A 3 kg block on a table (μ = 0.1) is joined over a pulley to a 2 kg weight. Find the acceleration and the tension — Physics problems | PhysLabo
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A block on a table and a hanging weight over a pulley

Medium 10 grade Mechanics / Newton's Laws
Problem statement

A block of mass $m_1 = 3$ kg lies on a horizontal table and is joined by a light inextensible thread over a fixed pulley to a weight of mass $m_2 = 2$ kg (see the figure). The coefficient of friction between the block and the table is $\mu = 0.1$. Neglect the masses of the pulley and the thread and the friction in the pulley. Find the acceleration of the bodies and the tension in the thread. Take $g = 10$ m/s².

Block m₁ on a table joined over a pulley to a hanging weight m₂; forces and accelerations are shown
Given
  • $m_1 = 3$ kg
  • $m_2 = 2$ kg
  • $\mu = 0.1$
  • $g = 10$ m/s²
  • $a = ?$
  • $T = ?$
Hints
Hint 1 / 2
Draw the forces on each body and write Newton's second law separately for the block (along the table) and for the weight (vertically).
Hint 2 / 2
The thread is inextensible and the pulley is ideal: the accelerations have the same magnitude and the tension is the same on both sides of the pulley. Add the two equations — $T$ cancels.
Solution

The thread is inextensible and the pulley is massless and frictionless, so both bodies move with the same acceleration $a$ and the tension $T$ is the same along the whole thread. The block moves to the right, the weight goes down.

Forces on the block and on the weight, and the chosen axes
Forces on the block and on the weight, and the chosen axes

1. The block, axis along the table. Vertically $N = m_1 g$, so $F_{fr} = \mu m_1 g$:

$$m_1 a = T - \mu m_1 g$$

2. The weight, axis downward:

$$m_2 a = m_2 g - T$$

3. Add the equations — the tension cancels:

$$(m_1 + m_2)\,a = m_2 g - \mu m_1 g \;\Rightarrow\; a = \frac{m_2 g - \mu m_1 g}{m_1 + m_2} = \frac{20 - 3}{5} = 3.4 \text{ m/s}^2$$

4. Find the tension from the equation for the weight:

$$T = m_2 (g - a) = 2 \cdot (10 - 3.4) = 13.2 \text{ N}$$

Check with the block: $T = m_1 a + \mu m_1 g = 10.2 + 3 = 13.2$ N — correct.

Answer

$a = 3.4$ m/s², $T = 13.2$ N

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