Problem 31 - College Physics

Question

Two crates, \(A\) and \(B,\) sit at rest side by side on a frictionless horizontal surface. The crates have masses \(m_{A}\) and \(m_{B} . A\) horizontal force \(\vec{\boldsymbol{F}}\) is applied to crate \(A\) and 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 force \(\vec{\boldsymbol{F}}\) is less than the total weight of the two crates, will it cause the crates to move? Explain.

Step-by-Step Solution

Answer

Yes, the crates will move because the force \( \vec{\boldsymbol{F}} \) is unopposed horizontally.

Step 1: Understanding the Scenario

We have two crates, A and B, on a frictionless surface with a force \( \vec{\boldsymbol{F}} \) applied to crate A. The crates are initially at rest and we want to analyze their motion under the force.

Step 2: Drawing Free-body Diagram for Crate A

On crate A, the external horizontal force \( \vec{\boldsymbol{F}} \) acts to the right. There is also a normal force \( N_A \) acting upwards and gravitational force \( m_A g \) acting downwards. Since crate B is in contact with crate A, there is a force exerted by crate B on crate A, let's call it \( F_{B \rightarrow A} \), acting to the left.

Step 3: Drawing Free-body Diagram for Crate B

On crate B, the only horizontal force acting is the contact force \( F_{A \rightarrow B} \) exerted by crate A on crate B, acting to the right. There is also a normal force \( N_B \) acting upwards and gravitational force \( m_B g \) acting downwards.

Step 4: Identifying Action-Reaction Pairs

According to Newton’s third law, action-reaction pairs include \( F_{B \rightarrow A} \) (force by B on A) and \( F_{A \rightarrow B} \) (force by A on B). These forces are equal in magnitude and opposite in direction: \( F_{B \rightarrow A} = -F_{A \rightarrow B} \).

Step 5: Assessing Motion with Force \( \vec{\boldsymbol{F}} \)

The applied force \( \vec{\boldsymbol{F}} \) causes the entire system (crates A and B) to move since the surface is frictionless. The weight of the crates provides no horizontal resistance to their movement. Thus, even if \( \vec{\boldsymbol{F}} \) is less than the total weight of the crates, the horizontal force is unopposed, allowing motion.

Key Concepts

Free-Body Diagram

A free-body diagram is a visual representation used in physics to illustrate all the forces acting on an object. It helps us analyze forces and predict the resulting motion. In the context of our exercise, we need to draw free-body diagrams for crates A and B.

For crate A, we consider several forces:

• The applied force \( \vec{\boldsymbol{F}} \) acts horizontally to the right.
• Gravity acts downward with a force of \( m_A g \).
• The normal force \( N_A \) acts upwards, counteracting gravity.
• There is also a contact force \( F_{B \rightarrow A} \) acting to the left, exerted by crate B.

Breaking it down for crate B:

• The force \( F_{A \rightarrow B} \) acts to the right, due to crate A pushing it.
• Gravity acts downward with force \( m_B g \).
• The normal force \( N_B \) acts upwards.

The free-body diagrams highlight how these forces interact, forming the basis for analyzing the motion of both crates.

Action-Reaction Pairs

Newton's third law of motion teaches us that every action has an equal and opposite reaction. This concept is illustrated through action-reaction pairs. In our scenario, the contact force between crates A and B showcases this principle.

Here's how:

• When crate A exerts a force on crate B (\( F_{A \rightarrow B} \)), crate B simultaneously exerts an equal and opposite force on crate A (\( F_{B \rightarrow A} \)).
• These forces are equal in magnitude but act in opposite directions: \( F_{B \rightarrow A} = -F_{A \rightarrow B} \).

This pair of forces does not cancel out in the system as a whole because they are acting on different objects. Understanding action-reaction pairs provides insight into how interactions between objects result in motion, particularly in contact scenarios like these two crates.

Frictionless Surface

The mention of a frictionless surface fundamentally changes how we analyze motion in physics problems. A frictionless surface means there is no resistance opposing the horizontal movement of the objects.

In our exercise:

• The applied force \( \vec{\boldsymbol{F}} \) on crate A can move both crates without any opposing frictional force.
• This absence of friction implies that even a small force can set the crates into motion, regardless of their weight.
• The total weight of the crates does not affect their ability to move horizontally because weight is a vertical force, countered by the normal force but irrelevant to horizontal motion on a frictionless surface.

This concept underscores why the crates can move under the influence of the force \( \vec{\boldsymbol{F}} \), illustrating that movement on a frictionless surface is solely determined by the horizontal force applied.