LAMI’S THEOREM.pptx

mechanical engineering

1. LAMI’S THEOREM shireen

2. Lami’s theorem relates the magnitudes of coplanar, concurrent and non-collinear forces that maintain an object in static equilibrium. The theorem is very useful in analyzing most of the mechanical as well as structural systems.

3. Lami’s Theorem Statement Lami’s Theorem states, “When three forces acting at a point are in equilibrium, then each force is proportional to the sine of the angle between the other two forces”. Referring to the above diagram, consider three forces A, B, C acting on a particle or rigid body making angles α, β and γ with each other. In the mathematical or equation form, it is expressed as,

4. Lami’s Theorem Derivation Now, let’s see how the theorem’s equation is derived. Let F A , F B , and F C  be the forces acting at a point. As per the statement of the theorem, we take the sum of all forces acting at a given point which will be zero. i.e. F A  + F B  + F C  = 0 The angles made by force vectors when a triangle is drawn are,

5. We write angles in terms of complementary angles and use triangle law of vector addition. Then, by applying the sine rule we get, So, we have,

6. Hence, it is clearly seen that by applying sine rule to complementary angles we arrive at the required result for Lami’s theorem . Now, we will see how Lami’s theorem is useful to determine the magnitude of unknown forces for the given system.

7. Lami’s Theorem Problems and Solved Examples Example 1 :  Consider an advertisement board hangs with the help of two strings making an equal angle with the ceiling. Calculate the tension in both the strings in this case.

8. Solution : The free body diagram of the same helps us to resolve the forces first. After resolving the forces we will apply the required theorem to get the value of tension in both the strings. Here, the weight of the signboard is in a downward direction, and the other force is the tension generated by the signboard in both the strings. In this case, the tension T in both the strings will be the same as the angle made by them with the signboard is equal.

9. Above figure represents the free body diagram of the signboard. Applying the Lami’s Theorem we get,

10. Since sin (180 – θ) = sin θ and sin (2θ) = 2sinθ cosθ So, we get, final tension force in the string T as, i.e,T =mg/2 cos θ The similar concept and equations can be applied for a boy playing on a swing, and we arrive at the same result.

11. THANK YOU