Let us draw free body diagram of each body and fix the...

Let us draw free body diagram of each body and fix the coordinate system, as shown in the figure. After analysing the motion of $M$ and $m$ on the basis of force diagrams, let us draw the kinematical diagram for accelerations (Fig.). As the length of threads are constant so, $d s_{m M}=d s_{M}$ and as $\vec{v}_{m M}$ and $\vec{v}_{M}$ do not change their directions that why $\mathbf{4 5}$ As $\vec{w}_{m}=\vec{w}_{m M}+\vec{w}_{M}$ so, from the triangle law of vector addition $w_{m}=\sqrt{2} w$ From the Eq. $F_{x}=m w_{x}$, for the wedge and block : $T-N=M w$ and $\quad N=m w$ Now, from the Eq. $F_{y}=m w_{y}$, for the block $m g-T-k N=m w$ Simultaneous solution of Eqs. (2), (3) and (4) yields : $m$$$ w=\frac{m g}{(k m+2 m+M)}=\frac{g}{(k+2+M / m)} $$ Hence using Eq. (1) $$ w_{m}=\frac{g \sqrt{2}}{(2+k+M / m)} $$

Let us draw free body diagram of each body and fix the coordinate system, as shown in the figure. After analysing the motion of $M$ and $m$ on the basis of force diagrams, let us draw the kinematical diagram for accelerations (Fig.). As the length of threads are constant so, $d s_{m M}=d s_{M}$ and as $\vec{v}_{m M}$ and $\vec{v}_{M}$ do not change their directions that why
$\mathbf{4 5}$
As $\vec{w}_{m}=\vec{w}_{m M}+\vec{w}_{M}$
so, from the triangle law of vector addition $w_{m}=\sqrt{2} w$
From the Eq. $F_{x}=m w_{x}$, for the wedge and block :
$T-N=M w$
and $\quad N=m w$
Now, from the Eq. $F_{y}=m w_{y}$, for the block $m g-T-k N=m w$
Simultaneous solution of Eqs. (2), (3) and (4) yields :
$m$$$
w=\frac{m g}{(k m+2 m+M)}=\frac{g}{(k+2+M / m)}
$$
Hence using Eq. (1)
$$
w_{m}=\frac{g \sqrt{2}}{(2+k+M / m)}
$$

Key Concepts

Vector Addition in Dynamics

Vector addition is the process of combining multiple vector quantities, such as forces or accelerations, to find a resultant vector. This concept is critical when dealing with components of motion or forces in different directions, allowing for a comprehensive analysis of the system’s behavior using the triangle law of vector addition.

Newton's Second Law

Newton's second law, formulated as F = ma, is the principle that relates the net force acting on an object to its acceleration, given its mass. It is applied separately to the different bodies and directions in the system to derive equations that describe the dynamics of the system.

Constraint Relationships

Constraint relationships arise in systems where the motion of connected bodies is interdependent due to conditions like constant thread length. These conditions create mathematical relationships between the displacements, velocities, and accelerations of the bodies involved, effectively reducing the number of independent variables in the problem.

Free-Body Diagram

A free-body diagram (FBD) is a simplified illustration that isolates an object to display all external forces acting upon it. This tool is crucial in mechanics for understanding how forces interact with a given body, forming the foundation for applying Newton's laws to analyze motion and equilibrium.

Kinematic Diagrams of Accelerations

Kinematic diagrams visually represent motion quantities such as displacements, velocities, and accelerations as vectors. These diagrams are used to determine the relationships between the accelerations of interconnected bodies, helping to establish how constraint conditions and vector directions influence the system's overall dynamics.

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