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.
is it B?
To determine the correct answer, we can use the formula for the frequency of an oscillating mass-spring system:
f = (1 / 2π) * √(k / m)
where f represents frequency, k represents the spring constant, and m represents the mass.
Given that mass A oscillates at twice the frequency of mass B, we can write the relationship as:
fA = 2 * fB
Using the above formula, we get:
(1 / 2π) * √(kA / mA) = 2 * (1 / 2π) * √(kB / mB)
Simplifying, we have:
√(kA / mA) = 2 * √(kB / mB)
Squaring both sides gives us:
kA / mA = 4 * (kB / mB)
Rearranging the equation, we find:
kA / kB = (4 * mB) / mA
This expression shows the relationship between the spring constants kA and kB. Now, let's evaluate the given answer choices:
A. The spring constant for B is one quarter of the spring constant for A.
This is incorrect because the expression above shows that the ratio is dependent on the mass ratio, not the inverse.
B. The spring constant for B is 4 times the spring constant for A.
This is also incorrect because the expression above includes a factor of (4 * mB) / mA, so the correct answer cannot be simply 4 times the spring constant of A.
C. The spring constant for B is half of the spring constant for A.
This is also incorrect because the factor in the expression is dependent on the masses, not just a simple halving.
D. The spring constant for B is 1.41 times the spring constant for A.
This is incorrect as well because 1.41 is an approximation of the square root of 2, which is not the factor in the expression.
E. The spring constant for B is twice the spring constant for A.
This answer choice matches the expression above, which states that the ratio of the spring constants is dependent on the ratio of the masses. Therefore, the correct answer is E.
To summarize, the correct statement is "The spring constant for B is twice the spring constant for A."