In the given diagram, we have two masses, $m_1 = 8 \, \text{kg}$ and $m_2 = 2 \, m_1 = 16 \, \text{kg}$. There is also a force $F = 50 \, \text{N}$ acting at an angle $\alpha = 60^\circ$ on $m_1$, and it appears to be connected to $m_2$ through a pulley system.

Steps to find $N$, the normal force on $m_1$:

1. Forces acting on $m_1$:

   • Weight: $W_1 = m_1 g = 8 \times 9.8 = 78.4 \, \text{N}$ downward.
   • Normal force: $N$, which is the force exerted by the surface on $m_1$ (vertical and upward).
   • Force $F = 50 \, \text{N}$ at an angle of $\alpha = 60^\circ$ to the horizontal.

2. Decomposing $F$:

   • The vertical component of $F$ is $F_{\text{vertical}} = F \sin \alpha = 50 \sin 60^\circ = 50 \times \frac{\sqrt{3}}{2} = 43.3 \, \text{N}$ upward.
   • The horizontal component is $F_{\text{horizontal}} = F \cos \alpha = 50 \cos 60^\circ = 50 \times \frac{1}{2} = 25 \, \text{N}$ (but this component doesn't affect the normal force).

3. Normal force calculation: The normal force $N$ will balance out the vertical forces acting on $m_1$. These forces are:

   • Downward: Weight of $m_1$, $W_1 = 78.4 \, \text{N}$.
   • Upward: Vertical component of $F$, which is $F_{\text{vertical}} = 43.3 \, \text{N}$.

   Hence, the normal force can be found by balancing the vertical forces: $N + 43.3 = 78.4$ Solving for $N$: $N = 78.4 - 43.3 = 35.1 \, \text{N}$

Thus, the normal force $N$ on $m_1$ is 35.1 N.

Do you want more details on the steps, or have any specific questions?

Here are 5 related questions to further expand:

1. How would friction affect the normal force if it were included in the problem?
2. What would change if the angle $\alpha$ were different?
3. How can the tension in the string between $m_1$ and $m_2$ be calculated?
4. What is the acceleration of the system if the pulley is frictionless?
5. How would you approach the problem if the surface under $m_1$ were inclined?

Tip: Decomposing forces into vertical and horizontal components simplifies the analysis of forces acting on objects.