Question

Two block's A and B of masses m and 2m respectively are held at rest such that the spring is in natural length. Find out the accelerations of blocks A and B respectively just after release (pulley string and spring are massless).

• A. $\frac{g}{3}\downarrow, \frac{g}{3}\uparrow$
• B. $g\downarrow, g\downarrow$
• C. $0,0$
• D. $g\downarrow, 0$

Answer 1

$\frac{g}{3}\uparrow, \frac{g}{3}\downarrow$

Answer 2

Initially, the spring is at its natural length, so the spring force is zero. Let's analyze the forces on block A (mass m) and block B (mass 2m). We'll consider upward direction as positive.

For block B: The forces acting on B are gravity (downwards, $2mg$) and tension from the string (upwards, $T$). The equation of motion for block B is: $T - 2mg = 2ma_B$ (1)

For block A: The forces acting on A are gravity (downwards, $mg$), and tension from the string (upwards, $T$). The spring force is initially zero. The equation of motion for block A is: $T - mg = ma_A$ (2)

Constraint relation: Since the string is inextensible and passes over a massless pulley, if block A moves up by a distance $x$, block B must move down by the same distance $x$. Therefore, their accelerations are related by $a_A = -a_B$.

Substitute $a_B = -a_A$ into equation (1): $T - 2mg = 2m(-a_A)$ $T - 2mg = -2ma_A$ $T = 2mg - 2ma_A$

Now substitute this expression for $T$ into equation (2): $(2mg - 2ma_A) - mg = ma_A$ $mg - 2ma_A = ma_A$ $mg = 3ma_A$ $a_A = \frac{g}{3}$

Since $a_A = \frac{g}{3}$, block A accelerates upwards with an acceleration of $\frac{g}{3}$. Using the constraint $a_B = -a_A$: $a_B = -\frac{g}{3}$

This means block B accelerates downwards with an acceleration of $\frac{g}{3}$.

Therefore, the accelerations of blocks A and B are $\frac{g}{3}$ upwards and $\frac{g}{3}$ downwards, respectively. This corresponds to $\frac{g}{3}\uparrow, \frac{g}{3}\downarrow$. None of the provided options exactly match this result. Option (2) is $\frac{g}{3}\downarrow, \frac{g}{3}\uparrow$, which has the directions reversed for both blocks. Assuming a typo in the options, the intended answer is likely derived from our calculation.