Find the constraint equation between the accelerations of...

Find the constraint equation between the accelerations of the blocks $m_{1}, m_{2}$ and $m_{3}$. Solution For string l: $x_{1}+c+x_{\mathbb{B}}=$ constant $\Rightarrow \frac{d^{2} x_{1}}{d t^{2}}|0| \frac{d^{2} x_{B}}{d t^{2}}-0 \quad \Rightarrow \quad a_{1} \mid a_{s}-0$ For string 2: $x_{2}-x_{s}+c_{1}+x_{3}-x_{B}=$ constant $\Rightarrow \quad x_{2}+x_{3}-2 x_{B}+c_{1}=$ constant $\Rightarrow \frac{d^{2} x_{2}}{d t^{2}}+\frac{d^{2} x_{3}}{d t^{2}}-\frac{2 d^{2} x_{B}}{d t^{2}}-0 \quad \Rightarrow \quad a_{2}+a_{3}-2 a_{n}-0 \quad \ldots$ From \Gammaeqs. (1) and (2), we get: $\Rightarrow a_{2}+a_{3}-2\left(-a_{1}\right)-0 \quad \Rightarrow 2 a_{1}+a_{2}+a_{3}-0$

Find the constraint equation between the accelerations of the blocks $m_{1}, m_{2}$ and $m_{3}$.
Solution
For string l: $x_{1}+c+x_{\mathbb{B}}=$ constant
$\Rightarrow \frac{d^{2} x_{1}}{d t^{2}}|0| \frac{d^{2} x_{B}}{d t^{2}}-0 \quad \Rightarrow \quad a_{1} \mid a_{s}-0$
For string 2: $x_{2}-x_{s}+c_{1}+x_{3}-x_{B}=$ constant
$\Rightarrow \quad x_{2}+x_{3}-2 x_{B}+c_{1}=$ constant
$\Rightarrow \frac{d^{2} x_{2}}{d t^{2}}+\frac{d^{2} x_{3}}{d t^{2}}-\frac{2 d^{2} x_{B}}{d t^{2}}-0 \quad \Rightarrow \quad a_{2}+a_{3}-2 a_{n}-0 \quad \ldots$
From \Gammaeqs. (1) and (2), we get:
$\Rightarrow a_{2}+a_{3}-2\left(-a_{1}\right)-0 \quad \Rightarrow 2 a_{1}+a_{2}+a_{3}-0$

Key Concepts

Inextensible String Constraints

In systems involving connected masses by a string, the string is typically assumed to be inextensible, meaning its total length does not change. This constraint leads to equations that express a constant sum of the displacements of various parts of the system. Such constraints are fundamental in relating the positions, velocities, and accelerations of the masses involved, ensuring that the motion of each mass is not independent of the others.

Differentiation of Constraint Equations

Because the string’s length is constant, one can differentiate the constraint equation with respect to time to establish relationships among the velocities (first derivative) of the masses, and differentiate a second time to relate their accelerations (second derivative). This process yields the necessary kinematic constraints that govern how the accelerations of one mass are related to those of the others.

Sign Conventions in Kinematic Relationships

When deriving the relationships between the motions of the connected masses, it is essential to choose appropriate sign conventions to indicate the direction of movement. Consistent use of positive and negative signs ensures that the constraint equations accurately represent the system's geometry and motion. Correctly applying these conventions is crucial for the validity of the acceleration relationships established from the differentiated constraint equations.

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