.s.i.
5.6 days
FULL SOLUTIONS FOR THE VARIATION PROBLEMS
MS273 Appendix C Solutions
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SECTION I
1.
The safe uniformly distributed load P which a beam simply supported at each end can carry varies
directly as its width w, and the square of its depth d and inversely as the distance L between the
supports. If a 6 in width by 12 in depth wooden beam having a length of 18 ft between the
supports can carry a safe load of 2400 lb, determine:
(a)
a relationship between P,w,d
numerical constant of proportionality)
2
P = k wd
L
2
2
2400 = k 6 % 12 P = 600 wdL
18 % 12
2400 = 4k
k = 600
(b)
the safe load for a 2 in depth by 8” width beam of the same material, having 10 ft between
supports
2
2
2%8
= 640 lb
P = 600 wdL = 600 10%12
2.
The velocity of sound in air varies as the square root of its absolute temperature T. If
the velocity of sound V in air at 273 K is 331 m/s, determine:
(a)
a relationship between V and T in the units given (final answer must contain a
numerical constant of proportionality)
v=k T
331 = k 273
k = 331 = 20.03
273
(b)
v = 20.03 T
the temperature necessary for a velocity of 350 m/s
350 = 20.03 T
T = 17.471
T = 305.24 K
3.
The velocity of sound V, in a medium through which it travels is directly proportional to its
acoustical impedance Z and inversely proportional to its density ρ. If the velocity of sound is
6000 m/s when the density of a medium is 8.00 g/cm3 and the acoustical impedance is 4.8x107
MS273 Appendix C Solutions
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Rayl, determine a relationship between V, Z and ρ. in the units given (your answer must contain a
numerical constant of proportionality)
V = kZ
!
7
6000 = k 4.8 % 10
8.00
g
6000 ms % 8.00 cm 3
m$g
k=
= 1.0 % 10 −3
7
4.8 % 10 Rayl
s $ Rayl $ cm 3
4
V = 1.0 % 10 −3 Z!
The frequency f of vibration of a piano wire is directly proportional to the square root of its
tension T and inversely proportional to its length L. If a wire 50 cm long vibrates at 256 Hz when
under a tension of 245 N, determine:
(a)
a relationship between f, T and L in the units given
T
L
245
256 = k
50
k = 256Hz % 50cm = 817.762 Hz $1/2cm
N
245N
f=k
(b)
T
L
the frequency of an identical wire 80 cm long and under a tension of 402 N
f = 817.762
(c)
f = 817.762
T
L
= 817.762
402
80
= 204.95Hz
the amount by which the length of wire in (a) must be increased to produce a frequency of
200 Hz (the tension is maintained at 402 N)
T
L
402
200 = 817.762
L
817.762 402
L=
= 81.98cm
200
increase = 81.98 − 80 = 1.98cm
f = 817.762
(d)
if instead of increasing the length as in (b), what tension could be applied to achieve the
200 Hz with a wire length of 80 cm?
MS273 Appendix C Solutions
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T
L
T
200 = 817.762
80
T = 80 % 200 = 19.566
817.762
T = 382.81 N
f = 817.762
5.
Rapid expansion of a gas follows a relationship in which the pressure P varies directly with the
1.4 power of volume V and inversely as the absolute temperature T.
If the volume is 100 cm3 when the pressure is 200 kPa and the temperature is 300 K, determine:
(a)
a relationship between P, V and T in the units given
1.4
P = kV
T
1.4
200 = k 100
300
k = 200kPa1.4% 300K
= 95.09 Pa $ 3K
3
cm
100 cm
(b)
1.4
P = 95.09 VT
the pressure when the volume is 600 cm3 and the temperature is 500 K
1.4
P = 95.09 V
T
1.4
= 95.09 600 = 1474kPa
500
(c)
the volume if the pressure is 750 kPa and the temperature if 475 K
P = 95.09 V
T
1.4
750 = 95.09 V
475
V 1.4 = 750 % 475 = 3746.450
95.09
1
V = 3746.450 1.4 = 356.9cm 3
Skip 6. The deflection D in a beam loaded at the centre and supported at the ends varies directly as the
load W at the centre, the cube of the distance L between the supports and inversely as the width b
MS273 Appendix C Solutions
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of the beam and the cube of its depth h. If a 2.5 in. wide by 3 in. deep beam has its supports 12 ft.
apart and a load of 250 lb. at its centre, its deflection is 4 in., determine:
(a)
3
D = k WL3
bh
250 % (12 % 12) 3
4=k
2.5 % 3 3
3
2
−4 in
%
k = 4in % 2.5in % 27in
=
3.936
10
lb
250lb % 144 3 in 3
(b)
3
D = 3.936 % 10 −4 WL
bh 3
the deflection when W is 400 lb., L is 10 ft., h is 4 in. and b is 2 in.
3
D = 3.936 % 10 −4 WL3
bh
???
3
(
)
400 % 12 % 10
= 3.936 % 10 −4
=
2 % 43
INDIRECT OR INVERSE PROPORTION/VARIATION
SECTION II
1.
The time taken to do a certain job is inversely proportional to the number of people on the job. If
eight men do a certain job in twelve days, how many days will it take ten men to do the same job?
1
T = k men
Or use the following method
12 = x
1
1
8
x=
2.
10
12
10
1
8
= 9.6
The time taken to empty a storage tank is inversely proportional to the area of the drain pipe. If a
tank is emptied in ten minutes by a drain pipe whose area is four square inches, how long will it
take to empty the tank if the area of the pipe is one and a half square inches?
MS273 Appendix C Solutions
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T = k A1
10 = x
1
1
4
1.5
10
1.5
1
4
x=
3.
= 26.7 min
A pump drains a storage tank in 25 minutes at a rate of 240 gallons per minute. How long will it
take to empty if the rate of pumping was increased to 330 gallons per minutes?
T = k R1
25 = x
1
1
240
330
x=
4.
25
330
1
240
= 18.18 min
A storage tank is filled with water in twelve hours when using four taps. How long will it take to
fill the same tank using nine taps?
1
T = k #taps
12 = x
1
1
4
9
12
9
1
4
x=
5.
= 5.33hr
A certain job is completed in five working days by three plumbers. How long would it
take to do the job using two plumbers?
1
T = k #plumbers
5 = x
1
1
3
x=
2
5
2
1
3
= 7.5days
JOINT VARIATION
SECTION III
MS273 Appendix C Solutions
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1.
The lost of head due to friction increases as the length of the pipe increases and decreases as the
diameter increases. If thirty feet of head are lost in 350 feet of two-inch diameter pipe, how much
will be lost in 560 feet of three-inch pipe?
H = k DL
30 = x
350
560
2
x=
2.
3
30%560
3
350
2
= 32ft
The volume of a cylindrical tank increases jointly as the length of the tank and the area of the
circular ends. If the volume is 6,280 cubic feet when the area of the circular ends is 314 square
feet and the length of the tank is twenty feet, calculate the volume of the tank when the area of the
end is 600 square feet and the length of the tank is twenty-five feet.
V = kLA
6280 =
x
25 % 314 20 % 600
x = 6280 % 25 % 600 = 15000 ft 3
20 % 314
3.
A certain job takes six men seven days at eight hours per day to complete. How long would it
take for ten men to complete the job working six hours per day?
1
T = k #men%#days%#hours
1
1
6%7%8
=
1
1
10%6%x
1
1
=
10 % 6 % x 6 % 7 % 8
x = 6 % 7 % 8 = 5.6 days
10 % 6
4.
The pressure in a liquid varies jointly as the depth and the specific gravity. The pressure in
mercury at a depth of three feet is 17.67 p.s.i. and the specific gravity of mercury is 13.6. Find the
pressure in gasoline at a depth of four feet if the specific gravity of gasoline is 0.64.
MS273 Appendix C Solutions
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P = k % d % sg
17.67 =
x
3 % 13.6 4 % 0.64
x = 17.67 % 4 % 0.64 = 1.11psi
3 % 13.6
5.
The time taken (days) to excavate a pine trench varies jointly as the length, width and depth of the
trench and inversely as the number of machines used to dig the trench. If six machines dig a
trench 100 m long, 1.4 m wide and 3 m deep in seven days, how long will it take ten machines to
dig a trench 320 m long, 1.1 m wide and 2.0 deep?
LWD
T = k #machines
7
100%1.4%3
6
=
x=
MS273 Appendix C Solutions
x
320%1.1%2
10
7%320%1.1%2
10
100%1.4%3
6
10
= 7.04days
31/01/04