Assuming pulleys and strings are light, find constraint...

Assuming pulleys and strings are light, find constraint relation between the accelerations of the bar $m_{1}$ and the block $m_{2}$ as in the given figure. Solution For 1, $x_{1}+x_{E}=$ constant $\frac{d^{2} x_{1}}{d t^{2}}+\frac{d^{2} x_{n}}{d t^{2}}-0 \quad \Rightarrow a_{1}+a_{B}-0$ For 2 , $\left(x_{1}-x_{n}\right)+\left(x_{c}-x_{n}\right)-$ constant $\Rightarrow x_{1}-2 x_{A}+x_{C}-$ constant $\Rightarrow \frac{d^{2} x_{1}}{d t^{2}}-2 \frac{d^{2} x_{B}}{d t^{2}}+\frac{d^{2} x_{C}}{d t^{2}}-0 \quad \Rightarrow \quad a_{1}-2 a_{B}+a_{C}-0$ For 3 , $\left(x_{1}-x_{c}\right)+\left(x_{2}-x_{C}\right)=$ constant (3) $\Rightarrow \quad x_{1}+x_{2}-2 x_{C}=$ constant $\Rightarrow a_{1}+a_{2}-2 a_{c}=0$ From Eqs. (1) and (2) $a_{1}-2\left(-a_{1}\right)+a_{c}=0$ $\Rightarrow a_{1}+2 a_{1}+a_{C}=0 \quad \Rightarrow \quad 3 a_{1}+a_{c}=0$ $\therefore \quad a_{\ell}=-3 a_{1}$ From ?q. (3) $\Rightarrow a_{1}+a_{2}-2\left(-3 a_{1}\right)=0$ $\Rightarrow a_{1}+a_{2}+6 a_{1}=0 \quad \Rightarrow \quad 7 a_{1}+a_{2}=0$

Assuming pulleys and strings are light, find constraint relation between the accelerations of the bar $m_{1}$ and the block $m_{2}$ as in the given figure.
Solution
For 1, $x_{1}+x_{E}=$ constant
$\frac{d^{2} x_{1}}{d t^{2}}+\frac{d^{2} x_{n}}{d t^{2}}-0 \quad \Rightarrow a_{1}+a_{B}-0$
For 2 ,
$\left(x_{1}-x_{n}\right)+\left(x_{c}-x_{n}\right)-$ constant
$\Rightarrow x_{1}-2 x_{A}+x_{C}-$ constant
$\Rightarrow \frac{d^{2} x_{1}}{d t^{2}}-2 \frac{d^{2} x_{B}}{d t^{2}}+\frac{d^{2} x_{C}}{d t^{2}}-0 \quad \Rightarrow \quad a_{1}-2 a_{B}+a_{C}-0$
For 3 , $\left(x_{1}-x_{c}\right)+\left(x_{2}-x_{C}\right)=$ constant
(3) $\Rightarrow \quad x_{1}+x_{2}-2 x_{C}=$ constant
$\Rightarrow a_{1}+a_{2}-2 a_{c}=0$
From Eqs. (1) and (2) $a_{1}-2\left(-a_{1}\right)+a_{c}=0$
$\Rightarrow a_{1}+2 a_{1}+a_{C}=0 \quad \Rightarrow \quad 3 a_{1}+a_{c}=0$
$\therefore \quad a_{\ell}=-3 a_{1}$
From ?q. (3) $\Rightarrow a_{1}+a_{2}-2\left(-3 a_{1}\right)=0$
$\Rightarrow a_{1}+a_{2}+6 a_{1}=0 \quad \Rightarrow \quad 7 a_{1}+a_{2}=0$

Key Concepts

Kinematic Differentiation

By differentiating the position constraint equations with respect to time, one can derive velocity relationships, and by differentiating once more, acceleration relationships are obtained. This process allows for establishing a connection between how fast and how the bodies accelerate relative to each other based on the unchanging string length.

Constraint Relations

In mechanical systems with pulleys and strings, constraint relations establish the fixed geometric relationships between different parts of the system. They arise from the fact that the total length of the connecting string remains constant, linking the subsequent movements (positions, velocities, and accelerations) of the connected bodies.

Inextensible Strings

Assuming the strings are inextensible means that they do not stretch. This assumption is fundamental as it ensures that the sum of the lengths of all segments of the string is constant over time, leading to strict relationships between the motion parameters of various connected masses.

Acceleration Constraints in Pulley Systems

In pulley systems, the acceleration of one mass is directly related to the accelerations of other masses through the derived constraint equations. This relationship is crucial for dynamically analyzing the system, as it ensures that the motion of interconnected components is consistently coupled according to the geometry and inextensibility of the connecting string.

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