A particle at the end of a spring executes simple harmonic motion with a period {t_1}, while the corresponding period for another spring is {t

Q: A particle at the end of a spring executes simple harmonic motion with a period ${t_1}$, while the corresponding period for another spring is ${t_2}$. If the period of oscillation with the two springs in series is $T$, then

b (b) ${t_1} = 2\,\pi \sqrt {\frac{m}{{{k_1}}}} $ and ${t_2} = 2\,\pi \sqrt {\frac{m}{{{k_2}}}} $ In series, effective spring constant, $k = \frac{{{k_1}{k_2}}}{{{k_1} + {k_2}}}$ $\therefore $ Time period, $T = 2\,\pi \sqrt {\frac{{m\,({k_1} + {k_2})}}{{{k_1}{k_2}}}} $ …..(ii) Now, $t_1^2 + t_2^2 = 4\,{\pi ^2}m\,\left( {\frac{1}{{{k_1}}} + \frac{1}{{{k_2}}}} \right) = \frac{{4\,{\pi ^2}m\,({k_1} + {k_2})}}{{{k_1}{k_2}}}$ $t_1^2 + t_2^2 = {T^2}.$ [Using equation (ii)]

SECTION - A [PHYSICS - MCQ]

GSEB - English Medium

A particle at the end of a spring executes simple harmonic motion with a period ${t_1}$, while the corresponding period for another spring is ${t_2}$. If the period of oscillation with the two springs in series is $T$, then

A$T = {t_1} + {t_2}$
B${T^2} = t_1^2 + t_2^2$
C${T^{ - 1}} = t_1^{ - 1} + t_2^{ - 1}$
D${T^{ - 2}} = t_1^{ - 2} + t_2^{ - 2}$

AIEEE 2004, Medium

STD 11 Science
NEET
STD 11 - 13. oscillations
Physics

Download our app
and get started for free

Experience the future of education. Simply download our apps or reach out to us for more information. Let's shape the future of learning together!No signup needed.*

Download App

Similar Questions

1
A particle executes simple harmonic motion with a frequency $f$. The frequency with which its kinetic energy oscillates is
View Solution
2
Time period of a particle executing $SHM$ is $8\, sec.$ At $t = 0$ it is at the mean position. The ratio of the distance covered by the particle in the $1^{st}$ second to the $2^{nd}$ second is :
View Solution
3
An $LCR$ circuit is equivalent to a damped pendulum. In an $LCR$ circuit the capacitor is charged to $Q_0$ and then connected to the $L$ and $R$ as shown below.

If a student plots graphs of the square of maximum charge $( Q_{Max} ^2 )$ on the capacitor with time$(t)$ for two different values $L_1$ and $L_2 (L_1 > L_2)$ of $L$ then which of the following represents this graph correctly? (plots are schematic and not drawn to scale)
View Solution
4
The period of a simple pendulum is doubled, when
View Solution
5
In an experiment for determining the gravitational acceleration $g$ of a place with the help of a simple pendulum, the measured time period square is plotted against the string length of the pendulum in the figure. What is the value of $g$ at the place? ...... $m/s^2$
View Solution
6
In forced oscillation of a particle the amplitude is maximum for a frequency $\omega_{1}$ of the force, while the energy is maximum for a frequency $\omega_{2}$ of the force, then
View Solution
7
An oscillator of mass $M$ is at rest in its equilibrium position in a potential $V\, = \,\frac{1}{2}\,k{(x - X)^2}.$ A particle of mass $m$ comes from right with speed $u$ and collides completely inelastically with $M$ and sticks to it . This process repeats every time the oscillator crosses its equilibrium position .The amplitude of oscillations after $13$ collisions is: $(M = 10,\, m = 5,\, u = 1,\, k = 1 ).$
View Solution
8
You are holding a shallow circular container of radius $R$, filled with water to a height $h ( h < < R )$. When yon walk with speed $v$, it is seen that water starts spilling over. This happens due to the resonance of the periodic impulse given to the container (due to walking) with the oscillation of the water in the container. If the time period of water oscillating in the container is inversely proportional to $\sqrt{ h }$, then $v$ is proportional to
View Solution
9
Two bodies performing $SHM$ have same amplitude and frequency. Their phases at a certain instant are as shown in the figure. The phase difference between them is
View Solution
10
A particle starts oscillating simple harmonically from its equilibrium position then, the ratio of kinetic energy and potential energy of the particle at the time $T/12$ is : ($T =$ time period)
View Solution

Download our appand get started for free

Similar Questions

Download our app
and get started for free