Application of Newton's Laws of motion

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To determine the minimum magnitude of force P required to hold a 20kg block against a vertical surface at a 30-degree angle, the frictional force must be considered, which depends on P. The correct approach involves balancing the forces, including gravitational force and the normal force created by P. The equation mg + Fr = mg cos(theta) is incorrect as it omits P's role in the normal force calculation. The frictional force (Fr) can be calculated using the coefficient of friction and the normal force. The correct minimum value for P to keep the block stationary is 202.2 N.
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A block of mass 20kg is pushed against a vertical surface at an angle of 30 degrees as shown. The coefficient of friction between the surface and the block is 0.2. What is the minimum magitude of P to hold the block still?


I got 271.5 when I tried but the answer is 202.2
 

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How did you come to the 271.5?
 
Have you taken into account the fact that the frictional force will depend also on P, ie how strongly it is being pressed against the wall?
 
I came up withthe equation

mg+ Fr = mg cos theta
 
That equation does not look right. Where is P?

Also how do you determine Fr. You need another equation right?
 
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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