Force needed to keep block from moving on frictionless triangular block

[Strukus]

Homework Statement

This is straight from the book.

A small block of mass m rests on the sloping side of a triangular block of mass M which itself rests on a horizontal table as shown in Fig. 4-50. Assuming all surfaces are frictionless, determine the force F that must be applied to M so that m remains in a fixed position relative to M (that is, m doesn't move on the incline).

(I forgot to draw the surface the triangular block is on and sorry for the big picture!)

The answer is:

Spoiler

(m + M) $$\ast$$ g$$\ast$$ tan($$\phi$$)

Homework Equations

Force along the x-axis: m $$\ast$$ g $$\ast$$ sin($$\phi$$)
Force along the y-axis: m $$\ast$$ g $$\ast$$ cos($$\phi$$)

The Attempt at a Solution

I know that there must be an equal and opposite force along the incline but I have no clue how to approach this problem.

 

[tiny-tim]

Welcome to PF!

Hi Strukus! Welcome to PF!

Force along the x-axis: m $$\ast$$ g $$\ast$$ sin($$\phi$$)
Force along the y-axis: m $$\ast$$ g $$\ast$$ cos($$\phi$$)

No, that's the wrong approach …

only one of those equations is correct …

start again … you know the vertical acceleration of m is zero, so what is N (the normal reaction force ) …

and then what is the horizontal component of N?

 

[baba89]

I would approach this problem by first identifying the forces acting on the system and then using the principles of Newton's laws of motion to find the necessary force to keep the block from moving. The forces acting on the system are the weight of the block (m), the weight of the triangular block (M), and the applied force (F). Since the surfaces are frictionless, there is no friction force to consider.

Next, I would draw a free body diagram to visually represent the forces acting on the system. This would help me to clearly see the direction and magnitude of each force.

To keep the block (m) from moving, the net force on it must be zero. This means that the force F applied to the triangular block (M) must be equal in magnitude and opposite in direction to the weight of the block (m). Using the equations provided, we can calculate the weight of the block (m) along the x-axis and y-axis. Then, we can set the equation for the force along the y-axis equal to the weight of the block (m) along the y-axis and solve for F.

In summary, the force needed to keep the block from moving on the frictionless triangular block is equal to the weight of the block (m) along the y-axis, which can be calculated using the given equations. This approach follows the principles of Newton's laws of motion and allows for a systematic and scientific solution to the problem.