Blocks A and B each have a mass m. Determine the largest horizontal force 𝐏 which can be applied

Blocks A and B each have a mass m. Determine the largest...

Key Concepts

Static Friction

Static friction is the force that resists the initiation of motion between two surfaces in contact. It has a maximum value determined by the normal force between the surfaces and the coefficient of static friction. Understanding static friction is crucial for analyzing conditions where one object must not slip against another even when external forces are applied.

Newton's Second Law

Newton's Second Law of motion states that the net force acting on an object is equal to the mass of the object times its acceleration. This principle is fundamental when determining the acceleration of systems and helps relate the applied force to the resulting motion, thereby playing a key role when dealing with problems involving multiple objects and applied forces.

Relative Motion and the No-slip Condition

The no-slip condition is a scenario where two objects in contact move together without any relative motion between them. This situation requires that the acceleration of the object on top does not exceed the maximum acceleration that static friction can sustain. Analyzing relative motion under this condition ensures that all parts of the system accelerate uniformly, preventing slipping.

Free-body Diagram Analysis

Free-body diagram analysis involves sketching all the forces acting on each component of a system. This method is essential for breaking down complex interactions into simpler parts by visually representing forces such as gravity, normal force, friction, and any applied forces. It is a critical step in systematically applying physical laws to solve for unknown quantities in mechanical problems.

Transcript

00:01 In this problem, we're going to be talking about force and acceleration, and the concepts we need in order to solve it are first newton's second law, which tells us that the sum of all the forces that act on a body, which we call the net force is equal to the mass of the body times the acceleration.

00:32 We also need to know that the friction force is equal.

00:36 To the coefficient of friction.

00:40 In our case, we're going to need the coefficient of static friction times the normal force.

00:49 Okay, so we are ready to dive into the exercise.

00:53 We have the system that's shown here, the figure.

00:56 We have to block, say, and b, both of them have mass m, and block b, both block b and block a are shaped in the form of right triangle.

01:09 And block b makes an angle of theta, the triangle, make the angle of theta, as shown in the picture.

01:17 And a force, p, is exerted on the right side of block b, such as that block b and block a, they travel together.

01:27 They have the same acceleration.

01:31 We have the information that the coefficient of static friction between the two blocks is mu -s, and are going to find what is the maximum value of p such that the blocks will still stick together.

01:49 In order to solve the exercise, we need to draw the free -body diagram for both blocks.

01:56 So we have first the a block, and the forces that act on this block are, like, where gravitational force, mg, we also have the normal.

02:21 And prime, that is the force that block b exerts on block a.

02:29 Okay.

02:31 And we also have the friction force.

02:34 Notice that block a would tend to slide in this direction.

02:42 So this means that the frictional force will be pointing in the opposite direction.

02:55 And now we can draw the force diagram on block b.

03:02 So if this is block b, mg points downwards.

03:10 The normal force that block a exerts in block b and prime has the same magnitude as the force that block b exerts in block a according to a newton's second law.

03:23 And the opposite direction.

03:25 We have the normal force that the ground exerts in block b and we have p.

03:31 Okay.

03:34 So now we can write the equations of motion for.

03:39 Both blocks.

03:42 First for block a in the y direction we have, remember that this angle here is theta, which means that this angle here, let me just write a little better, this angle here is also theta, which means that n prime, i'm going to use, i'm going to choose a coordinate system such the y -axis point sub -words and the x -axis, points to the left, we have m times the sign of theta.

04:33 I'm sorry, actually we should start with the y direction.

04:36 So we have n times the cosine of theta minus mg minus the frictional force times the sign of theta is equal to zero since the blocks are not moving in the y direction.

04:59 The x direction we have n prime times sine of theta plus the fictional force times cosign of theta and this is the mass times the acceleration.

05:19 Okay.

05:21 Then for block b we have again this here is theta.

05:32 Let me write it here actually it will be a little so for the y coordinate we have minus n prime times the cosine of theta minus m g plus n and force of course that points in this direction i'd forget them to write plus the force times v sine of theta and this is equal to zero.

06:24 In the x coordinate, we have minus n prime times sine of theta plus theta plus, plus sine of theta, is equal to mg.

06:41 Okay, so what i'm gonna do is try to calculate what is the maximum value of p...

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