We are given the following:

• Force $F = 90 \, \text{N}$, applied at an angle of $25^\circ$ to the horizontal.
• Mass $m_2 = 10 \, \text{kg}$.
• The acceleration $a = 3 \, \text{m/s}^2$ for both masses $m_1$ and $m_2$.
• The surface is frictionless, so we can ignore frictional forces.

To calculate the force that the horizontal surface exerts on mass $m_1$, we will find both the horizontal and vertical components of the applied force and use Newton's second law.

Step 1: Horizontal component of the applied force

The horizontal component of force $F_x$ is responsible for accelerating both masses:

$F_x = F \cdot \cos(25^\circ)$ This force acts on both $m_1$ and $m_2$, and thus we apply Newton’s second law:

$F_x = (m_1 + m_2) \cdot a$

Step 2: Solve for $m_1$

Rearranging the above equation to solve for $m_1$:

$m_1 = \frac{F_x}{a} - m_2$

Substitute $F_x$ and solve for $m_1$:

$m_1 = \frac{F \cdot \cos(25^\circ)}{a} - m_2$

Step 3: Vertical forces

The vertical component of the applied force $F_y$ is:

$F_y = F \cdot \sin(25^\circ)$

The normal force $N$ exerted by the horizontal surface on $m_1$ balances the vertical forces acting on $m_1$, i.e., the gravitational force $m_1g$ and the vertical component $F_y$:

$N = m_1 g - F_y$

Let’s calculate each of these steps.The mass $m_1$ is approximately $17.19 \, \text{kg}$, and the normal force that the horizontal surface exerts on mass $m_1$ is approximately $130.59 \, \text{N}$.

Do you want any more details or have further questions?

Here are some related questions:

1. How does the angle of applied force affect the normal force on the object?
2. What would change if there were friction between the masses and the surface?
3. How would the problem change if the masses were connected by a rope?
4. Can you calculate the total force exerted by the horizontal surface on both masses?
5. How would increasing the acceleration affect the forces in the system?

Tip: Breaking forces into components simplifies solving problems involving angles. Always resolve vectors into horizontal and vertical components.