Two crates connected by a rope lie on a horizontal surface (Fig. E5.375.37). Crate A has mass mAm_A, and crate B has mass mBm_B. The coefficient of kinetic friction between each crate and the surface is μkμ_k. The crates are pulled to the right at constant velocity by a horizontal force FF. Draw one or more free-body diagrams to calculate the following in terms of mAm_A, mBm_B, and μkμ_k: the magnitude of FF.

Ch 05: Applying Newton's Laws

Young & Freedman Calc - University Physics 14th Edition

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Young & Freedman Calc 14th Edition

Ch 05: Applying Newton's Laws

Problem 37a

Chapter 5, Problem 37a

Two crates connected by a rope lie on a horizontal surface (Fig. E $5.37$ ). Crate A has mass $m_{A}$ , and crate B has mass $m_{B}$ . The coefficient of kinetic friction between each crate and the surface is $μ_{k}$ . The crates are pulled to the right at constant velocity by a horizontal force $F$ . Draw one or more free-body diagrams to calculate the following in terms of $m_{A}$ , $m_{B}$ , and $μ_{k}$ : the magnitude of $F$ .

Verified step by step guidance

Step 1: Begin by analyzing the forces acting on each crate. For crate A, the forces include the tension in the rope (T), the kinetic friction force opposing motion (f_kA = μk * mA * g), and the normal force (N_A = mA * g). For crate B, the forces include the applied force (F), the tension in the rope (T), the kinetic friction force opposing motion (f_kB = μk * mB * g), and the normal force (N_B = mB * g).

Step 2: Since the crates are moving at constant velocity, the net force on the system is zero. This means the applied force F must balance the total frictional forces acting on both crates. Write the equation for the total frictional force: f_total = f_kA + f_kB = μk * mA * g + μk * mB * g.

Step 3: The tension in the rope (T) connecting the crates is internal to the system and does not affect the calculation of F. Therefore, the applied force F must equal the total frictional force to maintain constant velocity: F = f_total = μk * mA * g + μk * mB * g.

Step 4: Substitute the expressions for the frictional forces into the equation for F. This gives: F = μk * g * (mA + mB).

Step 5: Conclude that the magnitude of the applied force F required to pull the crates at constant velocity is determined by the combined mass of the crates, the coefficient of kinetic friction, and the acceleration due to gravity. The final expression for F is: F = μk * g * (mA + mB).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Free-Body Diagram

A free-body diagram is a graphical representation used to visualize the forces acting on an object. In this scenario, each crate (A and B) will have its own free-body diagram showing the applied force, gravitational force, normal force, and frictional force. This helps in analyzing the net forces and understanding how they affect the motion of the crates.

Friction

Friction is the resistive force that opposes the motion of two surfaces in contact. The coefficient of kinetic friction (μ_k) quantifies this resistance when the surfaces are sliding against each other. In this problem, friction acts on both crates, affecting the total force required to maintain constant velocity, which is crucial for calculating the applied force F.

Newton's Second Law

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 (F = ma). In this case, since the crates are moving at constant velocity, the net force is zero, meaning the applied force F must balance the total frictional forces acting on both crates. This principle is essential for determining the magnitude of F.

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Textbook Question

A pickup truck is carrying a toolbox, but the rear gate of the truck is missing. The toolbox will slide out if it is set moving. The coefficients of kinetic friction and static friction between the box and the level bed of the truck are $0.355$ and $0.650$ , respectively. Starting from rest, what is the shortest time this truck could accelerate uniformly to $30.0$ m/s without causing the box to slide? Draw a free-body diagram of the toolbox.

Two crates connected by a rope lie on a horizontal surface (Fig. E $5.37$ ). Crate A has mass $m_A$ , and crate B has mass $m_B$ . The coefficient of kinetic friction between each crate and the surface is $μ_k$ . The crates are pulled to the right at constant velocity by a horizontal force $F$ . Draw one or more free-body diagrams to calculate the following in terms of $m_A$ , $m_B$ , and $μ_k$ : the tension in the rope connecting the blocks.

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