(Consider mass of the bob = $20 \text{ g}$)
Pendulum Energy and Speed Calculation
The total mechanical energy ($E$) of a simple pendulum is the sum of its kinetic energy ($KE$) and potential energy ($PE$): $E = KE + PE$
We are given that the total energy is $E = 0.02 \text{ J}$.
At the equilibrium position, the potential energy ($PE$) is minimum, typically considered zero. Therefore, at this point, all the energy is kinetic energy ($KE_{max}$). $E = KE_{max} + 0$ $KE_{max} = E = 0.02 \text{ J}$
Calculating Speed at Equilibrium
The formula for kinetic energy is $KE = \frac{1}{2}mv^2$, where $m$ is the mass and $v$ is the speed.
Given:
- Total Energy, $E = 0.02 \text{ J}$
- Mass, $m = 20 \text{ g} = 0.02 \text{ kg}$ (converted to kilograms)
At equilibrium, $KE_{max} = \frac{1}{2}mv^2$. We set this equal to the total energy:
$0.02 \text{ J} = \frac{1}{2} \times (0.02 \text{ kg}) \times v^2$
Now, solve for $v$:
$0.02 = 0.01 \times v^2$
$v^2 = \frac{0.02}{0.01}$
$v^2 = 2$
$v = \sqrt{2} \text{ m/s}$
Calculating the square root:
$v \approx 1.414 \text{ m/s}$
Rounding to two decimal places, the speed is approximately $1.41 \text{ m/s}$.
Similar Questions
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 :
- For a simple pendulum, having time period T, the variation of kinetic energy (K.E.) with time (t) is represented by :
- 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 ______.
