To solve the problem of calculating the horizontal force \( F \) that must be applied so that masses \( m_1 \) and \( m_2 \) do not move relative to \( m_3 \), we need to analyze the forces acting on the system. Since all surfaces are frictionless, the only forces we consider are the gravitational forces acting on the masses.
Understanding the Forces
Each mass experiences a gravitational force given by \( F = m \cdot g \), where \( g \) is the acceleration due to gravity. For our scenario:
- Force on \( m_1 \): \( F_1 = m_1 \cdot g \)
- Force on \( m_2 \): \( F_2 = m_2 \cdot g \)
- Force on \( m_3 \): \( F_3 = m_3 \cdot g \)
Applying Newton's Second Law
To keep \( m_1 \) and \( m_2 \) stationary relative to \( m_3 \), the net force acting on the system must be zero. The applied force \( F \) must balance the gravitational forces acting on \( m_1 \) and \( m_2 \).
Calculating the Required Force
The total force required to keep \( m_1 \) and \( m_2 \) stationary can be expressed as:
F = (m_1 + m_2) \cdot g
Finding the Correct Option
Now, we need to relate this force to the options provided. The correct expression for the horizontal force \( F \) that keeps \( m_1 \) and \( m_2 \) from moving relative to \( m_3 \) is:
F = (m_1 + m_2) \cdot (m_2 \cdot g) / m_1
Thus, the correct answer is:
Option B: (m₁ + m₂) * (m₂ * g) / (m₁)
