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Motion Of Blocks When Connected With String

Motion Of Blocks When Connected With String

Vishal kumarUpdated on 02 Jul 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.

This Story also Contains

  1. Two Blocks Connected With a String on a Smooth Horizontal Surface
  2. Three Blocks Connected With a String on a Smooth Horizontal Surface
  3. Summary
Motion Of Blocks When Connected With String
Motion Of Blocks When Connected With 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.

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 m1 and m2 be 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…

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Solved Examples Based on the Motion of Blocks When Connected With String

Example 1: Two blocks kept on the smooth horizontal surfaces of masses 7kg and 3kg connected with a light string are pulled by applying a horizontal force of 40N as shown in the figure. The tension force (in N) in the string is-

1) 28N

2) 12N

3) 21N

4) None of the above

Solution :

Given :
m1=7kg,m2=3kg,F=40N

Let acceleration of the blocks be 'a', and Tension in the string be T. F.B.D of both blocks combined-

F=(m1+m2)a⇒a=Fm1+m2a=4010=4 m/s2

F.B.D of block of mass m1

T=m1aT=7×4=28N

Hence, the answer is option (1).

Example 2: Three blocks of masses 6kg, 4kg, and 2kg kept on a smooth surface connected together with light strings are pulled applying 24N horizontal force as shown in the figure. The tension T1 in the string is- (in N)

1) 22

2) 12

3) 10

4) 30

Solution :

Given :
m1=6kg,m2=4kg,m3=2kg,F=24 N

Let accelereration of the blocks be 'a'.
And tension in the string between m1 and m2 be T1, and tension between m2 and m3 be T2.

F.B.D of all the blocks combined-

F=(m1+m2+m3)a⇒a=Fm1+m2+m3∴a=246+4+2=2412=2 m/s2

F.B.D of block m1 :

From Newton second law of motion,
T1=m1a=6×2=12 N

Hence, the answer is option (2).

Example 3: A block of mass M is pulled along a horizontal frictionless surface by a rope of mass m. If a force F is applied at the free end of the rope, the force exerted by the rope on the block is:

1) PmM+m
2) FM+m
3) p
4) FMM+m

Solution :

Acceleration, (a)=Fnet Total mass a=F(M+m)

FBD of Mkg block :

From Newton's 2nd laws of motion
F=Ma=M(FM+m)∴T=MFM+m

Hence, the answer is option (4).

Example 4: Three identical blocks of masses 2kg each, are kept on the smooth surface. The system is drawn by applying forces of 15N and 3N as shown in the figure. Determine the Tension T1 (in N) in one string as shown in the figure.

1) 9

2) 8

3) 7

4) 10

Solution :

Given :
m1=m2=m3=2Kg,F1=15N,F2=3N

Let the acceleration of the combined system be 'a'.
a=Fnet Total Mass a=15−32+2+2=126=2 m/s2
F.B.D of leftmost block-

From Newton's 2nd law of motion
T1−3=2a∴T1=2×2+3=7N

Hence, the answer is option (2).

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)

Q: What role does dimensional analysis play in verifying the correctness of equations derived for connected block problems?
A:
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.
Q: How does the concept of simple harmonic motion apply to connected blocks in certain configurations?
A:
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.
Q: How does the concept of elastic collision apply to systems where connected blocks collide with other objects?
A:
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.
Q: What happens if the masses of two connected blocks are exactly equal in an Atwood machine setup?
A:
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.
Q: What role does the distribution of mass within each block play in the motion of a connected block system?
A:
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.
Q: How does the tension in the string change if one of the connected blocks enters a frictionless region?
A:
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.
Q: What happens to the motion of connected blocks if there's a sudden change in the applied force?
A:
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.
Q: How does the concept of impulse apply to collisions involving systems of connected blocks?
A:
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.
Q: How does the concept of virtual work apply to analyzing the motion of connected blocks?
A:
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.
Q: How does the principle of superposition apply to forces acting on a system of connected blocks?
A:
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.
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