Two blocks of 2 kg and 3 kg are joined by a thread, a 20 N force pulls the second one. Find the acceleration and the tension — Physics problems | PhysLabo
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Acceleration and thread tension for two connected blocks

Medium 9 grade Mechanics / Newton's Laws
Problem statement

Two blocks of masses $m_1 = 2$ kg and $m_2 = 3$ kg are joined by a light inextensible thread and lie on a smooth horizontal table. A horizontal force $F = 20$ N is applied to block $m_2$ (see the figure). Find the acceleration of the blocks and the tension in the thread.

Two blocks m₁ and m₂ on a smooth table joined by a thread; force F acts on m₂
Given
  • $m_1 = 2$ kg
  • $m_2 = 3$ kg
  • $F = 20$ N
  • $a = ?$
  • $T = ?$
Hints
Hint 1 / 2
The thread is inextensible, so both blocks have the same acceleration. First treat them as one body of mass $m_1 + m_2$.
Hint 2 / 2
The thread tension is the only horizontal force that accelerates the first block: $T = m_1 a$.
Solution

The thread is inextensible, so both blocks move with the same acceleration $a$. There is no friction and the vertical forces cancel, so only the horizontal forces matter.

Forces on each block separately
Forces on each block separately

1. Treat both blocks as one body of mass $m_1 + m_2$. The only external horizontal force is $F$ (the thread tensions are internal):

$$F = (m_1 + m_2)\,a \;\Rightarrow\; a = \frac{F}{m_1 + m_2} = \frac{20}{2 + 3} = 4 \text{ m/s}^2$$

2. The first block is accelerated only by the thread tension:

$$T = m_1 a = 2 \cdot 4 = 8 \text{ N}$$

Check with the second block: $F - T = m_2 a \Rightarrow 20 - 8 = 3 \cdot 4 = 12$ — correct.

Answer

$a = 4$ m/s², $T = 8$ N

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