The problem involves two blocks connected by a string, where a block of mass $M = 12 \, \text{kg}$ is on an inclined plane, and a smaller block of mass $m = 4 \, \text{kg}$ is hanging vertically. We are tasked with determining the angle $\theta$ of the plane such that neither block moves.

Step-by-step Solution:

We will use Newton’s second law for this system and consider that both masses are in equilibrium, meaning that the forces acting on them balance out.

Forces on the big block $M$:

For the block of mass $M$, which is on the inclined plane, the forces acting on it are:

• The gravitational force component along the incline: $M g \sin \theta$
• The tension in the string: $T$

Since the block does not move, the tension force must balance the gravitational force component along the incline. Therefore, the equation for the block on the incline is: $T = M g \sin \theta \quad \text{(1)}$

Forces on the smaller block $m$:

For the smaller block of mass $m$, the forces acting on it are:

• The gravitational force: $m g$
• The tension in the string: $T$

Since the smaller block is also not moving, the tension force must balance the weight of the block. Therefore, the equation for the smaller block is: $T = m g \quad \text{(2)}$

Equating both tensions:

From equations (1) and (2), we can equate the tensions: $M g \sin \theta = m g$ Canceling $g$ (acceleration due to gravity) from both sides: $M \sin \theta = m$

Now, solving for $\sin \theta$: $\sin \theta = \frac{m}{M} = \frac{4.0}{12.0} = \frac{1}{3}$

Finally, the angle $\theta$ is: $\theta = \arcsin\left(\frac{1}{3}\right)$

Using a calculator: $\theta \approx 19.47^\circ$

Final Answer:

The angle of the inclined plane, $\theta$, is approximately $19.47^\circ$.

Do you want more details, or do you have any further questions?

Here are 5 follow-up questions to consider:

1. How would the system change if the inclined plane were not smooth (i.e., there was friction)?
2. What would happen to the angle if the mass $M$ increased?
3. How would you calculate the tension in the string?
4. How would the setup change if both masses were on two inclined planes?
5. What happens if the masses were swapped, making $m = 12 \, \text{kg}$ and $M = 4 \, \text{kg}$?

Tip: When solving equilibrium problems, always identify all forces acting on each object and apply Newton's second law to balance them out.