Day: December 18, 2023

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202312181044 Exercise 21.2.27

The figure below shows two masses connected by a light inextensible string on a smooth incline where \sin\theta =\frac{1}{40}. Find the acceleration of the masses when the system is released from rest. (g=9.8)

Extracted from A. Godman & J. F. Talbert. (1973). Additional Mathematics Pure and Applied in SI Units.

Roughwork.

As always, begin with free-body diagrams. Hence, draw

\begin{aligned} m_Aa_A & =\textrm{Net }F_{A}=-T+W_{A}\sin\theta \\ m_Ba_B & =\textrm{Net }F_{B}=T+W_{B}\sin\theta \\ \end{aligned}

where

\begin{aligned} W_A & = m_Ag \\ W_B & = m_Bg \\ T & : \left\{\begin{array}{lr} =0 \textrm{ for }a_A\leqslant a_B & \text{} \\ >0 \textrm{ for }a_A>a_B & \text{}\\ \end{array} \\ \end{aligned}

Note that

\begin{aligned} a_A & = -\frac{T}{m_A}+g\sin\theta \\ a_B & = \frac{T}{m_B}+g\sin\theta \\ \cdots\cdots & \cdots\cdots\cdots \\ \because\quad a_A & \leqslant a_B \\ \therefore\quad T & = 0 \\ \end{aligned}

Thus,

a_A,a_B = g\sin\theta.

This problem is not to be attempted.