# A block on plank with spring

• Rikudo
In summary, the conversation discusses the possibility of a block stopping and reversing its direction of motion. It is noted that the initial velocity of the block, Vo, can be any value except 0. The question is raised whether there is a range of velocities for which the block does not reverse direction. It is also mentioned that the block and plank will eventually have the same speed, regardless of the initial velocity. The conversation then considers the event of the block subsequently moving in the other direction relative to the plank, and the ways in which the block can transfer its initial kinetic energy to the plank. Finally, it is concluded that the event of the block stopping and reversing its direction relative to the plank is possible, and the conversation ends with a thank

#### Rikudo

Homework Statement
A block of mass m is placed on a plank of mass M that rests on a frictionless floor. The block is connected to an obstruction on the plank with the help of a light spring of stiffness k as shown in the figure . Coefficient of friction between the plank and the block is μ. Initially when the spring is relaxed , the block is given a velocity Vo towards the obstruction . If the block stops on the plank before reversing its direction of motion relative to the plank , find the velocity Vo . Acceleration due to gravity is g.
Relevant Equations
Momentum conservation
Work - Energy theorem (?)

I don't understand the question. If it want the mass to stop and reverse its direction, then does that not means ##vo## can be anything? (obviously not 0 since it will make the system not moving at all).

Rikudo said:
If it want the mass to stop and reverse its direction, then does that not means ##vo## can be anything?
Could there be some range of velocities for which it does not reverse direction?

Lnewqban
Consider that initial velocity Vo is measured respect to both, the plank and the floor.
By the time the block stops respect to the plank ("the block stops on the plank before reversing its direction of motion"), the plank has acquired certain velocity respect to the floor; therefore, its velocity respect to the floor must match that certain plank's velocity.

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When they are in motion, the mass's speed will decrease while the plank's will increase. Eventually, they will have the same speed (which means at this instant, the mass is not moving with respect to the plank). No matter what Vo is, they will undergo this event.

Lnewqban
Rikudo said:
No matter what Vo is, they will undergo this event.
That is for the event that the block comes to rest relative to the plank. I asked about the event that the block subsequently moves in the other direction relative to the plank.

Rikudo
Consider as well that the sliding block has two simultaneous ways to transfer part of its initial KE to the plank: constant force of friction and variable elastic force of the spring, both reducing its velocity.

Now I understand. Thank you, all of you!

Lnewqban