Understand the application of Lami's Theorem. Click on the link to Watch the VIDEO explanation: Watch Video
Lets understand the application of Lami's theorem in solving problem
A weight is supported on a smooth plane of inclination alpha to the horizontal by a string inclined to the vertical at an angle gamma. if the slope of the plane be increased to beta and the slope of the string is unaltered, the tension of the string is doubled to support the weight. Prove that cot alpha minus cot gamma is equal to 2 cot beta.
Now let us see how to prove the given relation using Lami's theorem
Let R1 be the reaction on the weight W in case 1 and R2 be the reaction on the weight W in case 2
Click on the button Case 1
When the inclination is alpha. The forces R1, T and W acting at the weight are in the equilibrium
Appling Lami's theorem to the 3 forces, we have
R1 by sin of pie minus gamma is equal to T by sin of pie minus alpha is equal to W by sin of alpha plus gamma
This is equal to R1 by sin gamma is equal to T by sin alpha is equal to W by sin of alpha plus gamma. let this equation be equation 1
Click on the button Case 2
In this case the inclination of R2 with the weight W is beta. The forces R2, 2T and W acting at the weight are in the equilibrium
Therefore, Lami's theorem we have
R2 by sin of pie minus gamma is equal to 2T by sin of pie minus beta is equal to W by sin of beta plus gamma.
This implies R2 by sin gamma is equal to 2T by sin beta is equal to W by sin of beta plus gamma. Now let this equation be equation 2
From equation 1 we have T by W is equal to Sin alpha by sin of Alpha plus gamma and
From equation 2 we have T by W is equal to Sin beta by 2sin of beta plus gamma.
Now equating the two equations we have
Sin alpha by sin of Alpha plus gamma is equal to Sin beta by 2sin of beta plus gamma.
Simplifying the steps we have 2 cot beta is equal to cot alpha minus cot gamma Which is the required equation
Lets understand the application of Lami's theorem in solving problem
A weight is supported on a smooth plane of inclination alpha to the horizontal by a string inclined to the vertical at an angle gamma. if the slope of the plane be increased to beta and the slope of the string is unaltered, the tension of the string is doubled to support the weight. Prove that cot alpha minus cot gamma is equal to 2 cot beta.
Now let us see how to prove the given relation using Lami's theorem
Let R1 be the reaction on the weight W in case 1 and R2 be the reaction on the weight W in case 2
Click on the button Case 1
When the inclination is alpha. The forces R1, T and W acting at the weight are in the equilibrium
Appling Lami's theorem to the 3 forces, we have
R1 by sin of pie minus gamma is equal to T by sin of pie minus alpha is equal to W by sin of alpha plus gamma
This is equal to R1 by sin gamma is equal to T by sin alpha is equal to W by sin of alpha plus gamma. let this equation be equation 1
Click on the button Case 2
In this case the inclination of R2 with the weight W is beta. The forces R2, 2T and W acting at the weight are in the equilibrium
Therefore, Lami's theorem we have
R2 by sin of pie minus gamma is equal to 2T by sin of pie minus beta is equal to W by sin of beta plus gamma.
This implies R2 by sin gamma is equal to 2T by sin beta is equal to W by sin of beta plus gamma. Now let this equation be equation 2
From equation 1 we have T by W is equal to Sin alpha by sin of Alpha plus gamma and
From equation 2 we have T by W is equal to Sin beta by 2sin of beta plus gamma.
Now equating the two equations we have
Sin alpha by sin of Alpha plus gamma is equal to Sin beta by 2sin of beta plus gamma.
Simplifying the steps we have 2 cot beta is equal to cot alpha minus cot gamma Which is the required equation
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