**MOTION OF TWO BODIES CONNECTED BY A STRING OVER A PULLEY**

**Motion of Two Bodies Connected by a String over a pulley on different planes.**

Consider a light inextensible string passing over a smooth pulley, as shown in Fig. 1.25, so that the tension (T) in both the strings is same.

Let mass m1 is greater than mass m2. Since the string is inextensible, the upward acceleration of mass m2 will be equal to the downward acceleration of mass m1. This acceleration is given by

and tension in the string,

**Let us now consider the following cases of motion of two bodies connected by a string.**

1. First of all, let us consider the motion of two bodies connected by an inextensible string, one of which is hanging freely and the other is lying on a smooth horizontal plane

as shown in Fig. 1.26.

Since the string is inextensible, the tension (T) in both the strings will be equal . The acceleration of the system is given by

and

2. If instead of smooth plane, it is a rough horizontal plane, as shown in Fig, 1.27,

then frictional force equal to:

will act in the opposite direction to the motion of mass m2, where u is the coefficient of friction. In such a case,

3. When the plane is a smooth inclined plane, as shown in Fig. 1.28,

then

**Acceleration and Tension in string of bodies on inclined plane over pulley**

4. When the plane is a rough inclined plane, as shown in Fig. 1.29, then

## Motion of connected bodies

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The term “Motion of connected bodies” refers to the movement or motion of two or more objects that are physically linked or connected together. These objects can be rigid bodies or deformable bodies, and they move as a single entity due to their connection. Understanding the motion of connected bodies is essential in various fields of physics, engineering, and mechanics.

When connected bodies move, they can exhibit different types of motion, such as:

- Translational Motion: The connected bodies move as a whole, maintaining a constant relative position between each other. This type of motion is characterized by linear movement in a straight line or curved path.
- Rotational Motion: The connected bodies can rotate about a fixed axis or center point. In this motion, each body moves in a circular path around the common axis of rotation.
- General Curvilinear Motion: Connected bodies may undergo both translational and rotational motion simultaneously, resulting in complex paths or trajectories.

The motion of connected bodies is governed by the principles of physics, particularly Newton’s laws of motion and the conservation of momentum and energy. The behavior of connected bodies depends on various factors such as the nature of the connection (rigid or flexible), the forces acting on the bodies, and any external constraints or interactions they experience.

This concept is fundamental in many engineering applications, such as analyzing the movement of linked mechanical systems, understanding the dynamics of interconnected components in machines, and designing structures like bridges and trusses where multiple elements work together as a single system. Additionally, it plays a crucial role in understanding the mechanics of celestial bodies, where gravitational forces connect and influence their motions, like planets orbiting around the Sun or moons around their respective planets.

#### Question: A car travels from point A to point B at a constant speed of 60 km/h. The distance between the two points is 240 kilometers. How long will it take for the car to reach point B?

Answer: To find the time taken by the car to reach point B, we can use the formula:

Time (T) = Distance (D) / Speed (S)

Given that the distance (D) is 240 kilometers and the speed (S) is 60 km/h, we can plug these values into the formula:

T = 240 km / 60 km/h = 4 hours

Therefore, it will take the car 4 hours to reach point B.