Accelerating an inclined plane

[syang9]

hello,

consider an plane inclined at an angle theta with a block on it. the problem asks me to find a horizontal acceleration such that the block will not slide down the plane.
in part (a), friction is neglected, in part(b), friction is considered.

here is what i have done. i fixed the coordinate system so that the force of the push applied to the inclined plane is in the x-direction.

(a): Fnetx = Fnx (normal force in x direction) + Fp (force of push) = 0
Fnety = Fny + Fg = 0

solving for Fp, i get Fp = Fn(cos(theta)). this seems reasonable, but too easy. have i left something out?

(b): considering friction, i have fixed the coordinate system as in part a.

Fnetx = Fp + Fnx -Ffx (force of friction in x direction) = 0
Fnety = Fg + Ffy +Fny = 0

solving for Fp, i get Fp = -Fn(sin(theta))*(1-μs) (μs = coefficient of static friction) however, there should be a minimum and a maximum force.. the minimum force is with maximum friction, and the maximum force is the force that force that is on the verge of accelerating the block in the negative x direction. or am i wrong? have i left something out? i am pretty sure i have.

any help is greatly appreciated. (i have attached a picture of the problem)

 

[Doc Al]

here is what i have done. i fixed the coordinate system so that the force of the push applied to the inclined plane is in the x-direction.

(a): Fnetx = Fnx (normal force in x direction) + Fp (force of push) = 0
Fnety = Fny + Fg = 0

solving for Fp, i get Fp = Fn(cos(theta)). this seems reasonable, but too easy. have i left something out?

I assume you are analyzing the forces on the block. If so, two comments:

(1) The only forces acting on the block are weight and the normal force. What do you mean by Fp? (Note that the question asks for the acceleration, not the force on the plane that would produce such acceleration. Although once you have "a", it's easy to find "F".)

(2) Why do you assume equilibrium in the horizontal direction? After all, the question asks for the acceleration.​

Hint: If the block doesn't slide down the plane it must be in vertical equilibrium.

 

[kyle8921]

I would like to offer some insights and suggestions regarding your approach to this problem.

Firstly, your approach to solving the problem without friction seems reasonable. However, it is important to note that in real-world scenarios, friction is almost always present and cannot be neglected. Therefore, it is important to consider the effects of friction in your calculations.

In part (b), you have correctly considered the effects of friction. However, it is important to note that the coefficient of static friction, μs, is not a constant value and can vary depending on the materials involved. Additionally, the minimum and maximum forces you mentioned are not necessarily related to the coefficient of friction. The minimum force required to prevent the block from sliding down the plane will depend on the angle of inclination and the weight of the block, while the maximum force that can be applied before the block starts sliding will depend on the coefficient of friction.

To determine the minimum and maximum forces, you can use the equations for equilibrium in the x and y directions, as you have done in part (b). However, instead of solving for Fp, you can solve for the maximum and minimum values of Fp that satisfy the equations. This will give you a range of forces that can be applied without causing the block to slide.

Additionally, I would recommend using a free-body diagram to visualize the forces acting on the block and the inclined plane. This can help you better understand the problem and can also serve as a useful tool for double-checking your calculations.

I hope this helps and good luck with your problem-solving! Remember, as a scientist, it is important to always consider all factors and to continuously refine your approach as new information is gathered.