A 5 kg object A is placed on a rough surface with a coefficient of friction $\mu = \frac{1}{2}$. It is connected to object B by a light, inextensible string passing over a smooth pulley. We need to find the minimum mass of object B (in kg) required to initiate the movement of the system.

1. Problem Description

A 5 kg object A is placed on a rough surface with a coefficient of friction $\mu = \frac{1}{2}$. It is connected to object B by a light, inextensible string passing over a smooth pulley. We need to find the minimum mass of object B (in kg) required to initiate the movement of the system.

2. Solution Steps

Let $m_A$ be the mass of object A and $m_B$ be the mass of object B. We are given $m_A = 5$ kg and $\mu = \frac{1}{2}$. The weight of object A is $W_A = m_A g$, and the normal force acting on object A is equal to its weight, so $N = m_A g$. The friction force acting on object A is $f = \mu N = \mu m_A g$.

For the system to start moving, the tension in the string must be greater than or equal to the friction force. The tension in the string is equal to the weight of object B, which is $W_B = m_B g$.

Therefore, we have $m_B g \ge \mu m_A g$.

Dividing both sides by $g$, we get:

$m_B \ge \mu m_A$.

Substituting the given values, we have

$m_B \ge \frac{1}{2} \times 5 = 2.5$ kg.

The minimum mass of object B is therefore 2.5 kg.

3. Final Answer

2. 5