Question

In an under damped single degree of freedom system, the ratio of the first peak amplitude to the seventh peak amplitude is \[ \frac{x_1}{x_7}=64. \] Then, the ratio of two successive peak amplitudes is

• A. \(1.26\)
• B. \(1.50\)
• C. \(2.00\)
• D. \(6.40\)

Hint:

If \[ \boxed{ \frac{x_1}{x_{n+1}}=R, } \] then the ratio of two successive amplitudes is \[ \boxed{ r=R^{1/n}. } \]

Answer 1

The Correct Option is C

Step 1: Recall the logarithmic decrement relation. For an underdamped system, the ratio of amplitudes separated by \(n\) cycles is \[ \boxed{ \frac{x_1}{x_{n+1}}=r^n, } \] where \[ r \] is the ratio of two successive peak amplitudes.

Step 2: Substitute the given values. Given, \[ \frac{x_1}{x_7}=64. \] Since there are \[ 6 \] intervals between the first and seventh peaks, \[ r^6=64. \] Therefore, \[ r = 64^{1/6} = (2^6)^{1/6} = 2. \] Hence, \[ \boxed{2.00} \] is the correct answer. Thus, \[ \boxed{(C)} \] is the correct answer.