Why is the acceleration of wedge and block equal like this

Question

No friction at all anywhere. Let me explain the diagram in steps and the Q.

1. green colour let us start with. It represent a wedge on which there is mass of 10kg (red colour). That’s it the first picture.

2. dotted line in white represent when the block touches the bottom of the wedge.

3. represent the dotted yellow line which represents the direction of acceleration for 10 kg mass. Which is kind of like unknown or not in a symmetry.

4. Where the Q lies

The Dark yellow line represent displacements. Below one is AC I.e displacement of the wedge. Now , only thing in this Q what I don’t understand is that why is Displacement of mass 10kg BC even though the direction of acc for 10 kg is the dotted line.

If you wish to know why the dotted line , there is an example in my textbook which tells us that. Also , tells us that because of the dotted line kind of acceleration, we use pseudo force for solving such questions.

EDIT: Where a different concept and answer is also used.

Answer 1

The figure below shall answer your question.

Moving from point B to C, the sliding force $F_S = mgsin\theta = ma$,

since no friction, $a = gsin\theta$,

and $\theta = Tan^{-1}(L_{BC}/L_{AC})$

Answer 2

In order to find the acceleration the process is similar to this question.

For the inertial observer the angle of the acceleration ($\phi$) with the horizontal axis will be steeper than the angle of the inclined plane ($\theta$) (See red line in the kinetic diagram).

As you can see on the inertial Frame you have:

• $a_A$ the acceleration of A with respect to the inertial frame. (the direction is not known)
• $a_W$ is the acceleration of the wedge with respect to the inertial frame. Since the wedge will be accelerating then the acceleration vector is pointing to the right.

The remaining vector $a_{A|W}$ is the acceleration vector with respect to the frame of reference of the Wedge. For a person standing on the Wedge as it is moving, then the acceleration vector $a_{A|W}$ will be parallel to the inclined plane (it will form an angle $\theta$ with the horizontal plane).

So you have two vectors $a_W$ and $a_{A|W}$ , that you know their directions. Therefore because of the equation:

$$\vec{a}_A = \vec{a}_W + \vec{a}_{A|W}$$

you can add up the accelerations and obtain $a_A$. (apology for the similarities with the other question but the process is exactly the same)

The main difference know is that the motion of the wedge is dependent on the mass A, and on the angle $\theta$. So the diagrams for the wedge are

The resulting equations are the following $$\begin{cases} - N_{1x} = -m a_{Ax}\\ - mg + N_{1y} = -m a_{Ay}\\ N_{1x} = m a_{W}\\ - N_{1y} + N_2 - mg = 0\\ a_{Ax} = a_{Wx} + a_{(A|W)x} a_{Ay} = 0 + a_{(A|W)y} \end{cases}$$

which you can solve for the equations.