https://file.aiquickdraw.com/mathbot/file/1729927675108uwcp03ne.jpg

Let's analyze the problem step-by-step based on the provided diagram and the updated value \( P = 6 \, \text{kg} \).

---

### **Problem Setup:**
- Three masses are connected by ropes passing over pulleys:
  - \( M_1 = 10 \, \text{kg} \) (hanging on the left)
  - \( M_2 = 6 \, \text{kg} \) (on the inclined plane)
  - \( M_3 = 15 \, \text{kg} \) (hanging on the right)
- The inclined plane is 3 meters high and 4 meters in horizontal length (slope length: \( \sqrt{3^2 + 4^2} = 5 \, \text{m} \)).
- Gravitational acceleration: \( g = 10 \, \text{m/s}^2 \).
- Coefficient of friction between \( M_2 \) (the mass on the slope) and the incline: \( \mu_k = 0 \).

**Objective:**
Solve the following sub-questions from parts (a) to (d) about the motion, acceleration, tension, and distances.

---

### **Free-Body Diagrams and Equations of Motion:**

1. **For \( M_1 \) (hanging mass on the left):**
   \[ T_1 - M_1 g = -M_1 a \]
   where \( T_1 \) is the tension on the rope connected to \( M_1 \) and \( a \) is the system's acceleration.

2. **For \( M_2 \) (mass on the incline):**
   - Weight component along the incline: \( M_2 g \sin \theta = 6 \times 10 \times \frac{3}{5} = 36 \, \text{N} \).
   - Normal force: \( M_2 g \cos \theta = 6 \times 10 \times \frac{4}{5} = 48 \, \text{N} \).
   - No friction: \( f_k = \mu_k \times N = 0 \).

   Thus, the equation becomes:
   \[ T_2 - 36 = 6 a \]

3. **For \( M_3 \) (hanging mass on the right):**
   \[ M_3 g - T_2 = 15 a \]
   \[ 150 - T_2 = 15 a \]

---

### **System of Equations:**
We now have three key equations:

1. \( T_1 - 100 = -10a \) (from \( M_1 \)).
2. \( T_2 - 36 = 6a \) (from \( M_2 \)).
3. \( 150 - T_2 = 15a \) (from \( M_3 \)).

---

### **Step 1: Find the Acceleration \( a \):**

From equation (3):
\[
T_2 = 150 - 15a
\]
Substitute this into equation (2):
\[
150 - 15a - 36 = 6a
\]
\[
114 = 21a \quad \Rightarrow \quad a = \frac{114}{21} \approx 5.43 \, \text{m/s}^2
\]

---

### **Step 2: Find the Tensions \( T_1 \) and \( T_2 \):**

Using \( a = 5.43 \, \text{m/s}^2 \) in equation (3):
\[
T_2 = 150 - 15(5.43) \approx 68.5 \, \text{N}
\]

Now, use equation (1) to find \( T_1 \):
\[
T_1 - 100 = -10(5.43) \quad \Rightarrow \quad T_1 = 100 - 54.3 = 45.7 \, \text{N}
\]

---

### **Answering the Questions:**

**(a)** After 2 seconds, we need to calculate:
- **Acceleration**: \( a = 5.43 \, \text{m/s}^2 \).
- **Velocity after 2 seconds**: \( v = a \cdot t = 5.43 \times 2 = 10.86 \, \text{m/s} \).
- **Distance traveled**: \( s = \frac{1}{2} a t^2 = \frac{1}{2} \times 5.43 \times (2)^2 = 10.86 \, \text{m} \).
- **Tension in the rope**: \( T_1 = 45.7 \, \text{N} \), \( T_2 = 68.5 \, \text{N} \).

---

**(b)** If the rope on the right breaks after 2 seconds:
- Each mass will now move independently. The new distance traveled by each mass after 2 seconds can be analyzed further, but note that \( M_1 \) will continue moving downward under gravity.

---

Would you like me to proceed further with detailed calculations for the other parts (b–d)? 

---

### **Related Questions:**
1. What happens to the acceleration if friction is introduced between \( M_2 \) and the incline?
2. How would the motion change if \( M_3 \) were heavier?
3. What if the incline angle changes—how would it affect acceleration?
4. Can we analyze the energy of the system during the 2-second interval?
5. How do we calculate the tension if the rope is elastic?

---

### **Tip:**
When multiple objects interact, setting up proper free-body diagrams and equations of motion is key to solving the problem step-by-step.

Perhatikan gambar di bawah, diketahui 3 buah balok terhubung dengan tali melalui katrol. M1=10kg terhantung pada sisi sebelah kiri, M2=P kg berada pada bidang miring dan M3=15kg tergantung pada sisi kanan kemudian sistem dilepas dari keadaan diam. Jika percepatan gravitasi bumi g=10m/s², dan koefisien gesek antara M2 dengan bidang miring μk=0,P. (Catatan P= digit ke 3 NIM + Digit Terakhir NIM). Hitunglah beberapa variabel seperti percepatan, kecepatan, jarak tempuh, dan tegangan tali setelah 2 detik dilepas.

Explore a physics problem involving three masses connected by ropes and pulleys, calculating acceleration, speed, distance, and tension after 2 seconds. The problem incorporates Newton's Laws of Motion, inclined plane mechanics, and kinematic equations, ideal for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation