Understanding the dynamics of connected particle systems is essential in various physical contexts, from simple mechanisms like pulleys to complex real-world applications like vehicles towing trailers. This topic explores the interaction of forces and motion in such systems.

## Analysis of Systems with Connected Particles

- Connected particles: Motion of one affects others.
- Forces like tension or thrust are key.
- Follow Newton's laws for problem-solving.

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## Solving Tension in Inextensible Strings

- Used in pulleys; don't stretch, ensure uniform acceleration.
- Tension is constant in these strings.

## Thrust in Connecting Rods

- Rods act as rigid connectors.
- Experience compressive forces (thrust) of varying magnitudes.

## Motion Over Smooth Pulleys

- Change direction of tension, not magnitude.
- Assumed frictionless, maintain uniform tension.

## Vehicles and Trailers: Ropes and Rigid Tow-Bars

- Flexible connections (ropes) vs rigid (tow-bars).
- Rigid connections mean same acceleration for both vehicles.

## Equating Forces and Accelerations

- Use $F = ma$ for analysis.
- Balance forces to solve unknowns.

## Example Problems

### Example 1. Two Particles over a Smooth Pulley

Two particles, A (5 kg) and B (3 kg), are connected over a smooth pulley. Find the acceleration and the tension in the string.

**Solution:**

**Equations:**

- For A: (T - 5g = 5a)
- For B: (3g - T = 3a)
- Where $g = 9.81 \, \text{m/s}^2$, (T) is tension, and (a) is acceleration.

**Solving:**

**1. Add equations:** $(T - 5g) + (3g - T) = 8a$

**2. Simplify:** $-2g = 8a$

**3. Find (a): **$a = \frac{-2g}{8} = \frac{-2 \times 9.81}{8} \approx -2.45 \, \text{m/s}^2$

**4. Find (T)** using $T = 5g + 5a$: $T \approx 5 \times 9.81 - 5 \times 2.45 \approx 36.79 \, \text{N}$

**Conclusion:** $a \approx -2.45 \, \text{m/s}^2), (T \approx 36.79 \, \text{N}.$

### Example 2: Car Towing a Trailer

A car (mass = 1200 kg) tows a trailer (mass = 300 kg) with a rigid tow-bar, accelerating at 2 m/s². Find the tension in the tow-bar.

**Solution:**

**Total Force Exerted:**

- Total mass = 1200 kg + 300 kg = 1500 kg
- Total force exerted = $1500 \, \text{kg} \times 2 \, \text{m/s}^2 = 3000 \, \text{N}$

**Tension in the Tow-bar:**

- Tension = Total force exerted = 3000 N

**Conclusion:** The tension in the tow-bar is 3000 N.

**Example 3: Particle on an Incline**

A 4 kg particle on a 30° inclined plane is connected to a 6 kg hanging particle over a smooth pulley, with a coefficient of friction of 0.2. Find the system's acceleration and the tension in the string.

**Solution:**

**Given:**

- Mass on incline $(m_1)$ = 4 kg
- Hanging mass $(m_2)$ = 6 kg
- Incline angle $(\theta)$ = 30°
- Coefficient of friction $(\mu)$ = 0.2
- Acceleration due to gravity $(g)$ = 9.81 m/s²

**Analyzing Forces:**

- Force of gravity on $m_1$ down the incline: $m_1g\sin\theta$
- Normal force on $m_1$: $N = m_1g\cos\theta$
- Friction force on $m_1$
*:*$F{\text{friction}} = \mu N$ - Force of gravity on $m_2$: $m_2g$

**Equations for System's Acceleration **$(a)$** and Tension **$(T):$

- Without detailed steps, using the provided solution:
- Acceleration $a \approx 3.24 \, \text{m/s}^2$
- Tension $T \approx 39.39 \, \text{N}$

**Conclusion:** The system's acceleration is approximately 3.24 m/s², and the tension in the string is approximately 39.39 N.