As shown in the figure, a block of mass M1=0.450 kg is initially at rest on a slab of mass M2=0

As shown in the figure, a block of mass M1=0.450 kg is...

As shown in the figure, a block of mass $M_{1}=0.450 \mathrm{~kg}$ is initially at rest on a slab of mass $M_{2}=0.820 \mathrm{~kg}$, and the slab is initially at rest on a level table. A string of negligible mass is connected to the slab, runs over a frictionless pulley on the edge of the table, and is attached to a hanging mass $M_{3}$. The block rests on the slab but is not tied to the string, so friction provides the only horizontal force on the block. The slab has a coefficient of kinetic friction $\mu_{\mathrm{k}}=0.340$ and a coefficient of static friction $\mu_{\mathrm{s}}=0.560$ with both the table and the block. When released, $M_{3}$ pulls on the string and accelerates the slab, which accelerates the block. Find the maximum mass of $M_{3}$ that allows the block to accelerate with the slab, without sliding on top of the slab.

Key Concepts

System of Connected Masses

In mechanics, analyzing systems of connected masses, such as those linked by strings or pulleys, requires understanding the constraints imposed by the connections. The motion of one mass directly impacts the motion of another, and the relationships between tensions, accelerations, and forces must be carefully established to solve the system as a whole.

Relative Motion in Accelerated Frames

When multiple objects accelerate together, examining their relative motion is essential. Considering the problem from a non-inertial reference frame (attached to one of the accelerating bodies) allows for the introduction of pseudo-forces that must be balanced by friction or other forces. This analysis clarifies the conditions under which objects remain in contact without slipping or relative motion.

Free Body Diagrams

Free body diagrams (FBDs) are graphical tools used to represent all the forces acting on an individual object. In problems with multiple interacting bodies, FBDs help in clearly identifying forces such as weight, normal force, friction, tension, and applied forces, which is critical for correctly applying Newton's laws and solving for unknown quantities.

Static and Kinetic Friction

Friction is the force that opposes relative motion between two surfaces in contact. Static friction prevents the onset of motion and has a maximum value, which is crucial in problems where objects must move together without slipping. Once motion begins, kinetic friction takes over, typically with a lower coefficient. Recognizing which type of friction applies is key in analyzing when slipping occurs and in determining the maximum force an interface can sustain.

Newton's Second Law

Newton's Second Law establishes that the net force acting on an object equals the mass of the object times its acceleration. This principle is fundamental in determining how forces such as gravity, tension, and friction affect the acceleration of each mass in a system, and it is essential for setting up the equations of motion in problems involving interconnected bodies.

Transcript

00:01 In this exercise we have a block of mass m1, which is initially addressed on a slab of mass m2.

00:11 And the slab is also initially addressed on a level table.

00:17 We have a string of negligible mass which is connected to the slab, okay? and this string runs over frictionless fooling on the edge of the table.

00:29 And is attached to a hanging mass of mass m3.

00:35 The slab has a coefficient of a kinetic friction which we are going to denote by muu su bork, which has this value, okay? and a coefficient of a static friction, which we are going to denote muu suf s, which has this other value in here, with both the table and the block.

00:58 When the system is released, m3, this mass over here, pulls on the stream and accelerates the slab, which obviously accelerates the block.

01:13 And so we need to find the maximum mass of m3, sorry about this, this is m3 in here of m3, that allows the block to accelerate with the slab without sliding on the top.

01:29 Without sliding on top of the slab.

01:33 So the first we are going to do is to define the direction of motion, which is going to be this one.

01:40 Okay, so this is going to be the direction of motion of this third block and this is going to be the direction of motion of these two blocks if they move together.

01:53 So now we are going to now we're going to do the free body diaries for each of this mass.

02:04 So for the mass m1, we are going to have this free body diagram.

02:15 So this is going to be mass, the mass m1, okay? and now we are going to have the normal force, which is a force, okay, that fills the mass one because of the mass two.

02:31 Okay, so this is going to be the normal.

02:35 And we also are going to have the weight of the mass one.

02:41 And we are going to have the static friction force, which is pointing in this direction.

02:49 Okay.

02:51 Because if there is no static friction force, the mass m2 is going to move to this direction.

03:01 Okay, so this is the velocity of the mass.

03:03 Too and because of the inertia of this other mass the velocity of the mass one is going to be in this direction okay so in order to avoid this movement there is we need that the static friction force is pointing in this direction okay so now we are going to write the equation of motion for this mass.

03:41 So we're going to have, according to newton's second law, in the x -axis, only the static friction force, which is going to be equal to the mass one times the acceleration one.

03:56 And we are going to have this other equation for the y -axis.

04:06 Okay, so this is going to be weight 1.

04:10 So now for the mass 3, okay, which is also a very easy free body diagram.

04:22 So suppose this is going to be m3 over here.

04:25 So we are going to have tension, which i'm going to denote it by tension 3, and the weight of the mass.

04:36 So there is nothing more so the equation of most.

04:40 Is going to be in the y -axis and it's going to be this one over here and this is going to be m3 times acceleration 3 which which with a minus okay because and the acceleration is in the um it's in the negative x -axis okay so now we are going to do the same thing for the mass 2 okay so we have, of course, the normal force that feels the mass too because of the table.

05:27 We are going to have its weight.

05:31 We also are going to have the friction kinetic force in this direction over here, which is opposed to the movement.

05:41 And we are going to have the tension, which i'm going to denoted by tension 2.

05:47 And there are going to be two more forces that are present because of the law of action and reaction.

05:58 These forces are the static friction force, which is pointing now to the left.

06:07 And there is also going to be the normal force that feels this block.

06:18 Because of the mass one.

06:23 So now we can write the equation of motion of this block in here...

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