In the figure, ends P and Q of an inextensible string move...

In the figure, ends $P$ and $Q$ of an inextensible string move downwards with uniform speed $u$. Pullcys are fixed and massless. What is the speed $v$ of the mass $m$ in upward direction? Solution $x_{2}\left|c_{1}\right| 2 \sqrt{y^{2} \mid d^{2}}\left|c_{2}\right| x_{1}-$ constant $\Rightarrow \frac{d x_{2}}{d t}+0+2 \frac{1}{2 \sqrt{y^{2}+d^{2}}} 2 y \frac{d y}{d t}+0+\frac{d x_{1}}{d t}-0$ $\Rightarrow \frac{d x_{2}}{d t} \mid \frac{d x_{1}}{d t} ? 2\left(\frac{y}{\sqrt{y^{2} \mid d^{2}}}\right) \frac{d y}{d t}-0 \Rightarrow u \backslash u \backslash 2(\cos \theta) v-0$ $\Rightarrow 2 u+(2 \cos \theta) v-0 \quad \Rightarrow u+v \cos \theta-0$ $v-\left|\frac{u}{\cos \theta}\right|$ $\left\lfloor\Lambda s \frac{d x_{1}}{d t}-v_{\mathrm{r}}, \frac{d x_{2}}{d t}-v_{Q}, \frac{d y}{d t}-\quad v_{m}\right\rfloor \quad \therefore v-\frac{u}{\cos \theta}$

In the figure, ends $P$ and $Q$ of an inextensible string move downwards with uniform speed $u$. Pullcys are fixed and massless. What is the speed $v$ of the mass $m$ in upward direction?
Solution
$x_{2}\left|c_{1}\right| 2 \sqrt{y^{2} \mid d^{2}}\left|c_{2}\right| x_{1}-$ constant
$\Rightarrow \frac{d x_{2}}{d t}+0+2 \frac{1}{2 \sqrt{y^{2}+d^{2}}} 2 y \frac{d y}{d t}+0+\frac{d x_{1}}{d t}-0$
$\Rightarrow \frac{d x_{2}}{d t} \mid \frac{d x_{1}}{d t} ? 2\left(\frac{y}{\sqrt{y^{2} \mid d^{2}}}\right) \frac{d y}{d t}-0 \Rightarrow u \backslash u \backslash 2(\cos \theta) v-0$
$\Rightarrow 2 u+(2 \cos \theta) v-0 \quad \Rightarrow u+v \cos \theta-0$
$v-\left|\frac{u}{\cos \theta}\right|$
$\left\lfloor\Lambda s \frac{d x_{1}}{d t}-v_{\mathrm{r}}, \frac{d x_{2}}{d t}-v_{Q}, \frac{d y}{d t}-\quad v_{m}\right\rfloor \quad \therefore v-\frac{u}{\cos \theta}$

Key Concepts

Differentiation of the Constraint Equation

Once the constant-length constraint is established, differentiating this equation with respect to time transforms it into a relationship among the velocities of different parts of the system. This approach allows one to solve for the unknown velocities by incorporating the contributions from different segments of the string and any angular factors present in the geometry.

Geometric Relations in Pulley Systems

In many pulley problems, the geometry of the setup, particularly the angles between different segments of the string, plays an important role. The motion along the string often has components in different directions, and trigonometric functions (like cosine) are used to relate these components. Such geometric relations help in connecting the vertical and horizontal motions, or motions along the string, to the speed of the mass or endpoints.

Inextensible String

An inextensible string is one whose total length remains constant over time. This condition imposes a relationship among the movements of the different object points attached to the string. When segments of the string change their lengths due to motion of attached masses or endpoints, the changes must balance out so that the overall length does not vary, leading to constraint relations among the velocities.

Constraint Equation

For systems that involve interconnected masses and pulleys, the constant total length of the string can be expressed as an equation involving the individual lengths of its segments. This constraint equation is critical, as it can be differentiated with respect to time to create relationships between the velocities of various parts of the system.

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