How Does Increasing Inclination Affect Particle Acceleration?

AI Thread Summary
Increasing the inclination of a plane affects the acceleration of a particle resting in limiting equilibrium. At 30 degrees, the particle is on the verge of moving, with friction at its maximum value, which can be expressed as F = μR. To find the acceleration at 40 degrees, one must resolve the gravitational force into components parallel and normal to the incline, using the coefficient of friction obtained from the 30-degree scenario. As the angle increases, the parallel component of weight increases while the normal force decreases, impacting the net force and thus the acceleration. Understanding these forces and their relationships is crucial for solving the problem effectively.
CathyLou
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Hi.

Could someone please help me with the following M1 level question?

A particle rests in limiting equilibrium (that is the particle is on the point of moving and friction has its maximum value) on a plane inclined at 30 degrees to the horizontal. Determine the acceleration with which the particle will slide down the plane when the angle of inclination is increased to 40 degrees.

I know that limiting equilibrium means that F = miuR.

Any help would be really appreciated as I am completely stuck over what to do.

Thank you.

Cathy
 
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The point is that that mass is on an incline.

Resolve the weight (mg) into components that are parallel and normal to the incline. Use the information at 30° to find \mu. Remember friction is proportional to the normal force and acts opposed the direction of motion (in this case friction points up the incline).

Then with \mu, determine the net force when the incline is tilted to 40°. The component of weight parallel to the incline increases, while the normal force decreases.
 
Last edited:
I suspect that they want you to assume that the coefficients of static and kinetic friction are the same for this problem. The first step is to solve for that coefficient. Analyze the forces acting on the particle and apply the condition for equilibrium.
 
Thanks so much for your help!

Cathy
 
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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