What Is the Minimum Acceleration for a Mass to Fall Freely on a Wedge?

[anigeo]

Homework Statement

a block of mass m is resting on a wedge of angle θ as shown in fig. the wedge is given an accn. a towards left.what is the minimum value of a due to an external agent for which the mass will fall freely?

Homework Equations

N+m(asinθ) = 0.(N is the normal force)

The Attempt at a Solution

we draw the fbd of the mass m. we get the given equation.but without the mass of the wedge i cannot proceed. still my work is given in the adjacent image.

 

[anathema]

Thank you for your post. Your attempt at a solution is a good start. To find the minimum value of a, we need to consider the forces acting on the block of mass m and the wedge separately.

For the block, there are two forces acting on it: its weight (mg) and the normal force (N) from the wedge. As you correctly identified, the normal force is equal to -mgsinθ. This is because the wedge is accelerating towards the left, causing the normal force to act towards the right (opposite to the acceleration).

Now, for the wedge, there are three forces to consider: its weight (mg), the normal force (N) from the ground, and the force of friction (F) between the wedge and the ground. The wedge is accelerating towards the left, so the normal force from the ground must be greater than the weight (mg) in order to provide the necessary force to accelerate the wedge. This means that N > mg.

To find the minimum value of a, we need to consider the forces acting on the block and wedge together. The block will start to fall freely when the normal force from the wedge (N) becomes zero. This means that N = 0, which implies that -mgsinθ = 0. Solving for a, we get a = gsinθ.

Therefore, the minimum value of a for which the mass will fall freely is gsinθ. I hope this helps. Let me know if you have any further questions.