Boxes $A$ and $B$ are in contact on a horizontal, frictionless surface (Fig. E$4.23$). Box $A$ has mass $20.0$ kg and box $B$ has mass $5.0$ kg. A horizontal force of $250$ N is exerted on box $A$. What is the magnitude of the force that box $A$ exerts on box $B$?

Verified step by step guidance

1. Step 1: Identify the system and forces acting on it. Box A and Box B are in contact on a frictionless surface. A horizontal force of 250 N is applied to Box A. The goal is to find the force that Box A exerts on Box B.
2. Step 2: Calculate the acceleration of the system. Since the surface is frictionless, the net force acting on the system is the applied force. The total mass of the system is the sum of the masses of Box A and Box B. Use Newton's second law, \( F = ma \), to find the acceleration: \( a = \frac{F}{m_{\text{total}}} \), where \( m_{\text{total}} = m_A + m_B \).
3. Step 3: Analyze the forces acting on Box B. Box B is only influenced by the force exerted by Box A. This force is responsible for accelerating Box B with the same acceleration as the system. Use Newton's second law for Box B: \( F_{A \text{on} B} = m_B \cdot a \).
4. Step 4: Substitute the value of acceleration from Step 2 into the equation for \( F_{A \text{on} B} \). This will give the magnitude of the force that Box A exerts on Box B.
5. Step 5: Verify the solution conceptually. Ensure that the calculated force is consistent with the physical situation, where Box A pushes Box B forward due to the applied force.

Key Concepts

Newton's Second Law of Motion

Newton's Second Law states that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. This relationship is expressed by the formula F = ma, where F is the net force, m is the mass, and a is the acceleration. In this scenario, the total force applied to box A will determine the acceleration of both boxes A and B since they are in contact.

Newton's Third Law of Motion

Newton's Third Law states that for every action, there is an equal and opposite reaction. This means that when box A exerts a force on box B, box B exerts an equal and opposite force back on box A. Understanding this concept is crucial for determining the force that box A exerts on box B when a force is applied to box A.

Frictionless Surface

A frictionless surface is an idealized concept in physics where no frictional forces oppose the motion of objects. This allows for the simplification of calculations, as the only forces acting on the boxes are those applied externally and the normal forces between them. In this problem, the absence of friction means that the entire force applied to box A contributes to the acceleration of both boxes.