Two blocks are sliding to the right across a horizontal...

Two blocks are sliding to the right across a horizontal surface, as the drawing shows. In Case A the mass of each block is 3.0 $\mathrm{kg}$ . In Case $\mathrm{B}$ the mass of block 1 (the block behind) is 6.0 $\mathrm{kg}$ , and the mass of block 2 is 3.0 $\mathrm{kg}$ . No frictional force acts on block 1 in either Case A or Case B. However, a kinetic frictional force of 5.8 $\mathrm{N}$ does act on block 2 in both cases and opposes the motion. For both Case $A$ and Case $B$ determine (a) the magnitude of the forces with which the blocks push against each other and $(b)$ the magnitude of the acceleration of the blocks.

Key Concepts

Newton's Second Law

Newton's second law states that the net force acting on an object is equal to the mass of the object multiplied by its acceleration (F = ma). This principle serves as the foundation for analyzing the motion of objects when forces are applied, allowing the determination of acceleration if the net force and mass are known. It is widely used in dynamics to predict how objects will behave under various force conditions.

Kinetic Friction

Kinetic friction is the force that opposes the motion of objects sliding against each other. Its magnitude typically depends on the normal force between the two surfaces and the coefficient of kinetic friction. When analyzing dynamics problems, kinetic friction is considered as an external force that reduces the net force acting on a moving object, thereby influencing its acceleration.

Free-Body Diagram Analysis

Free-body diagrams are graphical representations that depict all the forces acting on a single object. They are crucial in dynamics as they allow the identification and isolation of forces such as gravitational, normal, frictional, and applied forces. By summing the forces depicted in a free-body diagram, one can apply Newton's laws effectively to solve for unknown quantities like acceleration or internal forces between objects.

Internal Forces and Action-Reaction Pairs

Internal forces are the forces that objects within a system exert on each other. According to Newton's third law, these forces always occur in equal magnitude and opposite directions, known as action-reaction pairs. Recognizing these pairs is essential when analyzing multi-body systems, as it helps in understanding how forces are transmitted between bodies and how they influence the overall motion of the system.

System Approach in Multi-Body Dynamics

In multi-body dynamics, analyzing the system as a whole versus individually is a key strategy. The system approach involves considering all interacting bodies together to determine the common acceleration using the net external forces acting on the system. This method simplifies the problem by neutralizing internal forces, allowing for the determination of overall motion before analyzing the internal force interactions between the individual bodies.

Transcript

00:01 For this question, we have to use newton's second law.

00:03 So let me adopt my reference frame as this one.

00:06 This is my y -axis and this is my x -axis.

00:10 Now, let us apply newton's second law to the case a for the block number one.

00:17 So we are in case a and then for block number one, x -axis, we have the following.

00:24 The net force acting in block number one in the x direction is equal to the mass of that block times its acceleration.

00:32 The net force on the x -axis that's acting block number one is composed by only one force, which is this contact force exerted by block number two on block number one.

00:42 Then, the contact force is equal to the mass of the block number one times its acceleration, which isn't now.

00:51 Now, we can apply newton second law on the y -axis.

00:57 By doing that, we get the following.

01:00 The net force, acting on the vertical direction is equal to the mass of block number one times its acceleration on the vertical direction.

01:09 Since it's not moving and will not move on the vertical direction, its acceleration is equal to zero.

01:15 Then normal number one minus weight number one is equal to zero, meaning that normal number one is equal to the weight number one.

01:25 Now we do the same for block number two.

01:28 For block number two on the x -axis we get the net -flexes, we get the net -flexes, force acting is equal to the mass of block number two times his acceleration, which is equal to the acceleration of the block number one because they are moving together.

01:45 Then the net force is composed now by two forces, the contact force and the frictional force.

01:53 So we get contact force that points to the positive direction, minus frictional force that points to the negative direction is equal to the mass of block number two times its acceleration.

02:09 Before moving further, let i correct something that i have forgot.

02:13 This force isn't positive because this is the contact force that block number two exerts on block number one and it points to the left.

02:22 So in our reference frame, it is a negative force.

02:25 So minus fc is equal to m1 times the acceleration in the x direction.

02:31 Now continuing for block number two, y direction, we get the following.

02:38 The net force acting on that block on that direction is equal to the mass of that block times its acceleration on that direction...

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