Problem 25 - University Physics with Modern Physics

Question

Crates \(A\) and \(B\) sit at rest side by side on a frictionless horizontal surface. They have masses \(m_A\) and \(m_B\), respectively. When a horizontal force \(\vec{F}\) is applied to crate \(A\), the two crates move off to the right. (a) Draw clearly labeled free-body diagrams for crate \(A\) and for crate \(B\). Indicate which pairs of forces, if any, are third-law action-reaction pairs. (b) If the magnitude of \(\vec{F}\)is less than the total weight of the two crates, will it cause the crates to move? Explain.

Step-by-Step Solution

Answer

Both crates will move, regardless of weight, if the force \(\vec{F}\) is applied.

Step 1: Free-Body Diagram for Crate A

For crate A, which has mass \(m_A\), draw a force diagram showing: the applied force \(\vec{F}\) acting to the right, the normal force \(\vec{N}_A\) acting upward, and the gravitational force \(\vec{w}_A = m_A g\) acting downward. Since crate A is in contact with crate B, there is also a contact force from crate B on crate A, denoted as \(\vec{f}_{BA}\), acting to the left.

Step 2: Free-Body Diagram for Crate B

For crate B, which has mass \(m_B\), draw a force diagram showing: the contact force from crate A on crate B, denoted as \(\vec{f}_{AB}\), acting to the right, the normal force \(\vec{N}_B\) acting upward, and the gravitational force \(\vec{w}_B = m_B g\) acting downward. No applied external force is acting directly on crate B.

Step 3: Identify Action-Reaction Pairs

According to Newton’s Third Law, \(\vec{f}_{AB}\) and \(\vec{f}_{BA}\) are an action-reaction pair; they are equal in magnitude and opposite in direction. \(\vec{N}_A\) and \(\vec{w}_A\), as well as \(\vec{N}_B\) and \(\vec{w}_B\), do not form action-reaction pairs within this system, as they interact with different bodies (the ground and crates respectively).

Step 4: Analyze the Effect of Force Magnitude on Motion

The magnitude of the applied force \(\vec{F}\) dictates the acceleration of the system. On a frictionless surface, any non-zero force will cause acceleration, regardless of the total weight of the system. Therefore, as long as \(\vec{F} > 0\), the crates will accelerate to the right because there is no friction to counteract the force applied.

Key Concepts

Free-Body Diagrams

Free-body diagrams are essential tools when analyzing forces acting on an object. They help visualize all the forces at play. For crate A with mass \(m_A\), imagine a diagram that includes:

• The applied force \(\vec{F}\) acting to the right.
• A normal force \(\vec{N}_A\) that acts upward from the ground.
• The gravitational force \(\vec{w}_A = m_A g\), acting downward towards the Earth's center.
• A contact force \(\vec{f}_{BA}\) that crate B exerts on crate A, acting to the left.

Crate B has its own forces:

• A contact force from crate A, \(\vec{f}_{AB}\), pushing to the right.
• A normal force \(\vec{N}_B\) upward.
• The gravitational force \(\vec{w}_B = m_B g\) downward.

These diagrams help you identify the interactions between the crates and other forces.

Action-Reaction Pairs

Newton's Third Law states that for every action, there's an equal and opposite reaction. In our crate scenario, this involves the forces between crates A and B. The force \(\vec{f}_{AB}\) exerted by crate A on crate B is matched by the force \(\vec{f}_{BA}\) exerted by crate B on crate A. They are equal in magnitude and opposite in direction, forming an action-reaction pair due to their interaction.Other forces like the normal and gravitational forces between each crate and the Earth are not considered as action-reaction pairs within the system of crates. They interact in different contexts (with the Earth) and not directly because of an interaction between two objects within our main system focus.

Frictionless Surface

A frictionless surface simplifies analysis in physics. It means zero friction between the crates and the ground. This allows all applied forces to convert entirely to motion without any resistance.On a frictionless surface, even a small force will cause the crates to move, because there is no opposing frictional force. The only requirement for motion is that the applied force \(\vec{F}\) must be greater than zero but doesn’t need to exceed the total weight or any particular threshold, since no energy is lost overcoming friction.

Force and Motion

Understanding the relationship between force and motion is key in physics. According to Newton's Second Law, the acceleration of an object depends on the net force acting on it and the object's mass, given by \(a = \frac{F_{net}}{m}\).In our exercise, the force \(\vec{F}\) applied to crate A leads to the movement of both crates. Because the surface is frictionless, this force isn't countered by friction, and thus fully contributes to the acceleration of the system. Regardless of the weight of the crates, as long as the force \(\vec{F}\) is greater than zero, both crates will accelerate and move to the right. This demonstrates how force influences motion without the complexity that comes with resistive forces like friction.