Question

The system shown adjacent is in equilibrium. Find the acceleration of the blocks A, B & C all of equal masses m at the instant when:

(A) The spring between ceiling & A is cut.(Also find the tensions in the strings when the system is at rest)

• A. $a_a=\frac{3g\downarrow }{2}=a_B;a_c=0;T=mg/2$
• B. $a_a=\frac{g\downarrow }{2}=a_B;a_c=g;T=mg/2$
• C. $a_a=\frac{g\uparrow }{2}=a_B;a_c=0;T=3mg/2$
• D. $a_a=\frac{g\uparrow }{2}=a_B;a_c=g;T=3mg/2$

Answer 1

The correct option is A $a_a=\frac{3g\downarrow }{2}=a_B;a_c=0;T=mg/2$

A) When the spring is cut the blocks A and B will have a pull equal to $k\times \Delta x$ where $\Delta x$ is extension in spring between B and C .
From force equilibrium ,before the spring is cut:
$k\Delta x=m_{C}g\Rightarrow \Delta x=\frac{mg}{k}$
The masses A and B will move with same acceleration as the string is taught between them
$a_{net}=\frac{totalforce}{totalmass}=\frac{mg+mg+k\times \frac{mg}{k}}{2m}=\frac{3g}{2}$
writing Equation of motion for block B
$mg+T=m{a}_{net}\Rightarrow T=3mg/2-mg=mg/2$
acceleration of C will be
$a_C=\frac{k\Delta x-mg}{m}=\frac{k\frac{mg}{k}-mg}{m}=0$