Find the spring constant and speed?

[sonutulsiani]

Homework Statement

A cube of mass m = 5 kg is released from rest and falls a height of h1 = 2 m before hitting a vertical spring which brings the cube to a stop (momentarily) after being compressed by h2 = 50 cm.

http://bcs.whfreeman.com/WebPub/Physics/tiplerphysics6e_bridge/question_bank_images/7-44.png?11

Assume the spring has no mass, heating of the spring and the cube is negligible, and there is no air resistance.1. Which of the following statements are true for the system which includes Earth, the cube, and the spring: (There are more than 1 true answers)

A. The mechanical energy of the system is preserved.
B. The gravitational potential energy of the cube is converted into elastic potential energy of the spring at maximum compression.
C. The kinetic energy of the cube at the time of impact is equal to the elastic potential energy of the spring at maximum compression.
D. At no point during the motion of the cube will the net force on it be zero.2. Calculate the spring constant k = ____ N/m (use g = 9.81 m/s^2 and round your answer to the nearest integer).3. What is the speed of the cube at a height h3 = 20 cm below the point of impact?

A. 6 m/s.
B. 0 m/s.
C. 21 m/s.
D. 980 m/s.

Homework Equations

The Attempt at a Solution

For the first question, I think it is A and B?
I don't know about the 2nd and 3rd question. Please help

 

[rock.freak667]

For the first question, I think it is A and B?
I don't know about the 2nd and 3rd question. Please help

I believe A and B are true.

For the 2nd part, re-read what the option of B in part 1 is saying and think about what it means in terms of equations.

 

[sonutulsiani]

So the equation is W = mgh and W = 0.5 kx^2 ?
So W = 5*9.81*2 = 98.1
And W = 0.5*k*(0.5m)^2 = 0.125 k ?

Is it like this

 

[sonutulsiani]

How should I proceed further??

 

[rock.freak667]

At the initial height is point A.

The point at the spring in equilibrium is B (this is our reference line, so heights are positive or negative depending on where it is in relation to this point)

The point at maximum compression is C.

At what is the energy at point A?


At point C, what energies is present?

 

[sonutulsiani]

At point A, there is potential energy = mgh is there.
At point C, potential energy of the spring is there = 0.5 k x^2 is present and I guess mgh is also present?

 

[rock.freak667]

At point A, there is potential energy = mgh is there.
At point C, potential energy of the spring is there = 0.5 k x^2 is present and I guess mgh is also present?

Yes and yes


So by conservation of energy we will have mgh₁=0.5kx²+mgh₂

What is h₂ here knowing how we are measure the heights?

 

[sonutulsiani]

H1 = 2m
H2 = 0.5 m
So then what is x? 2-0.5?

 

[rock.freak667]

H1 = 2m
H2 = 0.5 m
So then what is x? 2-0.5?

so mg(h₁-h₂)=0.5kx₂


ok we know h₁=2 and x=0.5.


So h₁ to the bottom of the spring is 2m and the distance between B and C is 0.5m, so what is the distance of C to the bottom of the spring?

 

[sonutulsiani]

How do I find that without knowing the height of the spring?
Is it 1.5? Or 0.5?

 

[sonutulsiani]

If you take h1 as the height of the spring, which is 2, then h2 will be 1.5

 

[rock.freak667]

How do I find that without knowing the height of the spring?
Is it 1.5? Or 0.5?

If you take h1 as the height of the spring, which is 2, then h2 will be 1.5

Ignore my last post.


See how I said to take the reference point as point B?

at maximum compression, in relation to point B, h₂=-0.5m

so your equation is mgh₁=0.5kx₂
mgh₂ and they told you x=0.5m

 

[sonutulsiani]

What? I didn't understand, what is h2 again?
How is h2= -0.5

H1 = 2, x = 0.5, and the equation is mg(h1-h2) = 0.5 k x^2 right?

I understood the H1, x and the equation part. I didn't understand h2 and how do you get it.

 

[rock.freak667]

What? I didn't understand, what is h2 again?
How is h2= -0.5

H1 = 2, x = 0.5, and the equation is mg(h1-h2) = 0.5 k x^2 right?

I understood the H1, x and the equation part. I didn't understand h2 and how do you get it.

ok I probably confused you with that. Let me try to clarify.

A
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B (0 potential energy or reference point...h=0)
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C


B is the top of the spring such that AB=2m (h₁)

C is the maximum compression of the spring such that BC=0.5m


Now at A, we know E₁=mgh₁.

Meaning that relative to the point B, E₁=mg(h₁-0)

Now at maximum compression, the height relative to B is h₂= -0.5m since it is on the other side of the reference point.

So the conservation of energy equation becomes mgh₁=(1/2)kx₂+mgh₂

 

[sonutulsiani]

I get it now.
So k = 981 N/m?

And for 3rd part, is it 0 m/s?

 

[rock.freak667]

I get it now.
So k = 981 N/m?

And for 3rd part, is it 0 m/s?

:uhh: that spring looks fishy


but anyhow, for the third part it is the same process. Now instead of the right side being elastic and graviational pe, there are these two plus what type of energy?

 

[sonutulsiani]

I don't know what you said. Can you please reply soon, I have to submit in 1 hour!

 

[rock.freak667]

I don't know what you said. Can you please reply soon, I have to submit in 1 hour!

ok well it's like this at h=2m, there is mgh right?

this is converted into gravitation pe at the height 0.2m (relative to B),elastic potential energy of the spring and what other type of energy? (it is moving so what energy does it have?)

 

[sonutulsiani]

kinetic?

 

[sonutulsiani]

So how will be the equation?

 

[sonutulsiani]

mgh = 1/2 k x^2 + 1/2 m v^2?

 

[sonutulsiani]

Then what is h and x?
h= 2m
x = 0.2 m?

 

[sonutulsiani]

If the values and the equation are right, then I get answer as 5.6 m/s.. Is it right?