EASY question: An object of mass M suspended from a spring vibrates with a frequency f. A second?

Question

An object of mass M suspended from a spring vibrates with a frequency f. A second mass B is added to mass M. The system now vibrates with a frequency f/2. The second mass is:

A. 4M

B. 3M (correct answer)

C. 2M

D. M

I don't know how to get this answer. Do you use...

f = (1/(2*pi)) * (K/M)^(0.5)

please how you got that. kthx

Answer 1

First, don't forget the 2pi, the equation you are using is correct because w = 2pi f, which is the angular velocity.  And the derivation of your equation comes from angular kinetic energy, which uses w as in KE = 1/2 Iw^2.  So to be consistent, the 2pi, as in 2pi f = w, is required.

The reason 2pi in or out gives the same result is because it cancels out when doing a ratio as in F = (1/2pi)(K/M)^1/2 and f = F/2 = (1/2pi)(K/M')^1/2.  When comparing different values of the same factors, the best approach is to use a ratio.  For example,

F/f = F//F/2 = 2 = (K/M)^1/2//(K/M')^1/2; so that 4 = M'/M and M' = 4M, which says the mass M' when the frequency is f = F/2 must be 4 times the mass M when the frequency is F.  Thus, M' = 4M = M + 3M and the "second mass" is 3M or three times the first mass.

The real lesson here is that using ratios to solve problems where the same factors are used but at different values is usually the best approach.

Answer 2

Forget about the 1/2pi. We just express frequency in radians per second, so;

w = (K/M)^0.5

Where;

w = frequency

K = spring constant

M = the original mass

Mf = the combined mass

B = the added mass

w = the original frequency

w/2 = the frequency with the combined mass

(1). Mf = M + B

(2). (w)^2 = K/M

(3). (w/2)^2 = K/Mf

Dividing Eq. (2) by (3);

1/(1/4) = 1/MMf

Mf = 4M

B = 4M - M = 3M

Answer 3

USE:

frequency= 1/ ( 2Pi * SQRT( M/K) )

yes 3M +1 M = 4M

with sqrt,, sqrt 4= 2 , so the frequency would be halfed

Answer 4

Correct eqn. To halve the frequency, you have to increase the mass by a factor of 4; i.e. M becomes 4M = M + 3M