# Blocks and Strings, Force Analysis

1. The problem statement, all variables and given/known data
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.

2. Relevant 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).

3. 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 $\frac{4mg}{17m+M}$