Solved Example & Practice Problems: Lami's theorem

Solved Examples for Understanding

Example 2.7.1 

Find components Fa and Fb of the 100 N force along the directions shown in Fig. 2.7.2 using Lami's theorem.

Solution : 

• To use Lami's theorem, convert the problem to an equilibrium problem using equilibrant.

• The 100 N force shown has to be resultant of Fa and Fb.

⸫ A 100 N force in opposite direction (equilibrant) will be able to keep Fa and Fb in equilibrium as shown in Fig. 2.7.2 (a).

Using Lami's theorem,

Note: 

Equilibrant is same in magnitude as that of resultant but opposite in direction.

Example 2.7.2 

If the resultant of the two forces shown in Fig. 2.7.3 is 700 N directed bertically upwards, find the angles α and β using Lami's theorem.

Solution: 

• As the resultant is 700 N vertically upward, the equilibrant will be 700 N vertically downward as shown in Fig. 2.7.3 (a). 

• By Lami's theorem,

Squaring, we get,

16 - 25 sin2α = 49 - 70 cos α + 25 cos2 α

70 cos α = 49 + 25 cos2 α + 25 sin2 α - 16

⸫ 70 cos α = 49 + 25 – 16

70 cos α = 58

From equation (1),

4 sin β = 5 sin 34.05

β = 44.42°                 ... Ans.

Note: The solution by Lami's theorem becomes tedious when angles are unknown.

Examples for Practice

Q.1

Determine components F and F of the 500 N force along the directions shown in Fig. 2.7.4 using Lami's theorem. 

[Ans. : Fa = 442.28 N, Fb = 542.53 N]

Q.2

If the resultant of the two forces shown in Fig. 2.7.5 is 90 kN directed vertically upwards, find the angles a and ẞ using Lami's theorem.

 [Ans. : α = 29°, β = 56.44°]