A marble block of mass m1=559.1 kg and a granite block of mass m2=128.4 kg are connected to ea

A marble block of mass m1=559.1 kg and a granite block of...

Key Concepts

Pulley System and Constraint Relations

In a system with a pulley connecting two masses, the motion of the bodies is linked by the constraint of the rope. Assuming the rope is inextensible and the pulley is frictionless, the acceleration of one mass is directly related to that of the other. Recognizing these constraint relations is vital for setting up the combined equations that govern the system’s dynamics.

Friction

Friction is the resistive force that opposes the motion of objects in contact. It is typically calculated by multiplying the normal force by the coefficient of friction. Understanding the role of friction, whether static or kinetic, is crucial in problems involving inclined planes, as it affects the net force and thus the acceleration of the bodies.

Newton's Second Law

This principle states that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. In problems involving connected bodies, it is used to set up equations for each object by summing the forces in the direction of motion, allowing the determination of unknown quantities like acceleration and tension in the connecting rope.

Free-Body Diagram

A free-body diagram is a visual representation of all the forces acting on an object. It is a essential tool for identifying forces such as gravity, normal force, friction, and tension. In multi-body problems, constructing free-body diagrams for each component helps in setting up the correct equations based on Newton’s laws.

Inclined Plane Mechanics

Inclined plane mechanics involves analyzing the components of gravitational force acting on a body placed on a slope. The weight of the object is resolved into components parallel and perpendicular to the plane. This decomposition is fundamental for calculating the net force that causes acceleration along the incline, as well as determining the normal force necessary for friction calculations.

Transcript

00:01 Okay, so in this exercise we have two blocks, one marble block of mass m1, and one granite block of mass m2.

00:09 These blocks are connected to each other by a road that runs over a pulley, okay? and the road glides over the pulley without friction, and the blocks are located on incline planes with angles alpha and vita.

00:26 Okay, so there is no friction over the pulley, but there is some friction.

00:30 Between the blocks and the two inclined planes.

00:34 The coefficient of friction between the block one and the client plane is mu one.

00:40 And the coefficient of friction between the block two and the inclined plane is new to.

00:45 So they asked to us what is the acceleration of the marble block, which is block number one.

00:53 So i'm going to suppose in here that these blocks are moving in this direction over here.

01:00 So they're moving like this okay and i'm going to identify all the forces that are acting on these plots so in this case we are going to have the normal force okay we are going to have the weight of the block we are going to have the tension which i'm going to denoted tension one for now we are going to have also the kinetic friction force one which is opposed to the movement so it's going to be in this direction if the block is moving in the other direction.

01:36 Okay, so in the case of the other block, we are going to have the normal force, the weight of the block, the tension, which i'm going to denote tension 2 by now, and we are going to have the kinetic friction force, which i'm going to denote like this.

01:57 Okay, so now i'm going to do the i'm going to do the free body diagrams for each of these blocks.

02:08 So in the case of the block 2, so i'm going to have this choice of free body diagram in here.

02:24 Okay, so this is going to be the block 2, okay, which is represented by this little dot in here.

02:34 And now i'm going to put all the 4.

02:36 In here so this is going to be the normal two this is sorry that looks like a w so this is going to be the normal two now the weight is going to be over here okay and because this angle over here okay like here this angle over here is going to be beta so then we are going to have that this angle over here it's going to be we are going to have also, sorry, this is not the color, the kinetic friction force in here, okay.

03:20 And we're going to have the tension, of course, tension too in here.

03:26 And now i'm going to write the equations of motions following the newton second law, okay? so for the mass two, we're going to have that the sum of the forces in x are equal to tension 2 minus the kinetic friction force minus the weight times the sign of beta okay which is this component over here this is going to be equal to the mass 2 times the acceleration 2 okay and the equations in the other axis so the sum of the forces in the y axis is going to be equal to the normal 2 minus the weight in this case is weight 2, the weight 2, cosine of beta, which is this component over here, and this is going to be equal to 0.

04:22 This is going to be equation 1, this is going to be equation 2.

04:27 And now i'm going to do the same thing, but for the other block, which is log number 1, i'm going to choose this orientation for the system of coordinates.

04:39 This is going to be the representation of the block one, which is the representation of the center of mass.

04:52 So this is going to be normal one.

04:57 There is going to be the weight in here, which is the weight one.

05:01 And because this is going to be alpha in here, this is going to be alpha in here.

05:08 This is going to be alpha in here.

05:10 This is going to be 9 -t minus alpha, so this is going to be alpha.

05:15 Okay.

05:16 So in this case, okay, so the other forces are the tension, which is over here, the tension always pulls.

05:26 So this is going to be tension one.

05:28 And there is also going to be the kinetic friction force one.

05:33 Okay.

05:34 So now we are going to grind the equation of motion for the mass tube.

05:42 Okay, i'm going to put it in here, a little bit more to the left.

05:48 And so the sum of the forces in the x -axis is going to be equal to minus tension 1, minus friction force 1.

05:59 The weight is going to be positive, the component of the weight is going to be positive and is going to be weight 1 times the sign of alpha, which is this component over here.

06:11 Sign of alpha and this is going to be m1 times the separation 1 and in the other axis we are going to have the normal one minus the weight 1 cosine of alpha and this is going to be 0 because there is no motion in this axis and we are going to name this equation 3 and this equation 4 so now we want to obtain the acceleration of the marble block which is block number 1 in this case, because there is just one rope and there is no friction over the pulley, we are going to have, we are, we, so we have, we have, we have that the acceleration, the modulus of the acceleration one is going to be equal to the modulus of the acceleration two, and this is going to be named it like this.

07:18 We have that the tension is the same, so we are going to have the detention modulus of the tension 1 is equal to the modulus of the tension 2, and this is going to be named like this.

07:31 Okay, so in this case, equation 1 is going to be tension, okay, now tension 2, and acceleration, now the acceleration 2, and the same for equation 3.

07:45 So now i'm going to obtain an expression for tension from equation 1.

07:55 So tension is going to be equal to the kinetic friction force 2, okay, plus the weight 2 times the sign of beta, i think.

08:15 This and this is going to be plus mass two times the acceleration.

08:23 Oh, sorry.

08:25 This is not tension one.

08:26 This is just tension...

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