Energy conservation with tension

[nysnacc]

Homework Statement

Homework Equations

Conservative of energy

mg(y2-y1) +1/2 k (s²-s₀²) = 1/2 mv₁² +1/2 mv₂²

The Attempt at a Solution

v1 = 0 at rest
y2 = 0 bottom

What I got is v2 = 8.20 m/s but not correct,
I don't know how I can take into account the tension..
F_spring = -ks = -4000 N/m * 0.2 m = 800 N at position 2, isn't it? why it says 500 N?

 

[TSny]

In the formulas F = -ks and U = (1/2)ks², does s represent the length of the spring or does it represent how much the spring is stretched?

 

[nysnacc]

U is for how much the string stretches.
And the force is also, but it does not say whether the string is unstretch when it is in position 2

 

[nysnacc]

If it is of lowest tension in pos 2, then in pos 1, the tension and U will be higher than it usually is.

 

[TSny]

U is for how much the string stretches.
And the force is also, but it does not say whether the string is unstretch when it is in position 2

If the spring were unstretched in position 2 then it would exert no tension force in position 2.

 

[nysnacc]

If the spring were unstretched in position 2 then it would exert no tension force in position 2.

you're right! and would it be 0.125 m for the length when it is unstretched? because I found that T= ks => 500 N= 4000 N/m * s
SO s = 0.125 (the string is stretched for 0.125 m in pos 2)

 

[nysnacc]

Oh no, should be 0.2 - 0.125 as s₀ (unstretch)

 

[TSny]

Just to be clear what you are saying, what are your answers for the following
(1) What is the natural (unstretched) length of the spring?
(2) How much is the spring stretched beyond its natural length in postion 2?

 

[nysnacc]

(1) 0.075 m
(2) 0.125 m

and so in pos 1, the length stretched will be length of the spring - 0.075 m
Correct ? Thanks

 

[TSny]

(1) 0.075 m
(2) 0.125 m

and so in pos 1, the length stretched will be length of the spring - 0.075 m
Correct ? Thanks

Yes, that's right.

 

[nysnacc]

Would the equation be:

mgy₁ + 1/2 mv₁² + 1/2k(s₁²-s₀²) = mgy₂ + 1/2 mv₂²+ 1/2k(s₂²-s₀²)

For the bold part, will it be zero, because it is perpendicular to the motion?

 

[TSny]

Would the equation be:

mgy₁ + 1/2 mv₁² + 1/2k(s₁²-s₀²) = mgy₂ + 1/2 mv₂²+ 1/2k(s₂²-s₀²)

For the bold part, will it be zero, because it is perpendicular to the motion?

The expressions for the potential energy of the spring at positions 1 and 2 are not correct. When the spring is in position 2, the spring is stretched a certain amount from its natural length. So, the potential energy of the spring at position 2 is not zero.

The fact that the spring is perpendicular to the motion at point 2 does not affect the potential energy at that point. The force of the spring is perpendicular to the motion at position 2, which means that at that instant the spring force is not doing any work. So, the potential energy of the spring is not changing at that one instant, but the spring does have nonzero potential energy at that instant.

 

[nysnacc]

mgy₁ + 1/2 mv₁² + 1/2k(s₁²-s₀²) = mgy₂ + 1/2 mv₂²+ 1/2k(s₂²-s₀²)

So I can assume the height y2 be zero, and y1 be 0.25 m
Given v1 is 0, v2 is what we are finding.

But do we need to know the spring at unstretch state? Cuz we need the change of energy?
ΔU_weight + ΔU_spring = ΔKE
mg(y2 -y1) + 1/2 k (s2²-s1²) = 1/2 m (v2²-v1²) ??
Loss in U_weight + Loss in U_spring = Gain in KE

 

[TSny]

You are not calculating the potential energy of the spring correctly. Hooke's law is F = -kx and the potential energy of the spring is (1/2)kx² where x is the amount by which the spring is stretched from its unstretched length. If s₀ is the unstretched length of the spring and if s is the length of the spring when it is stretched, how would you express x in terms of s and s₀? Then how would you express the potential energy in terms of s and s₀,

 

[nysnacc]

You are not calculating the potential energy of the spring correctly. Hooke's law is F = -kx and the potential energy of the spring is (1/2)kx² where x is the amount by which the spring is stretched from its unstretched length. If s₀ is the unstretched length of the spring and if s is the length of the spring when it is stretched, how would you express x in terms of s and s₀? Then how would you express the potential energy in terms of s and s₀,

So I can assume the height y2 be zero, and y1 be 0.25 m
Given v1 is 0, v2 is what we are finding.

But do we need to know the spring at unstretch state? Cuz we need the change of energy?
ΔU_weight + ΔU_spring = ΔKE
mg(y2 -y1) + 1/2 k (s2²-s1²) = 1/2 m (v2²-v1²) ??
Loss in U_weight + Loss in U_spring = Gain in KE

mgy₁ + 1/2 mv₁² + 1/2k(x₁²) = mgy₂ + 1/2 mv₂²+ 1/2k(x₂²)
where x1 is length of spring in pos 1 - unstretch
x2 is length of spring in pos 2 - unstretch?

 

[nysnacc]

So x1 = 0.245 and x2 = 0.125?
where the equation now be
mg(y₂ - y₁)+ 1/2k(x₂²-x₁²) = 1/2 mv₂²

(4)(-9.81)(0-0.25) + 1/2 (-4000) (0.125²-0.245²) = 1/2 (4) v²

9.81 +2000(0.444) = 2 v²
v = 7.02 m/s

Correct?

 

[TSny]

So x1 = 0.245 and x2 = 0.125?
where the equation now be
mg(y₂ - y₁)+ 1/2k(x₂²-x₁²) = 1/2 mv₂²

(4)(-9.81)(0-0.25) + 1/2 (-4000) (0.125²-0.245²) = 1/2 (4) v²

9.81 +2000(0.444) = 2 v²
v = 7.02 m/s

Correct?

You got the right answer, but there are sign errors in your first equation above.
g and k are positive numbers.

You wrote the energy equation correctly in post #15, but you did not rearrange it correctly when you solved for 1/2 mv₂².

 

[nysnacc]

So on the left hand side, 1 - 2 then on the right hand side 1/2 m (v2^2 - v1^2), this case 2 -1 ??

 

[TSny]

So on the left hand side, 1 - 2 then on the right hand side 1/2 m (v2^2 - v1^2), this case 2 -1 ??

Yes

 

[nysnacc]

Great thanks!
I think what I was off is the calculation of x1 and x2...
I thought it was 1/2 k (S1² - S0²) + the other energy = 1/2 k (S2² - S0²) + other energy

where S0 is the un-stretched distance

 

[TSny]

Great thanks!
I think what I was off is the calculation of x1 and x2...
I thought it was 1/2 k (S1² - S0²) + the other energy = 1/2 k (S2² - S0²) + other energy

where S0 is the un-stretched distance

Right. It should be 1/2 k (S1 - S0)² rather than 1/2 k (S1² - S0²) .

 

[nysnacc]

I see so I should calculate each "stretched distance" in pos 1 and pos 2... then find the difference between them using the equation you jjust provided.

 

[TSny]

I see so I should calculate each "stretched distance" in pos 1 and pos 2... then find the difference between them using the equation you jjust provided.

I think so, if I'm understanding your statement. The key point is that U = (1/2)kx² where x is the amount the spring has been stretched from its natural length. So, you just need to do whatever is necessary to find x. If you know the natural length (s₀) of the spring, then x = s - s₀ where s is the total length of the spring.