Tension and Acceleration Calculation for Pulley System with Blocks and Table

Question:

Find the tension and acceleration in the string in which the one end of the string is attached to a block of mass m lying on the table and the other end of the string with mass ‘2m’ is hanging off the table after running over a pulley, attached to the end of the table.

Solution: Tension and Acceleration in the string

Free body diagram of the lower block

Block on the table (of mass ‘m’)

• Tension T acts horizontally
• Normal force (due to the table) acts vertically upwards
• Gravitational force (mg) acts vertically downwards.
• Vertical acceleration of the mass ‘m’ is zero since downward and upward force cancel out.

Block of mass ‘2m’ (hanging off the table)

• Tension ‘T’ acts upwards due to string with a mass
• Gravitation force ‘2mg’ acts vertically downwards [F=mg =(2m)g=2mg]

Acceleration of both the blocks with string (First method)

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

This net force and net mass is for the pulley-block system

$${Acceleration =\frac{2mg}{3m}}$$

$${Acceleration=\frac{2g}{3}}$$

Tension in the string

$${2mg-T=2ma}$$

(Check the free body diagram)

$${T=2mg-2ma}$$

$${T=2mg-2m(\frac{2g}{3})}$$

$${T=2m(g-\frac{2g}{3})}$$

$${T=2m(\frac{g}{3})}$$

$${T=\frac{2mg}{3}}$$

Conclusion:

Acceleration of the both blocks,attached with string is 2g/3

Tension in the string is given by 2mg/3

Result shows the relationship between tension,gravitational force and the acceleration in the system in which the two bodies are connected by a string.