Problems on lamis theorem.pdf

These are the solved problems on Lami's theorem with some assignment questions.

1. Prepared by: Prof. V.V. Nalawade

1

Problems on Lami’s Theorem
Q1. Find the tension in each rope in fig.

ANS: Given data: W = 100 kg = 100*9.81= 981 N
θ = tan
−1
4
3
= 53.13ᵒ

Step 1: Consider the FBD of point C

General Calculation :
Angle between W and TBC = 90 – 30 = 60ᵒ
Angle between TBC & TAC = 30 + 90 + (90 - 53.13) =156.87ᵒ
Angle between TAC & W = 53.13 + 90 =143.13ᵒ

Step 2: Apply Lami’s Theorem ,

981
sin156.87°
=
���
sin60°
=
���
sin143.13°


Therefore, TAC = 2162.76 N
TBC = 1438.41 N

2. Prepared by: Prof. V.V. Nalawade

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Q2. Block P = 5kg and block Q of mass m kg is suspended through
the chord is in the equilibrium position as shown in fig.
Determine the mass of block Q.

ANS: Given data:
i. P = 5 kg = 5*9.81 = 49.05 N
ii. Q = m kg
iii. θ = tan
−1
4
3
= 53.13ᵒ
CASE 1: CONSIDER JOINT B
Step 1: Consider the FBD of Joint B

FBD of Joint B
General Calculation :
Angle between P and TBC = 90 + 30 = 120ᵒ
Angle between TBC & TAB= 180 – 53.13 - 30 =96.87ᵒ
Angle between TAB & P = 53.13 + 90 =143.13ᵒ
Step 2: Apply Lami’s Theorem ,

49.05
sin96.87°
=
���
sin120°
=
���
sin143.13°


Therefore, TAB = 42.79 N

3. Prepared by: Prof. V.V. Nalawade

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TBC = 29.64 N

CASE 2: CONSIDER JOINT C
Step 1: Consider the FBD of Joint C

FBD of Joint C
General Calculation :
Angle between Q and TCD = 90 + 70 = 160ᵒ
Angle between TCD & TBC= 180 – 70 + 30 =140ᵒ
Angle between TBC & Q = 90-30 =60ᵒ
Step 2: Apply Lami’s Theorem ,

�
sin140°
=
29.64
sin160°
=
���
sin60°


Therefore, Q = 55.70 N =
��.��
�.��
=�.�� ????????????
TCD = 75.05 N
Q3. A light string ABCDE whose extremity A is fixed, has weights
W1 and W2 attached to it at B and C. It passes round peg at D
carrying a weight of 300 N at the free end E as shown in fig. if
in the equilibrium position, BC is horizontal and AB and CD make
150
ᵒ
and 120
ᵒ
with BC, find (i) Tension in the portion AB, BC and
CD of the string and (ii) magnitude of W1 and W2.

4. Prepared by: Prof. V.V. Nalawade

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ANS: Given data:
i. Weight at E = 300 N
ii. TCD = TDE = 300 N
CASE 1: CONSIDER JOINT B
Step 1: Consider the FBD of Joint B & c and apply Lami’s
theorem at C,

Fig: FBD of Joint C

���
�???????????? ��
=
���
�???????????? ���
=
??????�
�???????????? ���

Hence, TBC =
���
�???????????? ��
?????? �???????????? ��� = 150N
Similarly W2 = 259.81 N
CASE 1: CONSIDER JOINT B
Step 2: Consider the FBD of Joint B and apply Lami’s
theorem at B,

5. Prepared by: Prof. V.V. Nalawade

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Fig: FBD of Joint B
���
�???????????? ��
=
���
�???????????? ���
=
??????�
�???????????? ���

We know that, TBC = 150N
Therefore, TAB =
���
�???????????? ���
?????? �???????????? �� = 173.21 N
Similarly, W1 = 86.60 N
Therefore, W1 = 86.60 N and W2 = 259.81 N respectively.
EXERCISE : 1

6. Prepared by: Prof. V.V. Nalawade

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Q.4 Two Cylinders P & Q rest in a channel as shown in fig. The
cylinder P has diameter of 100 mm and weighs 200 N, whereas
the cylinder Q has diameter of 180 mm and weighs 500 N.

SOL: Given data:
i. Diameter of Cylinder P = 100 mm
ii. Weight of Cylinder P = 200 N
iii. Diameter of Cylinder Q = 180 mm
iv. Weight of Cylinder Q = 500 N
v. Width of channel = 180 mm.

Step 1: Consider the FBD of Cylinder P

Fig: FBD of cylinder P
General Calculation :
From geometry of the figure, we can find that
ED = radius of cylinder P =
100
2
=50 ��
BF = radius of cylinder Q =
180
2
=90 ��

7. Prepared by: Prof. V.V. Nalawade

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And ∆ BCF = 60
ᵒ

Hence, CF =
��
�??????� 60
=52 ��
FE = BG = 180 – 52 – 50 = 78 mm
And AB = 50 + 90 = 140 mm
Hence ∆ ABG = COS
-1

��
��
=
78
100
=56.14°
Step 2: Apply Lami’s theorem at A,
��
�???????????? ( ��+��.��)
=
��
�???????????? ��
=
���
�???????????? (���−��.��)


Hence, R1 =134.19 N & R2 = 240.85 N

Step 3: Consider the FBD of Cylinder Q

Fig: FBD of cylinder Q

Step 4: Apply Lami’s theorem at B,

��
�???????????? (��+��.��)
=
���.�
�???????????? (��)
=
��−���
�????????????(���+��−��.��)


Hence, R3 = 155 N & R4 = 622.50 N

Q5. Three cylinders weighing 100 N each and of 80 mm diameter are

8. Prepared by: Prof. V.V. Nalawade

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placed in a channel of 180 mm width as shown in fig. Determine
the pressure exerted by i) the cylinder A on B at the point of
contact ii) the cylinder B on the base & iii) the cylinder B on the
wall.

SOL: Given data:
i. weight of each Cylinder = 100 N
ii. Diameter of Cylinder = 80 mm and
iii. Width of channel = 180 mm

Case 1: Pressure exerted by the cylinder A on the cylinder
B
Let, R1 = pressure exerted by the cylinder A on B.
Step 1: Consider the FBD of cylinder A

Fig: FBD of cylinder A
General Calculation :
From geometry of the triangle OPS, we can find that
OP = 40 + 40 = 80 mm

9. Prepared by: Prof. V.V. Nalawade

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PS = 90 – 40 = 50 mm
Therefore, ∆ POS = Sin
-1

��
��
=
50
80
=38.68°
Step 2: Apply Lami’s theorem at O,
��
�???????????? ( ���.�)
=
��
�???????????? ���.�
=
���
�???????????? (��.�)


Hence, R1 = R2 = 64 N
Case 2: Pressure exerted by the cylinder C on the base
Step 3: Consider the FBD of cylinder B

Fig: FBD of cylinder A
General Calculation :
Let, R3 = Pressure exerted by the cylinder B on the wall,
And R4 = Pressure exerted by the cylinder B on the base.
Step 4: Apply Lami’s theorem at P,

��
�???????????? ( ��)
=
��
�???????????? ���−��.�
=
(��−���)
�???????????? (��+��.�)


Hence, R3 = 40 N & R4 = 150 N

EXERCISE : 2
Q1. A right circular cylinder of diameter 40 cm, open at both

10. Prepared by: Prof. V.V. Nalawade

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ends, rests on a smooth horizontal plane. Inside the cylinder,
there are two sphere having weights and radii as given. Find the
minimum weight of the cylinder for which it will not tip over.

Ans : R1 = R2 = 212.16 N & W = 300 N
Q2. Three cylinders are piled up in a rectangular channel as
shown in fig. determine the reactions at point 6 between the
cylinder A and the vertical wall of the channel.

Ans : R6 = 784.8 N

Q3. Two identical rollers, each of weight 50 N, are supported
by an inclined and a vertical wall as shown in fig find the
reactions at the points of supports A,B & C. Assume all the
surfaces to be smooth.

11. Prepared by: Prof. V.V. Nalawade

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Ans : Ra = 43.3 N , Rb = 72 N & Rc = 57.5 N