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

Two crates connected by a rope lie on a horizontal surface (Fig. 5.55$) .$ 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 $\vec{F} .$ In terms of $m_{A}, m_{B},$ and $\mu_{k}$ , calculate (a) the magnitude of the force $\overrightarrow{\boldsymbol{F}}$ and $(\mathrm{b})$ the tension in the rope connecting the blocks. Include the free-body diagram or diagrams you used to determine each answer.

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

0:00 Hi there.

00:01 So for this problem, we have two crates connected by a rope.

00:07 And we, the information given is the mass of the crate a, the mass of the crate b, and the coefficient of kinetic friction, musica.

00:19 And the first thing that we are asked is to determine the magnitude of the force f that is being applied to these crates.

00:30 So to do that, the first thing to do is to draw the diagram of forces that are adding up these two bodies, and from there, obtain equations that allow us to determine that magnitude for the force f.

00:48 So in this first part, we're going to obtain the magnitude of the force f.

01:00 For the first grade, grade a, we have the following forces.

01:12 We know that there is a normal force.

01:16 I'm going to call it normal force a.

01:20 Remember that this is y and this is the x -a that is perpendicular to the surface, that normal force.

01:26 We also have the weight of the grade, which we call w -a.

01:31 And we also have attention in here is positive, as you can see in the picture.

01:39 And we also have a friction force that opposes the direction of motion.

01:47 Since the system is moving to the right, the friction force should be in a position to that.

01:54 And we are calling this friction force a.

01:57 Now, for the second crate or the crate b, we have, again, a normal force perpendicular to the surface.

02:08 In this case, we call nb.

02:10 Again, this is the y and this is the x assets.

02:15 In this case, we have again a weight b, the weight of the crate.

02:21 And we can see that it's been inserted at force f.

02:27 To the right so that we have a tension in the negative x component...

Please give Ace some feedback