Determine the smallest coefficient of friction

AI Thread Summary
To determine the smallest coefficient of static friction for a cylinder to start rolling up a 30° incline, the correct free body diagram (FBD) must be established. The FBD should depict the normal force acting perpendicularly to the incline and the frictional force acting parallel to the incline. Friction is tangential to the cylinder's radius and is proportional to the normal force. Understanding these forces is crucial for calculating the coefficient of friction needed for the cylinder to ascend the incline. Properly analyzing the forces will lead to the solution for the problem.
blackandyello
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Homework Statement


The horizontal force P acts on the rim of the homogeneous cylinder of radius R and weight W. Determine the smallest coefficient of static friction that enables the cylinder to start rolling up the 30◦ incline.


Homework Equations



*depends on the fbd. refer below

The Attempt at a Solution



since its being ask for the coefficient of friction, which FBD would be correct? should it be a vertical normal force with friction force going on left, or should it be the normal force facing perpendicularly the 30 degree include while the friction is along the 30 degree line also. which is the proper fbd? tnx
 
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Friction acts tangentially to the radius of a cylinder, and the force of friction is proportional to the normal force on a surface.
 
Thread 'Variable mass system : water sprayed into a moving container'

Thread 'Variable mass system : water sprayed into a moving container'

Starting with the mass considerations #m(t)# is mass of water #M_{c}# mass of container and #M(t)# mass of total system $$M(t) = M_{C} + m(t)$$ $$\Rightarrow \frac{dM(t)}{dt} = \frac{dm(t)}{dt}$$ $$P_i = Mv + u \, dm$$ $$P_f = (M + dm)(v + dv)$$ $$\Delta P = M \, dv + (v - u) \, dm$$ $$F = \frac{dP}{dt} = M \frac{dv}{dt} + (v - u) \frac{dm}{dt}$$ $$F = u \frac{dm}{dt} = \rho A u^2$$ from conservation of momentum , the cannon recoils with the same force which it applies. $$\quad \frac{dm}{dt}...
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