Two crates connected by a rope lie on a horizontal surface (Fig. E5.37). Crate A has mass mA, an

Two crates connected by a rope lie on a horizontal surface...

Two crates connected by a rope lie on a horizontal surface ($\textbf{Fig. E5.37}$). 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 $\mu_k$. 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 $\mu_k$: (a) the magnitude of $F$ and (b) the tension in the rope connecting the blocks. Figure e5.37(Figure Cant copy)

Key Concepts

Newton's Laws (Equilibrium Concept)

Newton's laws, particularly the first and second laws, underlie the analysis of objects moving at constant velocity. When an object moves at constant velocity, the net force acting on it is zero, meaning that all the forces (such as applied force, friction, and tension) balance each other. This principle is key in setting up the force equations for both crates.

Free-Body Diagrams

Free-body diagrams are essential tools in mechanics that involve drawing a simplified representation of an object and all the forces acting on it. They help in visualizing forces such as gravity, normal force, applied force, friction, and tension, allowing for the systematic setup of equilibrium equations to solve problems involving multiple forces.

Kinetic Friction

Kinetic friction is the force that opposes the relative sliding motion between surfaces in contact. It is calculated as the product of the coefficient of kinetic friction and the normal force acting on the object. In problems of constant motion, this resistive force must be overcome by the applied force, making its correct assessment crucial for accurate calculations.

Tension in a Rope

Tension is the force transmitted through a rope, string, or cable when forces are applied at both ends. In problems involving connected objects, such as crates, the tension determines how force is distributed between the objects. Analyzing tension is critical for setting up equations for each object along the rope and ensuring that the net forces result in constant velocity motion.

Transcript

00:01 In part a, we want to figure out what the applied forces on these crates so that we get a constant velocity.

00:07 So let's go ahead and draw the combo system force diagram.

00:13 And so we have our mass here.

00:15 In this case, it's ma plus mb, because this is the combo system.

00:20 We have the normal force up.

00:23 We've got gravity down, which i'm going to label the mass by m total, where m total is equal to mass of a plus mass of b.

00:34 I have our applied force, which i'm just going to call f, and it's to the right.

00:38 And then friction force is to the left, which i'm going to indicate by a subscript little f there.

00:45 Now, by newton's second law in the y direction, we get immediately that the normal force is equal to, i forgot my g here, it's equal to m total times g, because it's not accelerating in the y direction.

00:58 This means that the friction force is equal to m total times g times me a k.

01:04 Since the friction force is just this coefficient times the normal force.

01:09 Now, the crate in the horizontal direction is moving at constant velocity, which means its acceleration is zero, which means that the sum of the forces in the x direction is equal to zero.

01:21 This equation here implies that the applied force is equal to the friction force, which we found was equal to the total mass times g times the coefficient of kinetic friction.

01:38 And so that's the answer to part a.

01:44 In part b, i'm just going to do block b...

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