Motion Of Blocks When Connected With String

Edited By Vishal kumar | Updated on Jul 02, 2025 05:32 PM IST

When the motion of one is directly related to the motion of the other, these bodies are said to be in connected motion One end of a string that is wound around a solid body is connected to a point mass. When released, the mass descends vertically, and the solid body spins, unwinding the string.

In this article, we will cover the concept of motion of blocks when connected with string. This topic falls under the broader category of laws of motion, which is a crucial chapter in Class 11 physics. It is not only essential for board exams but also for competitive exams like the Joint Entrance Examination (JEE Main), National Eligibility Entrance Test (NEET), and other entrance exams such as SRMJEE, BITSAT, WBJEE, BCECE and more. It's an important topic for the JEE Main exam as well as the NEET exam.

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Two Blocks Connected With a String on a Smooth Horizontal Surface
Three Blocks Connected With a String on a Smooth Horizontal Surface
Summary

Motion Of Blocks When Connected With String

Let's read this entire article to gain an in-depth understanding of the Motion of blocks when connected with string.

Two Blocks Connected With a String on a Smooth Horizontal Surface

Let acceleration of the blocks be 'a', and Tension in the string be T.

F.B.D of both blocks combined

From FBD,Fnet =MsysaF=(m1+m2)a⇒a=Fm1+m2…

F.B.D of a block of mass m₁

From Fnet =Msys aT=m1a From equation (1) ⇒T=m1Fm1+m2… (2)

Three Blocks Connected With a String on a Smooth Horizontal Surface

Let acceleration of the blocks be 'a', tension in the string between m₁and m₂be T₁, and tension between m₂ and m₃ be T₂.

F.B.D of all the blocks combined

From FBD, Fnet =Msys aF=(m1+m2+m3)a⇒a=Fm1+m2+m3…

F.B.D of block m₁

_{From Fnet =Msys a
T1=m1a}

_{From equation (1) -
⇒T1=m1Fm1+m2+m3…}

F.B.D of m₃

From Fnet =Msys aF−T2=m3a⇒T2=(m1+m2)Fm1+m2+m3…

Summary

When a force F is applied vertically or horizontally to two objects connected by a string, the resultant tension in the string causes some acceleration. This phenomenon is known as connected motion. Another way to describe it is as items that move at the same speed and are connected by certain links. Attached to one end of a string that is wound around a solid body is a point mass. Unwinding the string, the solid body spins and the bulk falls vertically downward upon release.

Frequently Asked Questions (FAQs)

1. How does the mass of connected blocks affect their acceleration when a force is applied?

The total mass of the connected blocks determines their acceleration. According to Newton's Second Law (F = ma), for a given force, the acceleration is inversely proportional to the total mass. So, heavier combined masses will result in lower acceleration for the same applied force.

2. What role does friction play in the motion of connected blocks on a horizontal surface?

Friction opposes the motion of the blocks. It acts on each block in contact with the surface, reducing the overall acceleration of the system. The total frictional force depends on the normal force (related to the masses of the blocks) and the coefficient of friction between the blocks and the surface.

3. What is the significance of the "massless string" assumption in problems involving connected blocks?

The "massless string" assumption simplifies calculations by neglecting the string's mass and any associated inertia. This allows us to treat the tension as constant throughout the string and focus on the forces acting on the blocks themselves.

4. How does the concept of inertia apply to the motion of connected blocks?

Inertia, the resistance to change in motion, is proportional to mass. In a system of connected blocks, the total inertia is the sum of the individual block inertias. This combined inertia determines how the system as a whole resists changes in its motion when forces are applied.

5. What happens if one block in a connected system is much heavier than the other?

If one block is much heavier, it will dominate the system's behavior. The acceleration will be closer to what the heavier block would experience alone. The lighter block will be "dragged along" by the heavier one, experiencing higher acceleration than it would on its own.

6. Why do two blocks connected by a string move together when a force is applied to one block?

The blocks move together because the string transmits the force between them. When tension develops in the string, it pulls on both blocks equally, causing them to accelerate as a single unit. This is assuming the string is inextensible and remains taut.

7. What happens to the tension in a string connecting two blocks when one block is pulled?

When one block is pulled, the tension in the string increases. This tension force is transmitted through the string, causing both blocks to accelerate together. The tension is the same throughout the string, acting equally on both blocks.

8. How does the direction of motion affect the tension in the string connecting two blocks?

The direction of motion doesn't directly affect the tension. The tension is determined by the forces causing acceleration and the masses involved. However, the direction can indirectly affect tension by changing how other forces (like friction or gravity on an incline) act on the system.

9. What happens to the motion of connected blocks if the string becomes slack?

If the string becomes slack, the blocks are no longer constrained to move together. They will move independently based on the individual forces acting on each block. This can occur if the acceleration of the leading block is less than that of the following block.

10. What is the relationship between tension and acceleration in a system of connected blocks?

The tension in the string is directly related to the acceleration of the system. Higher tension generally results in greater acceleration. The exact relationship depends on the masses involved and other forces like friction or gravity components on inclines.

11. How does the angle of an inclined plane affect the motion of connected blocks?

The angle of the incline determines the component of gravity parallel to the plane, which drives the motion. A steeper angle increases this component, resulting in greater acceleration down the plane. The perpendicular component affects the normal force and thus the friction.

12. What is the difference between static and kinetic friction in the context of connected blocks?

Static friction prevents the blocks from starting to move when at rest, while kinetic friction opposes motion once the blocks are moving. Static friction can be greater than kinetic friction, which is why it often takes more force to start moving the blocks than to keep them moving.

13. How does air resistance affect the motion of connected blocks?

Air resistance opposes the motion of the blocks, reducing their acceleration. Its effect is more noticeable at higher velocities and for lighter objects with larger surface areas. In many introductory problems, air resistance is often neglected for simplicity.

14. How does the concept of mechanical advantage apply to systems of connected blocks and pulleys?

Mechanical advantage in pulley systems allows a smaller force to lift a larger weight. In connected block systems with pulleys, the arrangement of pulleys can change the effective force acting on the blocks, altering their motion and the tension in the connecting strings.

15. What happens if the connecting string wraps around a massive pulley with significant rotational inertia?

A massive pulley with significant rotational inertia resists changes in its rotational speed. This additional inertia in the system reduces the overall acceleration of the blocks. The pulley's moment of inertia must be considered in calculations, making the analysis more complex.

16. How does the principle of conservation of energy apply to connected blocks moving on an inclined plane?

As connected blocks move on an incline, gravitational potential energy is converted to kinetic energy (and vice versa). The total mechanical energy remains constant in the absence of non-conservative forces like friction. This principle can be used to analyze the blocks' motion without directly considering forces.

17. What is the significance of the normal force in problems involving connected blocks on inclined planes?

The normal force on an inclined plane is crucial as it determines the frictional force. It's perpendicular to the plane's surface and is less than the block's weight on an incline. Understanding how the normal force changes with incline angle is key to accurately analyzing the blocks' motion and the forces involved.

18. How does the presence of fluid resistance (like in water) affect the motion of connected blocks?

Fluid resistance, unlike simple friction, increases with velocity. In water, this can lead to a terminal velocity for the system where the fluid resistance balances the driving force. The motion becomes more complex, with the resistance force changing dynamically as the blocks move.

19. What is the effect of having one block submerged in a fluid while connected to another in air?

A submerged block experiences buoyant force in addition to other forces. This can reduce the effective weight of the submerged block, changing the system's behavior. The motion analysis must account for buoyancy, fluid resistance, and the different environments of the two blocks.

20. What role does the moment of inertia play if one of the connected blocks is free to rotate?

If a block can rotate, its moment of inertia becomes important. It determines the block's resistance to rotational acceleration. In such cases, the analysis must consider both translational and rotational motion, making the problem more complex as angular acceleration and torque come into play.

21. What happens to the motion of connected blocks if there's a time delay in force transmission through the string?

In real systems, force transmission is not instantaneous. A time delay, however small, can lead to complex behaviors, including oscillations or wave-like motions along the string. This is usually negligible in introductory problems but becomes important in more advanced analyses or real-world applications.

22. How does the presence of rolling motion (instead of sliding) affect the analysis of connected block problems?

Rolling motion introduces rotational kinetic energy and angular momentum considerations. The rolling block's linear velocity is related to its angular velocity, and its acceleration depends on both force and torque. This makes the analysis more complex but also more realistic for many practical scenarios.

23. What is the effect of a non-uniform gravitational field on the motion of connected blocks?

In a non-uniform gravitational field, the gravitational force on each block might differ. This can lead to varying tensions in the string and complex motion, especially if the blocks are far apart. Such scenarios are relevant in space physics or when dealing with large-scale systems.

24. How does the breaking strength of a string affect the analysis of connected block problems?

The breaking strength sets an upper limit on the tension the string can withstand. If the required tension to keep the blocks moving together exceeds this limit, the string will break, and the blocks will no longer move as a unit. This introduces a constraint in problem-solving and real-world applications.

25. How does the concept of Newton's Third Law apply to blocks connected by a string?

Newton's Third Law states that for every action, there's an equal and opposite reaction. In connected blocks, the tension force exerted by the string on one block is equal and opposite to the tension force on the other block. This ensures that the blocks interact as a system.

26. How does the length of the string affect the motion of connected blocks?

In ideal scenarios, the length of the string doesn't affect the motion, assuming it remains taut. However, in practical situations, a longer string might introduce more elasticity or allow more slack, potentially affecting the synchronization of the blocks' motion.

27. What happens if the string connecting two blocks is elastic instead of inextensible?

An elastic string can stretch, storing and releasing energy. This can lead to more complex motion, including oscillations. The blocks may not move perfectly in unison, and the system's behavior becomes more complicated, requiring consideration of the string's elastic properties.

28. What is the importance of drawing free body diagrams for connected block problems?

Free body diagrams visually represent all forces acting on each block, including tension, gravity, normal force, and friction. They are crucial for identifying all relevant forces, their directions, and magnitudes, which is essential for correctly applying Newton's laws and solving the problem.

29. How does the coefficient of friction between the blocks and the surface affect their motion?

The coefficient of friction determines the magnitude of the frictional force for a given normal force. A higher coefficient results in greater friction, which opposes motion more strongly. This can significantly reduce the acceleration of the system or even prevent motion if static friction is not overcome.

30. What is the significance of the "light pulley" assumption in problems involving connected blocks and pulleys?

The "light pulley" assumption, similar to the massless string assumption, neglects the mass and rotational inertia of the pulley. This simplifies the problem by allowing us to treat the tension in the string as constant around the pulley, focusing solely on the forces acting on the blocks.

31. What happens to the acceleration of connected blocks if mass is transferred from one block to another?

Transferring mass between blocks doesn't change the total mass of the system, so the overall acceleration remains the same if the applied force is constant. However, it can change how the tension force is distributed between the blocks and may affect friction if the blocks are on a surface.

32. How does the initial velocity of the blocks affect the analysis of their subsequent motion?

The initial velocity affects the blocks' kinetic energy and momentum at the start of the analysis. While it doesn't directly impact acceleration (which depends on force and mass), it's crucial for determining the blocks' position and velocity at any given time, especially in energy conservation problems.

33. What is the role of tension in maintaining the shape of a string connecting two blocks?

Tension keeps the string taut and maintains its shape. Without tension, the string would become slack and lose its ability to transmit forces between the blocks. The magnitude of tension determines how straight and rigid the string appears during the blocks' motion.

34. How does the concept of relative motion apply to systems of connected blocks?

Relative motion is important when considering how the blocks move with respect to each other and their environment. While connected blocks generally move together, their motion relative to other objects or reference frames can vary, which is crucial in more complex scenarios or when dealing with multiple connected systems.

35. What is the effect of having multiple strings connecting several blocks in a complex arrangement?

Multiple strings in a complex arrangement create a system where the motion of each block depends on all others. The tension in each string may differ, and the system's behavior becomes more intricate. Analyzing such systems often requires simultaneous equations considering all interconnected forces and motions.

36. How does the surface texture affect the motion of connected blocks sliding on it?

Surface texture influences the coefficient of friction between the blocks and the surface. A rougher texture generally increases friction, potentially slowing the blocks more or requiring a greater force to initiate motion. Smooth surfaces typically offer less resistance, allowing for easier movement.

37. What role does the distribution of mass within each block play in the motion of a connected block system?

While the total mass of each block is crucial for determining the system's overall motion, the distribution of mass within each block (its moment of inertia) becomes important if the blocks can rotate. In most translational motion problems, however, blocks are treated as point masses, ignoring internal mass distribution.

38. How does the tension in the string change if one of the connected blocks enters a frictionless region?

If one block enters a frictionless region while the other remains on a surface with friction, the tension in the string may change. The block in the frictionless region will tend to accelerate more easily, potentially increasing the tension as it "pulls" the other block along.

39. What happens to the motion of connected blocks if there's a sudden change in the applied force?

A sudden change in applied force results in a change in acceleration according to Newton's Second Law. This change propagates through the system via the tension in the string. There might be a brief transient period where the blocks adjust to the new force before settling into a new steady-state motion.

40. How does the concept of impulse apply to collisions involving systems of connected blocks?

Impulse, the product of force and time of impact, is crucial in analyzing collisions of connected blocks. In a collision, the impulse changes the momentum of the system. For connected blocks, the impulse experienced by one block is transmitted to the other through the connecting string, affecting the entire system's motion.

41. What happens if the masses of two connected blocks are exactly equal in an Atwood machine setup?

In an ideal Atwood machine with equal masses, the system remains in equilibrium if initially at rest. Any small disturbance could set the system in motion, but there would be no net acceleration due to gravity. This demonstrates the concept of neutral equilibrium.

42. How does the concept of virtual work apply to analyzing the motion of connected blocks?

Virtual work principle states that the total work done by applied forces for any virtual displacement consistent with constraints is zero. This principle can be used to analyze the equilibrium and motion of connected block systems, especially in more complex configurations where direct force analysis is challenging.

43. How does the principle of superposition apply to forces acting on a system of connected blocks?

The principle of superposition states that the net effect of multiple forces on an object is the vector sum of the effects of each force applied individually. In connected block systems, this principle allows us to analyze complex force scenarios by breaking them down into simpler components and then combining the results.

44. How does the concept of constraint forces apply to the analysis of connected block systems?

Constraint forces, like tension in an inextensible string, enforce the physical constraints of the system. They ensure that connected blocks move together and maintain their relative positions. These forces are crucial in determining the system's motion and are often solved for as part of the problem.

45. How does the concept of work-energy theorem apply to systems of connected blocks?

The work-energy theorem states that the work done on a system equals its change in kinetic energy. For connected blocks, this principle can be used to analyze motion by considering the work done by all forces (including tension) and relating it to the system's overall change in kinetic energy.

46. What is the significance of the center of mass in analyzing the motion of connected blocks of different sizes?

The center of mass is the average position of mass in a system. For connected blocks of different sizes, the center of mass motion can simplify analysis, especially in the absence of external torques. The system's translational motion can be described by treating it as a single particle located at the center of mass.

47. What happens to the tension in the string if one of the connected blocks encounters a sudden change in surface friction?

A sudden change in surface friction for one block can cause an abrupt change in the tension. If friction increases, that block tends to slow down, potentially increasing tension as it's "pulled" by the other block. Conversely, a decrease in friction might momentarily reduce tension as that block accelerates more easily.

48. How does the concept of elastic collision apply to systems where connected blocks collide with other objects?

In an elastic collision involving connected blocks, both energy and momentum are conserved. The analysis becomes more complex as the collision affects both blocks through the connecting string. The post-collision motion depends on how the impulse is transmitted through the system and can lead to interesting dynamics.

49. How does the concept of simple harmonic motion apply to connected blocks in certain configurations?

Connected blocks can exhibit simple harmonic motion in configurations like a mass-spring system or certain pulley arrangements. This occurs when there's a restoring force proportional to displacement from equilibrium. The analysis involves concepts of period, frequency, and amplitude of oscillation.

50. What role does dimensional analysis play in verifying the correctness of equations derived for connected block problems?

Dimensional analysis is a powerful tool for checking the validity of equations in connected block problems. It ensures that all terms in an equation have consistent units. This can help catch errors in derivations and provide insight into the physical meaning of different terms in the equations describing the system's motion.