All surfaces shown in figure are assumed to be frictionless and the pulleys and the string are light. The acceleration of the block of mass

To solve the problem concerning the acceleration of the block of mass on frictionless surfaces with pulleys and strings, we will follow the structured approach below.

1. Break Down the Problem

1. Identify the masses involved (e.g., $m_1$, $m_2$).
2. Determine the forces acting on each mass.
3. Analyze the roles of the pulleys and the connections via the string.

2. Relevant Concepts

1. Newton's Second Law of Motion: $F = ma$.
2. For a system with pulleys and connection strings, the tension in the string affects the acceleration of the masses.

3. Analysis and Detail

Let's denote:

• $m_1$: mass of the block on one side of the pulley
• $m_2$: mass of the hanging block on the other side
• $g$: acceleration due to gravity (approximately $9.81 \, \text{m/s}^2$)

1. For the block on the surface ($m_1$), the only horizontal force is the tension $T$ from the string: $T = m_1 a$

2. For the hanging block ($m_2$), the forces acting are gravitational weight and tension: $m_2 g - T = m_2 a$

Now, let's express $T$ from the first equation and substitute it into the second:

1. From Block $m_1$: $T = m_1 a$

2. Substitute $T$ into the second equation: $m_2 g - m_1 a = m_2 a$

4. Verify and Summarize

Now we simplify and solve for $a$: $m_2 g = m_2 a + m_1 a$ $m_2 g = a(m_1 + m_2)$ $a = \frac{m_2 g}{m_1 + m_2}$

Final Answer

The acceleration $a$ of the block of mass is given by: $a = \frac{m_2 g}{m_1 + m_2}$