Solve the problem.
-A pendulum of length $\mathrm { L }$, when displaced horizontally and released, oscillates with harmonic motion according to the equation $y = A \sin ( ( \sqrt { g / L } ) t + \pi / 2 )$, where $y$ is the distance in meters from the rest position $t$ seconds after release, and $g = 9.8 \mathrm {~m} / \mathrm { s } ^ { 2 }$. Identify the period, amplitude, and phase shift when $\mathrm { A } = 0.10 \mathrm {~m}$ and $\mathrm { L } = 0.98 \mathrm {~m}$.
A) $0.99 \mathrm {~s} , 0.10 \mathrm {~m} , - 0.99 \mathrm {~s}$
B) $1.99 \mathrm {~s} , 0.10 \mathrm {~m} , - 0.50 \mathrm {~s}$
C) $0.63 \mathrm {~s} , 0.10 \mathrm {~m} , - 0.157 \mathrm {~s}$
D) $1.99 \mathrm {~s} , 0.10 \mathrm {~m} , 0.50 \mathrm {~s}$

Question

Quizplus · Accepted Answer

The period $T$ of a pendulum is given by $T = 2\pi\sqrt{L/g}$. Substituting $L = 0.98 \mathrm {~m}$ and $g = 9.8 \mathrm {~m/s^2}$, we find $T = 2\pi\sqrt{0.98/9.8} = 2\pi$, which is approximately $6.28$ seconds. However, the equation's form suggests a different interpretation for the period directly related to the angular frequency $\omega = \sqrt{g/L}$, leading to $T = 2\pi/\omega$. The amplitude $A$ is given directly as $0.10 \mathrm {~m}$. The phase shift can be found by equating the phase inside the sine function to zero at $t = 0$, which gives $\pi/2$, but since the question format doesn't directly ask for the phase shift in radians or its interpretation in terms of time, the calculation of the exact phase shift in seconds from the given information might have been misunderstood. The correct period calculation should consider the angular frequency formula and its relation to the period, which is $T = 2\pi\sqrt{L/g}$, leading to a period that matches one of the provided choices when correctly calculated. The amplitude is correctly identified as $0.10 \mathrm {~m}$ in all options. The phase shift, typically expressed in terms of the phase angle, seems to have been misinterpreted in the explanation. The correct choice should reflect the accurate calculation of the period based on the pendulum's length and the gravitational constant, the given amplitude, and a correct understanding of the phase shift's representation in the context of the equation.

Solve the Problem L\mathrm { L }L , When Displaced Horizontally and Released, Oscillates with Harmonic Motion

Solve the Problem $\mathrm { L }$ , When Displaced Horizontally and Released, Oscillates with Harmonic Motion