Force needed to push an incline with a mass on it

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Homework Help Overview

The problem involves an incline with a mass on it, where the goal is to determine the force required to push the entire system while keeping the mass at rest relative to the incline. The scenario is described as frictionless, and the masses involved are denoted as M and m, with the incline at an angle theta.

Discussion Character

  • Exploratory, Mathematical reasoning, Problem interpretation

Approaches and Questions Raised

  • The original poster attempts to understand the derivation of the force equation F=(m+M)gtan(theta) and expresses confusion after initial guidance from their teacher. Some participants suggest drawing free body diagrams (FBD) for both masses and breaking forces into components to analyze the system. There are discussions about the forces acting on the mass m and the incline M, including gravitational forces and normal forces.

Discussion Status

Participants are actively engaging with the problem, with one offering detailed steps on how to approach the force analysis through FBDs. There is a recognition of the need to equate forces in different directions and the relationship between the accelerations of the two masses. However, there is no explicit consensus on the final solution or method, and the discussion remains open-ended.

Contextual Notes

The problem is framed within a homework context, and the original poster expresses a desire for clarification without providing a complete solution. The discussion includes assumptions about the system being frictionless and the relationship between the forces acting on the masses.

nagaromo
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Homework Statement


Incline with the mass of M and angle of theta. mass on the incline with the mass of m. Find the force needed to push the entire system such that the mass remains at rest relative to the incline. Everything is frictionless

Homework Equations


F=ma

The Attempt at a Solution


well my teacher has gone over the problem where a=F/(M+m) but after that, I was completely lost and we arrived with the answer of F=(m+M)gtan[tex]\theta[/tex]

I'll love you if you could please explain to me how to arrive to that answer! <33 THANK YOUU
 

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1. draw an FBD for m on a tilted axis x prime and y prime.
2. convert that FBD to a normal axis by splitting the force arrows into components.
3. draw an FBD for M.

1. first, you need to draw all the force arrows. You got part of the forces right in your diagram. There is Fgsin(theta) and Fg(costheta). Don't forget, there is also a normal force acting on the block perpendicular to the incline. Also, The reason why the block does not slide down the ramp is because M exerts a force F(Mm) on it. Therefore, it points in the opposite direction of Fgsin(theta).

2. Next, you need to break up all these slanted force arrows into their x and y normal axis directions. after this, you will notice that the net force in the positive y direction is F(Mm)sin(theta) - Fncos(theta). This is equal to the net force in the negative y direction, Fg, as the block does not accelerate in the y direction. The net force in the positive x direction is Fgsin(theta)cos(theta). in the opposite direction you have F(Mm)cos(theta). Fgsin(theta)cos(theta) is bigger because the block does accelerate in the x direction.

3. now you draw an FBD for M. The net force in the positive x direction is Fa, the applied force. there is a force that m exerts on M, which is slanted toward the bottom left corner. When you break up this force into the x and y perpendicular forces, you get the following: a force pointing in the negative x direction F(mM)cos(theta); a force pointing in the negative y direction F(mM)sin(theta). On top of this, you have a normal force and a force of gravity acting in opposite directions. They are not equal! Fn = Fg + F(mM)sin(theta).

the acceleration of M is equal to the acceleration of m.

am = Fnet m
= Fgsin(theta)cos(theta) - F(Mm)cos(theta)

aM = Fnet m
= Fa - F(mM)cos(theta)

F(mM) = F(Mm) (Newton's third law)

then you use the equations to solve for a.

btw, i used components and not vectors. hope this helps.
 
oops, aM= Fnet M
 
thank you! :D
 

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