Physics · Oscillations and Simple Harmonic Motion

The correct answer is A) 2πxo/ao. In simple harmonic motion (SHM), the maximum velocity (vmax) is related to the amplitude (xo) and angular frequency (ω) as vmax = ωxo. The period of oscillation (T) is described by the equation T = 2π/ω. By substituting the expression for ω from vmax = ωxo, we get T = 2π/(vmax/xo) = 2πxo/vmax. Since the maximum velocity (vmax) is given as ao, we substitute it into the equation to get T = 2πxo/ao. Other options do not correctly reflect this relationship: B) inverts the placement of xo and ao, C) incorrectly multiplies ao and xo, and D) suggests dividing by both ao and xo, none of which mathematically represent the period of SHM.

The correct answer is A) 2πx_o/a_o.

In simple harmonic motion (SHM), the maximum velocity (vmax) is related to the amplitude (xo) and angular frequency (ω) as v_max = ωx_o.

The period of oscillation (T) is defined as the time taken for one complete cycle of the motion. It is related to the angular frequency as T = 2π/ω.

To find the relationship between the maximum velocity (vmax) and the period (T), we can substitute vmax = ωx_o into the equation T = 2π/ω:

T = 2π/ω = 2π/(vmax/x_o) = 2πx_o/vmax.

Given that the maximum velocity (vmax) is a_o, the equation becomes:

T = 2πx_o/a_o.

Incorrect. This option incorrectly places ao in the numerator and xo in the denominator, which does not match the derived formula for the period T = 2πx_o/a_o.

Incorrect. This option multiplies ao and xo, which is not consistent with the formula for the period. The correct relationship involves dividing xo by ao.

Incorrect. This option suggests dividing by both ao and xo, which does not reflect the correct formula for the period T = 2πxo/ao.