Question:

Three blocks of masses 6 Kg, 3 Kg and 1 Kg are connected with light string kept on a smooth surface. A horizontal force F= 14 N is applied as shown below.Calculate the tension in each string.

Solution:

To identify the forces acting on each mass:

• Gravitational force on the 6 Kg mass acting downward.
• Gravitational force on the 3 Kg mass acting downward.
• Gravitational force on the 1 Kg mass acting downward.
• Applied force of 14 N acting right on the 1 Kg mass.
• Tension T in the string for the 1 Kg mass is acting in the direction opposite to the applied force.
• Tension T₁ in the string for the 6 kg mass is acting in the direction of motion

Acceleration of all the blocks and the strings:

Acceleration of all the blocks and the string is

$${Acceleration =\frac{net \ force}{net \ mass}}$$

$${Acceleration =\frac{14 \ N}{10 \ Kg}}$$

$${Acceleration = 1.4 \ \frac{m}{s^2}}$$

Free body diagram of the 1 Kg block:

Equation using Newton’s second law:

Net force on the horizontal direction is 14 N

$${F-T=1a}$$

$${14 – T =1(1.4)}$$

$${T=14-1.4=12.6}$$

The tension between the 1 Kg and 3 Kg block is 12.6 N

Free body diagram of the 6 Kg block:

Equation using Newton’s second law:

Applying Newton »s second law T₁=6(1.4) = 8.4 N

Conclusion:

Tension in the string joining the 1 Kg and 3 Kg blocks is 12.6 N

Tension in the string joining 3 Kg and 6 kg block is 8.4 N

When the mass of the string is zero, then the tension throughout each string will be same.

Note:

When the mass of a string is considered zero(ideal string), The tension is uniform throughout the entire length of the string.

• When the string is massless and inextensible (ideal), means the string has no mass and does not stretch or contract under tension.This simplifies the analysis because the string does not contribute any forces acting on the system.
• The net force acting on any segment of the string must be zero,On the converse if any force acting on the segment of the string would result in infinite acceleration if there were any force imbalance.
• In the system at equilibrium(where forces are balanced), if the string is connected between two objects, the force exerted by the string on both objects must be equal in magnitude and opposite in direction.This force is the tension, since the string is massless. the tension at every point on the string must be the same to satisfy equilibrium.