How Much Should a Spring Be Compressed to Launch a Brick 3.6m High?

[Dschumanji]

Homework Statement

An ingenious bricklayer builds a device for shooting bricks up to the top of the wall where he is working. He places a brick on a vertical compressed spring with force constant k=350N/m and negligible mass. When the spring is released, the brick is propelled upward. If the brick has mass 1.80kg and is to reach a maximum height of 3.6m above the initial position on the compressed spring, what distance must the spring be compressed initially?

Homework Equations

W=Fs
W_total=K_f-K_i
W_total=0.5kx_f²-0.5kx_i²

The Attempt at a Solution

I set up the problem so that everything moves in the positive y direction. To simplify everything I let the uncompressed spring have a length of d, which is the distance the spring must be compressed to satisfy the height the brick must go. The total work done on the brick is zero so the work done on the brick by the spring must be the negative of the work done by the force of gravity when the brick moves up through the air after losing contact with the spring. The spring is initially compressed so the brick starts at a height of zero; on the ground. Here is the math to figure out the value of d:

Work done by the spring on the brick
W_total=0.5kd²

Work done by the force of gravity on the brick
W=-mg(3.6-d)

0.5kd²=mg(3.6-d)

The above equation is a quadratic equation so applying the quadratic formula gives d a value of 0.55m or -0.65. The value of d must be positive so 0.55m is the only answer that makes sense.

The answer given in my book is that the spring must be compressed by 0.60m. I get that answer too only if I suppose the block has a displacement of 3.6m above the uncompressed spring. The problem states that the displacement the block undergoes is 3.6m above the compressed spring. Am I doing something wrong or is the book just wrong?

I hate Young and Freedman

 

[PhanthomJay]

The total work done on the brick is zero so the work done on the brick by the spring must be the negative of the work done by the force of gravity when the brick moves up through the air after losing contact with the spring.

Why do you say that the work done by gravity only occurs after it loses contact with the spring?

 

[Dschumanji]

Why do you say that the work done by gravity only occurs after it loses contact with the spring?

It does do work, it is just crappy wording on my part. A better way to have put it is that the majority of the net force on the brick comes from the spring during its contact with it.

After the brick loses contact with the spring the net force acting on it is only the weight.

Besides that, it doesn't really answer my question.

 

[alphysicist]

Hi Dschumanji,

I believe you misunderstood PhanthomJay's post, because he was pointing out the error in your work. In your equation for the work done by gravity, you are not including the work done by gravity while the brick is in contact with the spring.

 

[Dschumanji]

I see! I was treating the problem as if compressing a spring horizontally, in which the work done by gravity can be ignored when it decompresses. It must be included here since the spring is compressed vertically and doing work on the brick in the positive y direction.

Sorry PhanthomJay and thanks for clarification alphysicist!

I really need to pay attention more. I also need to appologize to Young and Freedman.