Urban and Civil

Nov 26, 2013 (4 years and 5 months ago)

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1

D
ITIONAL QUESTIONS

Question 1

In each of the following situations a torque is acting. In each case, identify the axis of
rotation or pivot point about which the torque acts, estimate the size of the lever arm, and
estimate the angle betwe
en the lever arm and the direction of the force.

a turning on a tap

b lifting a wheelbarrow by the handles

c picking up an object with a pair of tongs

d swinging back on your chair so that it is supported only by two legs.

Question 2

A child
-
safe
fridge lock uses a Velcro device to increase the torque required to open the
door. Estimate the torque required to open a refrigerator door if a force of 100 N is
needed.

Question 3

Building wreckers wish to knock over a concrete wall. They plan to us
e a ball and chain
which exerts 5000 N to hit the wall at a point that is 3.0 m above the ground. What torque
is developed on the wall around its base? In what ways could the wrecker increase the
torque exerted on the wall?

Question 4

Which of the fol
lowing situations are in translational equilibrium?

A a stationary elevator

B an elevator 'going
-
up' with constant velocity

C an aeroplane during take off

D a billiard ball rolling with constant velocity

E a draw
-
bridge rising at a constant rate.

Que
stion 5

Two window
-
cleaners work on a platform which is supported by four cables. The
platform has a mass of 50 kg, and the cleaners weigh 600 N and 850 N. Assuming that all
the weight is evenly distributed, calculate the tension in each of the cables.

2

Question 6

Three cars are crossing a beam bridge of mass 500 kg in a single line. At one instant, the
left pillar (X) is required to exert a force of 20 000 N upwards, what is the size and
direction of the force exerted by the second pillar (Y)?

Question 7

A bridge over a river is made from steel trusses that have a mass of 5000 kg. The bridge
is supported by two pillars which each support half of the load. The bridge is designed to
support a further load of 20 tonnes with a safety factor of

8. What force must each of the
two supporting pillars exert?

Question 8

A rectangular advertising sign is supported from its upper corners by two cables, each
making an angle of 40
o

to the vertical. The sign has a mass of 5.0 kg. Calculate the
tensio
n in each cable.

Question 9

A picture is hung as shown in the following diagram. If the hanging wire has a breaking
strength of 40 N, what is the maximum possible mass of the picture?

Question 10

A 60 kg tight
-
rope walker carries a long beam
with a mass of 30 kg across a 10 m long
wire. When she is at the centre of the wire (i.e. 5 m across) each section of the wire
makes an angle of 5
o

to the horizontal. Calculate the tension in the wire.

3

Question 11

A 100 g electric light is suppo
rted by two cables, one at an angle of 60
o

with the ceiling,
and the other being perpendicular to the wall. Assuming the mass of each cable is
negligible, calculate the tension in each cable.

Question 12

The keystone in an arch is supported the
two stones adjacent to it, and these each produce
reaction forces N1 and N2. If each reaction force is 80 N, and q is 15
o

, determine the
mass of the keystone.

Question 13

Two children are balanced on a see
-
saw which is supported in the middle
. One child
weighs 200 N and is 1.2 m from the axis, while the other child is seated 1.5 m from the
axis. How much does the second child weigh? Ignore the mass of the see
-
saw.

Question 14

Two children wish to make a see
-
saw from a 5.0 m plank of wood
. The children weigh
25 kg and 20 kg. They each wish to sit right on the ends of the plank. Where should the
plank be supported in order for it to balance?

4

Question 15

A 1200 kg car is crossing a 30 m bridge which is supported by two columns X and
Y
producing forces and . The bridge beam has a mass of 10 000 kg and the centre of
mass of the car is 10 m from column X.

a Write an equation to represent the relationship between the vertical forces acting in the
system.

b Consider the force to

be a pivot point. Write an expression relating the clockwise
torques produced by the weight of the car and the weight of the bridge beam.

c Write an expression for the anticlockwise torque about X due to the column Y.

d What is the relationship betwee

e Determine the forces Fx and Fy .

Q
uestion 16

A 1200 kg car is crossing a 30 m bridge which is supported by two columns X and Y
producing forces Fx and Fy . The bridge beam has a mass of 10 000 kg and the centre of
m
ass of the car is 10 m from column X.

A 10 tonne truck is one
-
third of the way across the bridge (i.e. 10 m from column X).

Calculate the reaction force exerted by each column.

Question 17

A makeshift shelf in a farm shed is constructed using a 10 m

beam of mass 50 kg
supported at each end. If the shelf supports three loads of wheat each having a mass of
100 kg, 150 kg and 200 kg at the positions shown, calculate the support forces at each
end of the beam.

Question 18

A train engine passes

over a 20 m bridge span which is supported by two columns X and
Y. The engine has a total mass of 5.0 tonnes. At one instant the column Y produces a
reaction force of 30.6 kN. If the spanning beam is uniform and has a mass of 5.0 tonnes,
where is the cent
re of mass of the train?

5

Question 19

A 300 kg marble slab, 1.0
x

1.0 m square in cross
-
section, stands 5 metres high. If a
force of 700 N is applied at right angles to the slab in the exact centre of one side, will it
topple over?

Question 2
0

A 4.0 m cantilever
-
type verandah is constructed from the roof of a shop with a mass of
900 kg. The cantilever is supported by two supporting columns X and Y, which each
produce reaction forces Fx and Fy.

a Write an equation which shows how all vertical

forces are balanced.

b Write an expression relating the torques about the point X.

c Determine the force supplied by column Y, Fy .

d Write an expression relating the torques about the point Y.

e Determine the force Fy and indicate whether X and Y

are in compression or tension.

Question 21

a A 5.0 m long painter's platform has a mass of 20 kg and is supported by two ropes as
shown. The 70 kg painter stands 1.50 m from the left. Calculate the tension in each
supporting rope.

b The

painter now alters his position such that the left
-
hand rope experiences a tension of
557 N and the other a tension of 325 N. Where is the painter now standing in relation to
the left
-
hand rope?

6

Question 22

A pedestrian bridge over a small cree
k is made from two identical cantilevers, each of
400 kg.

a Calculate the reaction forces produced by the pillars A, B, C and D,

(i) when there are no pedestrians on the bridge, and

(ii) when a 70 kg person stands at position P on the left
-
hand can
tilever.

b What happens to the value of the forces in A and B as the person walks from A past B
to P?

Question 23

A 2.5 m scenic viewing platform on a cliff is constructed from a 300 kg horizontal
concrete beam and is supported by a metal strut. T
he strut provides a supporting force F
which acts along the line of the strut.

a Write expressions for each of the horizontal and vertical components of F.

b Write an expression for the clockwise torques that act about the point where the
platform mee
ts the rock face, X.

c Write an expression for the anticlockwise torques that act around X.

d Determine the magnitude for the force F provided by the strut.

e What force acts to balance the horizontal component of the force provided by the strut?

7

Question 24

A 2.5 m scenic viewing platform on a cliff is constructed from a 300 kg horizontal
concrete beam and is supported by a metal strut. The strut provides a supporting force F
which acts along the line of the strut.

A 60 kg person st
ands at the end of the viewing platform, 2.5 m from the cliff. Determine
the size of the new force that is required from the strut.

Question 25

A 10 m drawbridge is supported by two cables which extend from two holes either side
of a door in a ca
stle wall. The bridge has a mass of 700 kg and the tension is the same in
both cables. The bridge is just about to touch the ground and the cables make an angle of
30
o

to the horizontal.

a Write an expression for relating the torques that act around
the axis of rotation for the
bridge.

b Write an expression for the horizontal and vertical components of the tension.

c Calculate the size of the tension in each cable.

8

Question 26

A uniform 5.0 kg beam, 1.8 m long, extends from the side of
a building and is supported
by a cable which is attached 1.2 m from the wall at an angle of 45
o

. Determine the
tension in the cable.

Question 27

The end
-
post of a wire fence is held in position by a back
-
stay
which is under a tension of 800 N at a
n angle of 60
• to the horizontal. The geometry of
the situation is shown in the diagram.

a Determine values for the horizontal and vertical components of the tension in the back
-
stay wire.

b By considering the base of the post to be a pivot point, de
termine the size of the
tension in the fence wire, T.

9

a (i) The axis of rotation is the spindle.

(ii) The size of the lever arm is approximately 5 cm.

(iii) The angle is approximately 90
•.

b (i)
The axis of rotation is the front wheel axle.

(ii) The size of the lever arm is usually between 1 m and 5 m.

(iii) The angle is usually between 120
• and 150•.

c (i) The axis of rotation is the point at which the tongs are held.

(ii) The lever arm is
usually between 10 cm and 20 cm.

(iii) The angle is usually between 45
• and 90•.

d (i) The axis of rotation is where the two back legs are in contact with the floor.

(ii) The size of the lever arm is approximately 50 cm.

(iii) The angle is approxim
ately 90
•.

Answer: The length of the lever arm ( the width of the refrigerator door ) is usually
around 80 cm.

Then t = Frsin q

= (100 N)(0.80 m)sin90
o

= 80 Nm.

a t = Frsin q

= (5000 N)(3.0 m)sin90
o

= 1.5
x

104 N m.

b

The torque exerted on the wall can be increased by increasing either the length of the
chain, the mass of the ball, or both.

Answer: For a body to be in translational equilibrium, the vector sum of the forces acting
on it must be zero.

This

situation applies to any stationary body, or one moving with constant velocity.

The following situations are therefore examples of translational equilibrium:

A, B, D and E.

10

A
nswer: The total weight supported by the four cables

= 600 N + 8
50 N + (50 kg) (9.8 N kg
-
1) = 1940 N,

then the tension in each cable = 1940 / 4 = 485 N.

Answer: The total weight of the bridge plus cars

= (1000 kg + 1500 kg + 2000 kg + 500 kg)(9.8 N kg
-
1
)

= 4.9
x

104 N downwards.

The force exerted b
y pillar Y

= 4.9
x

104 N
-

2.0
x

104 N

= 2.9
x

104 N upwards.

= (5000 kg + 20000 kg) (9.8 N kg
-
1)

= 2.45
x

105 N down.

The total upward force that both pillars must provide to supp
ort this weight with a safety
factor of 8

= 8(2.45
x

105 N)

= 1.96
x

106 N

then each pillar must exert a force of

(1.96
x

106 N) / 2

= 9.8
x

105 N upwards.

A
nswer: Let T = tension in each cable

then 2(Tcos40
o
) = (5.0 kg)(9.8 N kg
-
1)

and T = 32 N.

o

= m (9.8 N kg
-
1)

and m = 5.2 kg.

11

a
nswer: Let T = tension in the wire

then 2(Tcos85
O
) = (60 kg + 30 kg)(9.8 N kg
-
1) = 882 N

and T = 5060 N.

in translational equilibrium.

Let T1 = the tension in the 60
O

cable, and let T2 = the tension in the other cable.

Then T1(sin30
O
) = T2

and T1(cos30
O
) = (0.100 kg)(9.8 N kg
-
1)

so T1 = 1.13 N, T2 = 0.57 N.

a
nswer: The vector sum of the ve
rtical components of N1 and N2 support the weight of
the keystone.

Then 2(80 N)sin15
O

= m(9.8 N kg
-
1)

and m = 4.2 kg.

Answer: Let the weight of the second child = W.

Since the see
-
saw is in rotational equilibrium,

then (200 N)(1.2 m) =
W(1.5 m)

and W = 160 N.

Answer: Let x = the distance of the pivot point from the heavier child.

Then (25 kg)(9.8 N kg
-
1)x = (20 kg)(9.8 N kg
-
1)(5.0 m
-

x)

and x = 2.2 m from the heavier child.

a The sum of the upward verti
cal forces provided by the two columns must equal the
total weight of the bridge plus car.

Then Fx + Fy = (1200 kg + 10 000 kg)(9.8 N kg
-
1) = 109 760 N

12

b tclockwise

= (1200 kg)(9.8 N kg
-
1 )(10 m) + (10 000 kg)(9.8 N kg
-
1 )(15 m)

c t anti
-
clockwise = (F
y )(30 m)

d Since the bridge is in rotational equilibrium,

then tclockwise = t anti
-
clockwise

e Fx + Fy = 109 760 N

and Fy (30 m) = 1 587 600 N

then Fy = 52 920 N

and Fx = 109 760 N
-

52 920 N = 56 840 N.

= (10
x

1
03 kg + 10
x

103 kg)9.8 N kg
-
1 = 196 000 N

and (Fy)(30 m)

= (10
x

103 kg)(9.8 N kg
-
1 )(10 m) + (10 000 kg)(9.8 N kg
-
1 )(15 m).

Then Fy = (2.45
x

106 N) / 30

= 8.17
x

104 N up

and Fx = 196 000 N
-

81 667 N

= 1.14
x

105 N up.

A
n
swer: Let the support forces at either end = FA and FB respectively.

Then FA + FB

= (100 kg + 150 kg + 200 kg + 50 kg)(9.8 N kg
-
1 ) = 4900 N

and (FB)(10 m) = (100 kg)(9.8 N kg
-
1 )(3 m) + (50 kg)(9.8 N kg
-
1)(5 m) + (150 kg)(9.8
N kg
-
1 )(7 m) + (200 kg)(9
.8 N kg
-
1 )(8.5 m).

Then FB = 3234 N up

and FA = 4900 N
-

3234 N = 1666 N up.

Answer: Since the bridge is in rotational equilibrium, then:

(30.6
x

103 N)(20 m)

= (5.0
x

103 kg)(9.8 N kg
-
1 )(10 m) + (5.0
x

103 kg) )(9.8 N kg
-
1) d

where d = the distance of the centre of mass of the train from column X.

Then d = 2.5 m = 17.5 m from column Y.

13

Answer: The torque produced by the 700 N force

= (700 N)(2.5 m) = 1750 N m.

The counter torque produced by the weight of the
column as it is being tilted

= (300 kg)(9.8 N kg
-
1)(0.50 m) = 1470 N m.

Since 1750 N m is greater than 1470 N m,

the column will topple over.

a Fx + Fy = (900 kg)(9.8 N kg
-
1) = 8.82 ¥ 103 N

b (Fy)(1.8 m) = (900 kg)(9.8 N kg
-
1)(2 m)

c F
y = 17640 N m / 1.8 m = 9800 N up

d (Fx)(1.8 m) =
-
(900 kg)(9.8 N kg
-
1)(2 m)

where a negative torque produces a clock
-
wise rotation

e 9800 N up.

Y is under compression, while X is under tension.

a Let the tension in the ropes = T1 and T2

respectively.

Then T1 + T2 = (20 kg + 70 kg)(9.8 N kg
-
1) = 882 N.

(70 kg)(9.8 N kg
-
1)(1.5 m) + (20 kg)(9.8 N kg
-
1)(2.5 m)

= T2(5.0 m)

and T2 = 304 N up,

T1 = 882 N
-

303.8 N = 578 N up.

b Let d = distance of painter from t
he left
-
hand rope.

Then taking moments about the left
-
hand rope:

(70 kg)(9.8 N kg
-
1)d + (20 kg)(9.8 N kg
-
1)(2.5 m)

= (325 N)(5.0 m)

and d = 1.7 m.

a (i) Each cantilever is in rotational equilibrium.

FA(
1.2 m) = (400 kg)(9.8 N kg
-
1)(0.3 m)

and FA = 980 N down.

From the symmetry of the situation FA = FD.

14

FB (1.2 m) = (400 kg)(9.8 N kg
-
1)(1.5 m)

and FB = FC = 4900 N up.

(ii) Each cantilever is in rotational equilibrium.

Then

FA (1.2 m) = (400 kg)(9.8 N kg
-
1)(0.3 m) + (70 kg)(9.8 N kg
-
1)(1.8 m)

and FA =2009 N down.

From the symmetry of the situation FA = FD.

FB(1.2 m) = (400 kg)(9.8 N kg
-
1)(1.5 m) + (70 kg)(9.8 N kg
-
1)(3 m)

and FB = FC = 6615 N up.

b The magnitude of the forces increase, in order to produce a greater torque, to counter
balance the increase in torque produced as the person moves further away from the pivot
point.

a Fx = F(sin55
o
)

Fy = F(co
s55
o
)

b The entire mass of the platform can be considered to act through the centre of mass, so:

t = (300 kg)(9.8 N kg
-
1)(1.25 m)

c t = (Fcos55
o
)(2.5 m)

d Since the platform is in rotational equilibrium:

(300 kg)(9.8 N kg
-
1)(1.25 m)

= (Fcos55
o
)(2.5 m
)

and F = 2563 N

e The reaction force of the wall.

Answer: The platform is in rotational equilibrium, therefore:

(Fcos55
o
)(2.5 m)

= (300 kg)(9.8 N kg
-
1)(1.25 m) + (60 kg)(9.8 N kg
-
1)(2.5 m)

and F = 3588 N.

a Let TV = ver
tical component of the tension T in each cable, and TH be the horizontal
component. Then:

2(TV)(10 m) = (700 kg)(9.8 N kg
-
1)(5.0 m)

b TV = T(sin30
o
), TH = T(cos30
o
)

c TV = (700 kg)(9.8 N kg
-
1)(5.0 m) / (20 m)

= 1715 N

and T = TV / sin30
o

= 3430 N.

15

Answer: The beam is in rotational equilibrium:

(Tcos45
o
)(1.2 m) = (5.0 kg)(9.8 N kg
-
1)(0.90 m)

and T = 52 N.

a TH = (800 N)cos60
o

= 400 N,

TV = (800 N)sin60
o

= 693 N

b (400 N)(0.85 m) = T(1.0 m),

then the tension T in the

fence wire = 340 N.