To solve this problem, we will consider the forces acting on the block and use Newton's second law of motion.

Step 1: Equilibrium Condition

Initially, the block is in equilibrium, meaning that the net force acting on it is zero. Let:

• $T_u$ be the tension in the upper spring.
• $T_l$ be the tension in the lower spring.
• $W = mg$ be the weight of the block, where $g = 10 \, \text{m/s}^2$ is the acceleration due to gravity.

Since the block is in equilibrium, the sum of forces is zero, meaning: $T_u + T_l = W = mg$

Step 2: After Cutting the Upper Spring

When the upper spring is cut, the block accelerates downward with an acceleration of $6 \, \text{m/s}^2$. This acceleration is caused by the net force acting on the block, which is now just its weight minus the tension from the lower spring, because the upper spring no longer provides any force.

Using Newton's second law, the net force acting on the block is: $F_{\text{net}} = mg - T_l$ Thus, the equation for acceleration is: $ma = mg - T_l$ Substitute $a = 6 \, \text{m/s}^2$ and solve for $T_l$: $m(6) = m(10) - T_l$ $6m = 10m - T_l$ $T_l = 4m$

Step 3: Cutting the Lower Spring

Now, if the lower spring is cut instead of the upper one, the block will accelerate upward, as the only force acting on it initially will be the tension from the upper spring, $T_u$.

Using Newton's second law again, the net force is: $F_{\text{net}} = T_u - mg$ Thus, the equation for acceleration becomes: $ma = T_u - mg$ We already know that $T_u + T_l = mg$ and that $T_l = 4m$, so: $T_u = mg - T_l = mg - 4m = 6m$ Substitute $T_u = 6m$ into the equation: $ma = 6m - mg$ $a = 6 - 10 = -4 \, \text{m/s}^2$ Thus, the acceleration of the block just after the lower spring is cut is $4 \, \text{m/s}^2$ upwards.

Final Answer:

The acceleration of the block just after the lower spring is cut is $4 \, \text{m/s}^2$ upwards.

Would you like more details on any part of the process, or have any questions?

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Here are five follow-up questions to deepen your understanding:

1. How do the spring tensions relate to the weight of the block in equilibrium?
2. What would happen if both springs were cut at the same time?
3. How would the problem change if the block were not in equilibrium initially?
4. What is the significance of the net force acting on the block after cutting the springs?
5. How would this problem differ if the springs had different spring constants?

Tip: Always check the direction of the net force when solving for acceleration, as it helps determine whether the acceleration is upwards or downwards.