Is it possible to combine two springs with different constants?

[Fontseeker]

Homework Statement

Homework Equations

T = 2 * pi * (m / k)^(1/2)

The Attempt at a Solution

I was thinking of treating both springs as a single system and adding their constants to be 3k (90 N/m), would this thought be incorrect?

 

[Charles Link]

This one can be solved with a little algebra. Apply a force $F$ to the system. How far does each spring stretch? Add the two distances stretched, and determine the spring constant.

 

[Fontseeker]

This one can be solved with a little algebra. Apply a force $F$ to the system. How far does each spring stretch? Add the two distances stretched, and determine the spring constant.

So if I apply a force F, then I would have an additional unknown

Net force = (F) - (kx) - (2k * x)

 

[tnich]

Homework Statement

View attachment 225096

Homework Equations

T = 2 * pi * (m / k)^(1/2)

The Attempt at a Solution

I was thinking of treating both springs as a single system and adding their constants to be 3k (90 N/m), would this thought be incorrect?

If you apply a downward force F to the to the bottom of the lower spring, what is the tension on the lower spring? On the upper spring? How much does each spring stretch under its tension? What is the total stretch of both springs?

 

[Fontseeker]

If you apply a downward force F to the to the bottom of the lower spring, what is the tension on the lower spring? On the upper spring? How much does each spring stretch under its tension? What is the total stretch of both springs?

Wouldn't the force applied be the same for both the lower and upper spring? And if that assumption is correct, it would be stretched x = F / (2k) for the lower spring and x = F / (k)

 

[Charles Link]

You should use the letter $x$ with subscript 1 for spring 1 and subscript 2 for spring 2. Then you can write $x_{total}=x_1+x_2$. In any case, you are on the right track. $\\$ Also, the $k's$ really should be distinguished in a similar way. Call them $k_1$, $k_2$, and $k_{total}$. You don't need to put in any numbers until after you get an algebraic expression for $k_{total}$.

 

[tnich]

Wouldn't the force applied be the same for both the lower and upper spring? And if that assumption is correct, it would be stretched x = F / (2k) for the lower spring and x = F / (k)

OK. Now find the total stretch and you will have everything you need to calculate the force per unit stretch.

 

[Fontseeker]

You should use the letter $x$ with subscript 1 for spring 1 and subscript 2 for spring 2. Then you can write $x_{total}=x_1+x_2$. In any case, you are on the right track. $\\$ Also, the $k's$ really should be distinguished in a similar way.

An addition of x_1 and x_2 would give me x_{total} = (2F)/(3k). So k = (2F) / (x_{total}). Should I use conservation of energy to find the total stretch?

 

[tnich]

An addition of x_1 and x_2 would give me x_{total} = (2F)/(3k). So k = (2F) / (x_{total}). Should I use conservation of energy to find the total stretch?

You need to check your addition.

 

[Fontseeker]

You need to check your addition.

xtotal=x1+x2 = [F / (k)] + [F / (2k)] = (2F/3k)

 

[Charles Link]

xtotal=x1+x2 = [F / (k)] + [F / (2k)] = (2F/3k)

Try again. It's incorrect.

 

[Fontseeker]

Try again. It's incorrect.

Are my equations for x1 (F / k) and x2 (F / 2k) correct? Not sure what I am doing incorrectly or how to get rid of the F.

 

[tnich]

Are my equations for x1 (F / k) and x2 (F / 2k) correct? Not sure what I am doing incorrectly or how to get rid of the F.

You can't just add the numerators together and add the denominators together. You have to get the denominators to agree and then add the numerators.

 

[Charles Link]

Are my equations for x1 (F / k) and x2 (F / 2k) correct? Not sure what I am doing incorrectly or how to get rid of the F.

Yes. So far, so good. Factor out $\frac{F}{k}$ and then add the two numbers together. You don't get 2/3 when you do that.

 

[Fontseeker]

You can't just add the numerators together and add the denominators together. You have to get the denominators to agree and then add the numerators.

I am dumb, xtotal = (3F / 2k)

 

[tnich]

I am dumb, xtotal = (3F / 2k)

Right. Now, what does the spring constant k (N/m) represent?

 

[Charles Link]

Now, you should be able to solve for $k_{total}$ if you can write the expression relating $F, k_{total}$, and $x_{total}$. How are $F, k_{total}$, and $x_{total}$ related. (In this case the applied $F$ is opposite the force from the spring, so there is no minus sign).

 

[Fontseeker]

Now, you should be able to solve for $k_{total}$ if you can write the expression relating $F, k_{total}$, and $x_{total}$.

So:
F = k_t * x_t = k_t * [(3F) / (2k)]
k_t = F / [(3F) / (2k)] = (F * 2k) = (3F) = (2/3) * k = (2/3) * 30 = 20 N/m?

 

[tnich]

So:
F = k_t * x_t = k_t * [(3F) / (2k)]
k_t = F / [(3F) / (2k)] = (F * 2k) = (3F) = (2/3) * k = (2/3) * 30 = 20 N/m?

Yes.

 

[Charles Link]

So:
F = k_t * x_t = k_t * [(3F) / (2k)]
k_t = F / [(3F) / (2k)] = (F * 2k) = (3F) = (2/3) * k = (2/3) * 30 = 20 N/m?

Your 3rd equal sign needs to be a "/" , (division sign), but otherwise correct.

 

[Charles Link]

@Fontseeker Incidentally, when two or more springs are placed side by side, that's when the spring constants add.

 

[Fontseeker]

Incidentally, when two or more springs are placed side by side, that's when the spring constants add.

Wow, how convenient it would have been if it was horizontal.

 

[jishnu]

Homework Statement

View attachment 225096

Homework Equations

T = 2 * pi * (m / k)^(1/2)

The Attempt at a Solution

I was thinking of treating both springs as a single system and adding their constants to be 3k (90 N/m), would this thought be incorrect?

This can be easily solved using the equation for series combination of springs, which indeed can be derived using the extension of both the springs
ie;
extension in spring 1(top)=x1
extension in spring 2(bottom)=x2
and
x1 =F /k1
x2 =F /k2
Using Hooke's law
Now the total extension
x = x1 + x2
x = F / k_eq

F / k_eq = F /k1 + F /k2
from which we get,

F1/ k_eq = 1 /k1 + 1/k2
This equation easily solves your problem
Hope that helps... [emoji4]