The string between blocks of mass m and 2m is massless and inextensible. The system is suspended by a massless spring as shown. If the string is cut find the magnitudes of accelerations of mass 2m and m (immediately after cutting)

Step 1: Analyze the system in initial equilibrium.

Before the string is cut, the system is at rest. Let $T$ be the tension in the string and $T_s$ be the tension in the spring.

For the lower block of mass $m$:

$$T = mg$$

For the upper block of mass $2m$:

$$T_s = 2mg + T$$

Substitute $T = mg$ into the equation for $T_s$:

$$T_s = 2mg + mg = 3mg$$

Step 2: Determine the forces immediately after the string is cut.

Immediately after the string is cut, the string tension instantly drops to zero.

$$T = 0$$

Since the spring cannot change its length instantaneously, the spring force remains unchanged.

$$T_s = 3mg$$

Step 3: Calculate the acceleration of the block of mass $2m$.

Applying Newton's second law to the block of mass $2m$ in the upward direction:

$$T_s - 2mg = 2m a_{2m}$$

Substitute $T_s = 3mg$:

$$3mg - 2mg = 2m a_{2m}$$ $$mg = 2m a_{2m}$$ $$a_{2m} = \frac{g}{2}$$

Step 4: Calculate the acceleration of the block of mass $m$.

Applying Newton's second law to the block of mass $m$ in the downward direction:

$$mg = m a_m$$ $$a_m = g$$

Answer: The magnitudes of the accelerations of mass $2m$ and $m$ immediately after cutting are $\dfrac{g}{2}$ and $g$, respectively.