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Statics of rigid bodies

Chapter Assessment

1.

Overhead cables for a tramway are supported by uniform, rigid, horizontal beams

of weight 1500 N and length 5 m. Each beam, AB, is freely pivoted at one end A

and supports two cables which may be modelled by ver

tical loads, each of

1000 N, one 1.5 m from A and the other at 1 m from B. The beam is supported by

a light wire, attached at one end to the beam at B and at the other to the point C

which is 3 m vertically above A, as shown in the diagram below.

(i)

Calculate the tension in the wire.

[5]

(ii)

Find the magnitude and direction of the force on the beam at A.

[6]

2.

A uniform ladder of length 8 m and weight 180 N rests against a smooth, vertical

wall and stands on a rough, horizontal

surface. A woman of weight 720 N stands

on the ladder so that her weight acts at a distance

x

m from its lower end, as

shown in the diagram.

The system is in equilibrium with the ladder at 20° to the vertical.

(i)

Show that the f

rictional force between the ladder and the horizontal surface is

F

N, where

90(1 ) tan20

F x

.

[5]

720 N

180 N

x

m

8 m

20°

A

B

C

3 m

beam

wire

1.5 m

2.5 m

1 m

1000 N

1000 N

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(ii)

Deduce that

F

increases as

x

increases and hence find the values of the

coefficient of friction between the ladder and the surface for whi

ch the

woman can stand anywhere on the ladder without it slipping.

[5]

3.

A simple lift bridge is modelled as a uniform rod AD of length 4.2 m and weight

5000 N. The rod is freely hinged at B and rests on a small support at C;

AB = 1.5 m and BC = 2.4 m,

as shown in the diagram below. The bridge closed is

represented by the rod being horizontal.

(i)

Calculate the forces acting on the bridge due to the hinge at B and support at

C.

[5]

A lump of concrete of mass

M

kg is placed at A to ‘counterbalan

ce’ the bridge to

make it easier to open. For the bridge to stay firmly closed, the force at C must be

25 N vertically upwards.

(ii)

Calculate the value of

M

.

[4]

With the lump of concrete attached, the bridge is held open at 60° to the horizontal

by m

eans of a light rope of negligible mass attached to D. The rope pulls upwards

at an angle of 10° to the horizontal, as shown in the diagram below.

(iii)

Calculate the tension in the rope.

[6]

4.

A uniform beam AB of length 3 m and weight 80 N i

s freely hinged at A.

Initially, the beam is held horizontally in equilibrium by a small, smooth peg at C

A

B

C

D

1.5 m

2.4 m

A

B

D

60°

10°

rope

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where the distance AC is 2.5 m, as shown in the diagram below.

(i)

Calculate the force on the beam from the peg at C.

[3]

The peg is now moved so that the beam is in equilibrium with AB at 60° to the

vertical, as shown in the diagram below. AC is still 2.5 m.

(ii)

Calculate the new force on the beam due to the peg.

[4]

A light string is now att

ached to the beam at B. The string is perpendicular to the

beam. The beam is in equilibrium with a tension of 20 N in the string, as shown in

the diagram below.

(iii)

Calculate the new force on the beam due to the peg.

[2]

The

peg is now removed and the string attached to a point D vertically above A so

that angle ABD is 50°, as shown in the diagram below.

(iv)

Calculate the new tension in the string. Calculate also the vertical component

of the forc

e acting on the hinge at A.

[5]

A

B

60°

3 m

50°

A

B

C

60°

2.5 m

20 N

A

C

3 m

2.5 m

B

A

B

C

60°

2.5 m

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Total 50 marks

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Solutions to

Chapter Assessment

1.

(i)

Taking moments about A:

sin 5 1000 4 1500 2.5 1000 1.5 0

3

5 9250

34

1850 34

3596

3

T

T

T

The tension in the wire is

3596

N (4 s.f.)

(ii)

Resolving horizontally:

cos 0

9250

1850 34 5

3 34 3

X T

X

Taking moments about B:

5 1000 3.5 1500 2.5 1000 1 0

5 8250

1650

Y

Y

Y

Magnitude of force on beam at A

2

2

9250

1650 3497

3

N (4 s.f.)

9250

3

4950

1650

tan

9250

28.2

Y

X

The direction of the force at A is 28.2° (3 s.f.) above t

he horizontal.

A

B

C

3 m

wire

1.5 m

1

.5 m

1 m

T

1000 N

1000 N

X

Y

5

3

34

1 m

1500 N

Y

X

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2.

(i)

Resolving vertically:

720 180 0

900

R

R

Taking moments about B:

720sin20 (8 ) 180sin20 4 sin20 8 cos20 8 0

720(8 ) 720 7200 sin20 8 cos20 0

900 90 90(8 ) tan20

90( 9 8 )tan20

90(1 )tan20

x R F

x F

F x

F x

F x

(ii)

Since all terms in the expression for x are constant except for x, and x is

positiv

e, then as x increases F must increase.

If the woman stands at the top of the ladder,

x

= 8.

The maximum frictional force required

90 9tan20 810tan20

810tan20

900

0.328

F R

F

R

3.

(i)

720 N

180 N

x

m

8 m

20°

S

F

R

A

B

A

B

C

D

1.5 m

2.4 m

2

.

1

m

5000 N

Y

R

Since all other forces

are

vertical, the force at the

hinge must be vertical.

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Taking moments about B:

2.4 0.6 5000 0

1250

R

R

Resolving vertically:

5000 0

5000 1250 3750

Y R

Y

The force at the hinge at B is 3750 N vertically upwards.

The force at C is 1250 N vertically upwards.

(ii)

Taking moments about B:

25 2.4 1.5 5000 0.6 0

1.5 2940

200

Mg

Mg

M

(iii)

Taking moments about B:

200 cos60 1.5 5000cos60 0.6 sin 70 2.7 0

2.7 sin 70 30

11.8

g T

T

T

The tension in the rope is

11.8

N (3 s.f.)

A

B

C

D

1.5 m

2.4 m

2

.

1

m

5000 N

Y

25 N

Mg

A

B

D

60°

10°

200

g

50

00

T

Y

X

1.5 m

0.6 m

2.1 m

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4.

(i)

Taking moments about A:

80 1.5 2.5 0

48

R

R

The force from the peg is 48 N.

(ii)

Taking moments about A:

80sin60 1.5 2.5 0

41.6

S

S

The force from the peg is 41.6 N (3 s.f.)

(iii)

Taking moments about A:

80sin60 1.5 2.5 20 3 0

2.5 120sin60 60

17.6

Z

Z

Z

The force from the peg is 17.6 N (3 s.f.)

A

B

C

60°

2.5 m

20 N

Z

0

.5 m

80 N

1

.5 m

A

C

1.5

m

2.5 m

B

80 N

R

A

B

C

60°

2.5 m

1

.5 m

80 N

S

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(iv)

Taking moments about A:

sin50 3 80sin60 1.5 0

120sin60

45.2

3sin50

T

T

The new tension is 45.2 N (3 s.f.)

Resolving vertically:

cos 70 80 0

80 45.22cos 70 64.5

Y T

Y

The vertical component

of the force at A is 64.5 N

upwards

(3 s.f.)

A

B

60°

80 N

50°

T

X

Y

7

0°

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