Calculate the weight of body and the coefficient of friction, Mechanical Engineering

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Calculate the weight of body and the coefficient of friction:

Q: Bodies resting on rough horizontal plane required pull of 24N inclined at 30º to the plane just to move it. It was found that a push of 30N at 30º to plane was just sufficient to cause motion to impend. Calculate the weight of body and the coefficient of friction. Sol.:     ∑H = 0; F = P1cosθ

∑V = 0; R = W - P1sinθ

Also F = µR

µ(W - P sin θ ) = P1cosθ

or P1 = µ W / (cosθ  + µsinθ )                                                                                               ...(i)

With reference to the free body diagram when push is applied

∑H = 0; F = P2cosθ

∑V = 0; R = W + P2sinθ

Also  F = µR

µ(W + P2sinθ   ) = P2cosθ

P2 = µW/(cosθ   - µsinθ )                                                                                                  ...(ii)

From equation (i) and (ii),

P1/ P2 = (cosθ    - µsinθ)/ (cosθ + µsinθ)

24/30 =   (cos 30°- µ sin 30°)/ (cos 30°+ µsin 30°)

= (0.866 - 0.5µ)/(0.866 + 0.5µ)

0.6928 + 0.4µ = 0.866-0.5u

By solving

µ = 0.192                                                                     .......Ans

Putting value of µ in the equation (i) we get value of W

W = 120.25N                                                                .......Ans

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