Simple Pendulum Kinetic Energy vs. Time Graph Analysis
Understanding Kinetic Energy in Simple Harmonic Motion
For a simple pendulum undergoing Simple Harmonic Motion (SHM), the kinetic energy ($K.E.$) varies cyclically with time ($t$). Key characteristics of this variation are:
- $K.E.$ is maximum at the mean (equilibrium) position where the pendulum's speed is greatest.
- $K.E.$ is zero at the extreme positions (maximum displacement) where the pendulum's speed momentarily drops to zero.
- Since $K.E. = \frac{1}{2} m v^2$, it is always non-negative ($K.E. \ge 0$).
- The kinetic energy pattern repeats every half of the time period ($T/2$). This means the frequency of $K.E.$ variation is double the frequency of the pendulum's oscillation. The mathematical form is often related to $\sin^2(\omega t)$ or $\cos^2(\omega t)$.
Analyzing the K.E. vs. Time Graph
The correct graph representing $K.E.$ versus $t$ must satisfy these conditions:
- Non-Negative Values: The graph must always lie above or on the time axis ($K.E. \ge 0$). Graphs showing negative $K.E.$ are incorrect.
- Zero Points: $K.E.$ should be zero at times corresponding to the extreme positions of the pendulum.
- Maximum Points: $K.E.$ should reach its peak value at times corresponding to the mean position.
- Periodicity: The energy cycle (e.g., from zero to maximum and back to zero) should repeat twice within one full oscillation period ($T$), indicating a period of $T/2$ for the $K.E.$ variation itself.
Considering these points, if the pendulum starts at an extreme position ($t=0$), its $K.E.$ is $0$. It increases to a maximum at $t=T/4$ (mean position), decreases to $0$ at $t=T/2$ (other extreme), increases to a maximum again at $t=3T/4$, and returns to $0$ at $t=T$. Option B accurately displays this behavior, showing a wave that starts at zero, rises to a peak, falls back to zero, and repeats this pattern, staying entirely within the non-negative domain and exhibiting the required periodicity.
Similar Questions
- The sum of kinetic energy and potential energy of a simple pendulum bob is $0.02 \text{ joule}$. The speed of the simple pendulum bob at equilibrium position is approximately :
(Consider mass of the bob = $20 \text{ g}$) For a travelling harmonic wave $y(x, t) = 2.0 \cos 2\pi(10 t - 0.0080 x + 0.35)$, where $x$ and $y$ are in cm and $t$ in s. The phase difference between oscillatory motion of two points separated by a distance of $0.5 \text{ m}$ is :
- A cylindrical cork of uniform density floats in a liquid of density $\rho_1$. If the cork is depressed slightly and released, it oscillates harmonically with time period T. If the same cork floats in another liquid of density $\rho_2$, then the similar oscillation has time period 2T. The value of $\rho_2/\rho_1$ is :
- Consider a spring-mass simple harmonic oscillator in one dimension. The mass of the particle is $m\text{ kg}$ and the spring constant is $k\text{ Nm}^{-1}$. At a given instant, the extension of the spring is $x\text{ meter}$ and the speed of the particle is $v\text{ ms}^{-1}$. On the $x-v$ plane, if the graph of $v$ as a function of $x$ is a circle, then the correct option is :
- For sound waves, if the number of nodes for the $5^{\text{th}}$ harmonic of an open-ended pipe is $n$ and that for the $9^{\text{th}}$ harmonic of the same pipe with one of its ends closed is $m$, the ratio $\frac{n}{m}$ is :
Important Questions from Oscillations and Waves
- The displacement of a particle, executing simple harmonic motion with time period $T$, is expressed as $x(t) = A\sin\omega t$, where $A$ is the amplitude. The maximum value of potential energy of this oscillator is found at $t = T/2\beta$. The value of $\beta$ is ________.
A simple pendulum has a bob with mass $m$ and charge $q$. The pendulum string has negligible mass. When a uniform and horizontal electric field $\vec{E}$ is applied, the tension in the string changes. The final tension in the string, when pendulum attains an equilibrium position is _________.
(g: acceleration due to gravity)In an open organ pipe $\nu_3$ and $\nu_6$ are $3^{\text{rd}}$ and $6^{\text{th}}$ harmonic frequencies, respectively. If $\nu_6 - \nu_3 = 2200 \text{ Hz}$ then length of the pipe is _________ mm.
(Take velocity of sound in air is $330 \text{ m/s}$.)- Using a simple pendulum experiment g is determind by measuring its time period T. Which of the following plots represent the correct relation between the pendulum length L and time period T ?
- A cylindrical block of mass $M$ and area of cross section $A$ is floating in a liquid of density $\rho$ and with its axis vertical. When depressed a little and released the block starts oscillating. The period of oscillation is ______.
