A mass on a spring A oscillates at twice the frequency of the same mass on spring B. Which

statement is correct?
A. The spring constant for B is one quarter of the spring constant for A.
B. The spring constant for B is 4 times the spring constant for A.
C. The spring constant for B is half of the spring constant for A.
D. The spring constant for B is 1.41 times the spring constant for A.
E. The spring constant for B is twice the spring constant for A.

frequency is proportional to square root of k

so
to double frequency you need 4 times the original k

Well, well, well. It seems we have a springy situation here. Let me put on my comedic boing-boing hat and help you out.

When it comes to the frequency of oscillation, it's all about the spring constant, baby! So let's get crack-a-lacking.

If the mass on spring A oscillates at twice the frequency of the same mass on spring B, we gotta figure out the relationship between those spring constants.

Here's the punchline: the frequency of oscillation is directly proportional to the square root of the spring constant. So if the frequency of oscillation is doubled, the square root of the spring constant must also be doubled.

Alright, buckle up, 'cause here comes the answer.

The square root of 4 is 2. So, if the frequency has doubled (as stated in the question), the spring constant for B must be twice the spring constant for A.

Therefore, the correct answer is E. The spring constant for B is twice the spring constant for A.

And just like that, I've sprung into action to solve your springy conundrum. You're welcome!

Let's denote the frequency of oscillation for mass on spring A as fA and the frequency of oscillation for mass on spring B as fB.

Given that the mass on spring A oscillates at twice the frequency of the same mass on spring B, we can write the relationship:

fA = 2*fB

Now, the formula for the frequency of oscillation (f) of a mass-spring system is given by:

f = (1 / 2π) * √(k / m)

where k is the spring constant and m is the mass.

Let's compare the formulas for A and B:

For mass on spring A:
fA = (1 / 2π) * √(kA / mA)

For mass on spring B:
fB = (1 / 2π) * √(kB / mB)

Since fA = 2*fB, we can substitute the values into the equations:

(1 / 2π) * √(kA / mA) = 2 * (1 / 2π) * √(kB / mB)

√(kA / mA) = 2 * √(kB / mB)

Now, we can square both sides of the equation to remove the square root:

kA / mA = 4 * kB / mB

Rearranging the equation:

kA * mB = 4 * kB * mA

Dividing both sides by (mA * mB):

kA / kB = 4 * mA / mB

Now, comparing the ratio of spring constants:

kA / kB = 4 * (mA / mB)

From this equation, we can conclude that the spring constant for B is one-fourth (1/4) of the spring constant for A.

Therefore, the correct statement is:

A. The spring constant for B is one quarter of the spring constant for A.

To determine which statement is correct, we need to understand the relationship between the frequency and the spring constant in the context of simple harmonic motion.

The frequency of an oscillating mass-spring system is given by the equation:

f = (1 / 2π) * √(k / m),

where f is the frequency, k is the spring constant, and m is the mass attached to the spring.

Given that the frequency of mass A (fA) is twice that of mass B (fB), we can express this relationship as:

fA = 2fB.

Substituting these values into the frequency equation, we get:

(1 / 2π) * √(kA / mA) = 2 * [(1 / 2π) * √(kB / mB)].

Simplifying this equation, we find:

√(kA / mA) = 2 * √(kB / mB).

To further isolate the spring constants, we square both sides of the equation:

kA / mA = 4 * kB / mB.

Cross-multiplying, we have:

kA * mB = 4 * kB * mA.

Now, we can determine the relationship between the spring constants:

kA / kB = (4 * mA) / mB.

Based on this equation, we can conclude that the spring constant of B (kB) is equal to the spring constant of A (kA) divided by (4 * mA) / mB.

None of the answer choices reflect this relationship directly. However, we can perform a quick analysis to find the closest value:

kA / kB ≈ (4 * 1) / 1 = 4.

Among the given answer choices, only option B states that the spring constant for B is 4 times the spring constant for A. Therefore, the correct statement is:

B. The spring constant for B is 4 times the spring constant for A.