https://file.aiquickdraw.com/mathbot/file/17280995209169wfv5x8u.jpg

Let's break down the parts of the question shown in the image:

### 4a) **State Newton’s Laws.**
Newton's laws of motion are:
1. **First Law (Law of Inertia):** An object at rest will remain at rest, and an object in motion will continue in motion at constant velocity, unless acted upon by a net external force.
2. **Second Law (Law of Acceleration):** The net force acting on an object is equal to the mass of the object multiplied by its acceleration. Mathematically, \( F = ma \).
3. **Third Law (Action and Reaction):** For every action, there is an equal and opposite reaction.

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### 4b) **Analyze the forces on the system (Objects A and B).**
Here, objects A and B are connected by a rope over a pulley. Object A is on a slope inclined at \( 60^\circ \), while object B is hanging vertically. 

Let’s identify the forces:
- **For Object A (10 kg, on the incline):**
  1. Gravitational force: \( F_g = m_A g \) acting vertically downward.
  2. Component of gravitational force parallel to the incline: \( F_{g\parallel} = m_A g \sin \theta \).
  3. Component of gravitational force perpendicular to the incline: \( F_{g\perp} = m_A g \cos \theta \).
  4. Tension \( T \) in the rope acts upward along the slope.
  5. Normal force \( N \) acts perpendicular to the surface of the incline.
  
- **For Object B (25 kg, hanging vertically):**
  1. Gravitational force: \( F_g = m_B g \) acting downward.
  2. Tension \( T \) in the rope acts upward.

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### 4c) **Copy the figure and indicate the forces.**
I can guide you in drawing this:
1. For **Object A**:
   - A downward arrow for gravity \( F_g \).
   - Two components of gravity: one along the slope (\( F_{g\parallel} \)) and one perpendicular to the slope (\( F_{g\perp} \)).
   - A tension \( T \) force along the rope.
   - A normal force \( N \) perpendicular to the slope.

2. For **Object B**:
   - A downward arrow for the weight \( m_B g \).
   - An upward arrow for the tension \( T \).

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### 4d) **If there is no movement, calculate the tension of the rope.**
In this scenario, the system is in equilibrium, meaning the forces on both objects balance.

- For **Object A** on the incline:
  \[
  T = m_A g \sin \theta
  \]

- For **Object B** hanging vertically:
  \[
  T = m_B g
  \]

Since there’s no movement, the tensions in the rope must be equal. Therefore, equate the two expressions for \( T \).

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### 4e) **If the 25 kg object starts to move downward at \( 3 \, \text{m/s}^2 \), calculate the frictional force between object A and the slanted floor.**

To calculate the frictional force, we first apply Newton's second law.

1. For **Object B**:
   \[
   m_B g - T = m_B a
   \]
   Rearranging to find \( T \):
   \[
   T = m_B (g - a)
   \]

2. For **Object A** on the incline:
   \[
   T - m_A g \sin \theta - f = m_A a
   \]
   Here, \( f \) is the frictional force we are solving for.

By solving these two equations, you can find the frictional force \( f \).

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Let me know if you'd like more details for any of these solutions or have further questions!

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### Related questions:
1. What are the values for tension in case of equilibrium?
2. How can we calculate the normal force acting on Object A?
3. What role does the incline angle play in determining the forces on Object A?
4. How would the situation change if the pulley had friction?
5. What if the acceleration of object B was zero, but object A experienced friction?

### Tip:
Always start by resolving forces into components along the axes that align with motion—typically parallel and perpendicular to inclined planes.

State Newton's laws and solve the problem involving objects A and B connected by a rope over a frictionless pulley, with one object on an inclined plane and the other hanging vertically.

Understand how to apply Newton's laws of motion to a system involving a pulley, two masses, and an inclined plane. Learn how to calculate the tension in the rope, frictional forces, and acceleration, suitable for high school physics students.

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