Blocks and Strings, Force Analysis

[phoenixXL]

Homework Statement

A block of mass M is connected with a particle of mass m by a light in-extensible string as shown in the figure. Assuming all the contacts as smooth, find the acceleration of the block after releasing the system.

Homework Equations

F = Mass * Acceleration
Normal Constraint : The acceleration of two particles in contact is same along the line perpendicular to their line of contact.
String Constraint : The length of the string is constant and could be used to find out the acceleration of the corresponding components using differentiation(using geometry).

The Attempt at a Solution

Using the FBD and Translational equations

The particle will have the acceleration both in x and y direction. The block will have the acceleration in x direction due to tension in the string and contact force .

for block M
$$\ T\ -\ N\ =\ Ma_x$$

The particle's acceleration in the x direction will be the same as that of the block in the x direction in accordance to the normal constraint.

for particle m
$$mg\ -\ T\ =\ ma_y\\ \ N\ =\ ma_x\\ \implies\ T\ =\ (m+M)a_x$$

Using string constraint we get than the particle's acceleration in the y-direction would be 4times the acceleration of the particle in x-direction
$$4a_x\ =\ a_y\\ \implies\ mg\ -\ T\ =\ 4ma_x\\ \implies\ mg\ -\ (m+M)a_x\ =\ 4ma_x\\ \implies\ a_x\ =\ \frac{mg}{5m+M}$$

But the answer I get is wrong and the correct answer is $\frac{4mg}{17m+M}$

Please help me out.
Thanks for your time

 

[haruspex]

$$\ T\ -\ N\ =\ Ma_x$$

Look at all those horizontal sections under tension. What are they pulling on?

 

[phoenixXL]

Right that won't be T but 4T.
ie $$4T\ -\ N\ =\ Ma_x$$

 

[phoenixXL]

Thanks for your input.
I get the correct answer

 

[HalfLostAndSearching]

, but I am not sure what you are asking for. Can you provide more context or information about the problem? I am happy to help, but I need more information in order to understand the issue you are having.