Circular Motion Problems: Maximum Angular Speed and Slip Calculation

AI Thread Summary
The discussion focuses on calculating the maximum angular speed of a block on a rotating ruler without slipping, considering friction and centripetal forces. For part (b), the block experiences both centripetal and tangential forces, which must be combined using Pythagorean theorem to determine the total force acting on the block. As the angular speed increases, the static friction can only provide a limited force before reaching its maximum value, which ultimately determines when the block will begin to slip. Understanding the balance of these forces is crucial for solving the problem. The key takeaway is that the static friction's limiting value is critical in calculating the angular speed at which slipping occurs.
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A block of mass m is kept on a horizontal ruler.Friction coefficient is k.The ruler is kept at one end and the block is at a distance L from the fixed end.The ruler is roated about the fixed end in the horizontal plane thru the fixed end.(a)what can the maximum angular speed be for which the block does not slip.(b) If the angular speed of the ruler is uniformly increased from zero at an angular acceleration a at what angular speed will the block slip..

I've already figured out part (a) please give me some hints as how to work out part (b)
 
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You have two forces on the block now. One is the centripetal force which you already worked out. The other is at right angles, accelerating the block (increasing its angular speed). You have to add these two vectors to get the total force.

As the forces are at right angles, you can use Pythagoras to add them.

Total force = sqrt((centripetal force)^2 + (tangential force)^2)
 
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As the angular speed increases the frictional force will also increase to provide the centipetal force, but the static friction can only increase up to a certain limiting value! Which will limit the maximum angular speed before it starts to slip.
 
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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