Answer to Question #132548 in Mechanics | Relativity for Shruti

Question #132548
determine the horizontal force P required to start moving the 500N block up the inclined surface.assume static friction is 0.3 and angle is 30
1
Expert's answer
2020-09-11T09:15:22-0400


On this picture mg\mathbf{mg} denotes the gravity force, Ffric=μN\mathbf{F_{fric}} = \mu\mathbf{N} - static frictional force and N\mathbf{N} - normal force. According to the problem, mg=500Nmg = 500 N, μ=0.3\mu = 0.3 and θ=30°\theta = 30\degree.

According to the Newton's second law, the projections on the y axis will be:


NmgcosθPsinθ=0N=mgcosθ+PsinθN - mg\cos\theta - P\sin\theta = 0\\ N =mg\cos\theta + P\sin\theta


The projections on the x axis will be:


Ffric+mgsinθPcosθ=0μN+mgsinθPcosθ=0F_{fric} + mg\sin\theta - P\cos\theta = 0\\ \mu N + mg\sin\theta - P\cos\theta = 0\\

Substituting NN from the first expression, get:


μ(mgcosθ+Psinθ)+mgsinθPcosθ=0\mu (mg\cos\theta + P\sin\theta) + mg\sin\theta - P\cos\theta = 0

Expressing PP, obtain:


P=mgμ+tanθ1+μtanθ P=5000.3+tan30°1+0.3tan30°373.9NP = mg\dfrac{\mu + \tan\theta}{1 + \mu\tan\theta}\\ \space\\ P = 500\cdot \dfrac{0.3 + \tan30\degree}{1 + 0.3\cdot \tan30\degree} \approx 373.9 N

Answer. 373.9 N.


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