Block A has a mass mA and is attached to a spring having a...

Block $A$ has a mass $m_{A}$ and is attached to a spring having a stiffiness $k$ and unstretched length $l_{0-}$ If another block $B$, having a mass $m_{B}$, is pressed against $A$ so that the spring deforms a distance $d,$ show that for separation to occur it is necessary that $d>2 \mu_{k} g\left(m_{A}+m_{B}\right) / k,$ where $\mu_{k}$ is the coefficient of kinetic friction between the blocks and the ground. Also, what is the distance the blocks slide on the surface before they separate?

Key Concepts

Separation Conditions

The separation condition sets a threshold for the deformation of the spring beyond which the blocks will separate. This involves a balance between the restoring force of the spring and the resisting force due to kinetic friction, with the derived inequality ensuring that the stored energy is sufficient to overcome the frictional forces acting on both blocks.

Work-Energy Principle

The work-energy principle involves the transformation of stored potential energy in the spring into the kinetic energy of the masses, while accounting for energy losses due to friction. This concept is important for determining how much energy is available to overcome friction and for calculating the sliding distance before separation occurs.

Hooke's Law

Hooke's Law states that the force exerted by a spring is proportional to its displacement from equilibrium, expressed as F = -kx. This fundamental concept is essential for understanding how the spring stores potential energy when compressed and how it subsequently exerts a restoring force on the blocks.

Newton’s Second Law and Dynamics

Newton’s Second Law, which relates force, mass, and acceleration (F = ma), provides the basis for analyzing the motion of the blocks. It is used to determine how the net forces — including the spring force and friction — affect the acceleration and motion of each block after the spring is deformed and released.

Kinetic Friction

Kinetic friction, characterized by the coefficient ?_k and the normal force, opposes the sliding motion of objects in contact with a surface. In problems like this one, kinetic friction plays a key role in dissipating energy and affecting the acceleration of the blocks, thereby influencing the conditions required for the blocks to separate.

Transcript

00:01 In this exercise, we have a spring that has a spring constant k and an unstretched length l0 that's connected to a block a that has a mass ma, which is close to a block b that has a mass mb.

00:16 Now, block b is pressed against block a such that the spring is compressed by a distance d, as shown here in figure, and then they are released.

00:28 Are going to show that in order for block a and block b to be separated after some time, we need that d be greater than two times the sum of the masses of the blocks times mu k times g divided by k.

00:48 And actually, i'm sorry, there shouldn't be a d here.

00:54 And also, we need to find what is the distance that the blocks slide on the surface before, they separate.

01:06 In the formula for d, mu k is the kinetic coefficient friction.

01:16 Kamps connect kinetic friction coefficient.

01:23 Okay, what we need to know in order to solve the exercise is that the elastic force f is equal to minus k times delta x.

01:35 Delta x is the difference between the unstretched length of the the spring and the compressed length or the stretched length.

01:48 And the minus sign is to show that when the spring is compressed, the force points in the opposite direction as the compression.

01:57 When the spring is stretched, the force points in the opposite direction as well.

02:03 Okay, so what we need to do first is to draw the free body diagram on the blocks.

02:11 So let's start by block a.

02:16 Block a subject to the forces of the elastic force, f.

02:29 We also have the friction force, ff, and we also have the force that block b exerts in block a, which i'm going to call n.

02:45 Okay.

02:46 And according to newton's second law, we have that f minus n minus f is equal to the math.

02:55 Of block a times the acceleration.

03:00 Now the force, which now is the magnitude of the elastic force, is k times d minus x, where x is the distance traveled by the block.

03:17 So let me draw here the spring that has an unstretched length l0.

03:27 It's compressed by distance d.

03:30 Okay this is block a now consider that actually block a traveled a certain distance x after being compressed so block a traveled this distance here okay notice that this distance is equal to d minus x so that the force that block a experiences is equal to k times d minus x so going back to the equation, we have k times d minus x minus n minus a frictional force, which is mu k times the normal force that the surface exerts on the block a.

04:32 The normal force is just mag in this situation.

04:39 So this is the first equation that we obtain.

04:47 And now we can carry on to block b.

04:51 So for block b we have the force n.

04:59 We also have the frictional force, and that is all we have...

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