(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 :
- 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 :
- 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 :
Important Questions from Oscillations and Waves
- Two tuning forks $A$ and $B$ are sounded together giving rise to $8$ beats in $2\text{ s}$. When fork $A$ is loaded with wax, the beat frequency is reduced to $4$ beats in $2\text{ s}$. If the original frequency of tuning fork $B$ is $380\text{ Hz}$ then original frequency of tuning fork $A$ is _________ $\text{Hz}$.
- A simple pendulum of string length 30 cm performs 20 oscillations in 10 s. The length of the string required for the pendulum to perform 40 oscillations in the same time duration is ___________ cm. [Assume that the mass of the pendulum remains same.]
- The velocity of sound in air is doubled when the temperature is raised from $0^\circ\text{C}$ to $\alpha^\circ\text{C}$. The value of $\alpha$ is ________.
- The equation of a transverse wave is $y = y_0 \sin 2\pi(ft - \frac{x}{\lambda})$. If the maximum particle velocity be four times that of wave velocity then
- The velocity of a particle executing simple harmonic motion along $x$-axis is described as $v^2 = 50 - x^2$, where $x$ represents displacement. If the time period of motion is $\frac{x}{7}~\text{s}$, the value of $x$ is ________.
