State whether the following statements are True or False:
$1.$ If a spring is cut into two equal pieces,the spring constant of each piece decreases.
$2.$ As the displacement of a Simple Harmonic Oscillator $(SHO)$ increases,its acceleration decreases.
$3.$ $A$ system can oscillate with more than one natural frequency.
$4.$ The periodic time of Simple Harmonic Motion $(SHM)$ depends on amplitude,energy,or phase constant.
$1.$ If a spring is cut into two equal pieces,the spring constant of each piece decreases.
$2.$ As the displacement of a Simple Harmonic Oscillator $(SHO)$ increases,its acceleration decreases.
$3.$ $A$ system can oscillate with more than one natural frequency.
$4.$ The periodic time of Simple Harmonic Motion $(SHM)$ depends on amplitude,energy,or phase constant.
(A-D) $1.$ False. When a spring is cut into two equal pieces,the spring constant of each piece doubles $(k' = 2k)$.
$2.$ False. In $SHO$,acceleration $a = -\omega^2 x$. As the magnitude of displacement $|x|$ increases,the magnitude of acceleration $|a|$ increases.
$3.$ True. Complex systems (like coupled oscillators or vibrating strings) can have multiple natural frequencies (normal modes).
$4.$ False. The periodic time $T = 2\pi \sqrt{m/k}$ of an ideal $SHM$ is independent of amplitude,energy,and phase constant.
$2.$ False. In $SHO$,acceleration $a = -\omega^2 x$. As the magnitude of displacement $|x|$ increases,the magnitude of acceleration $|a|$ increases.
$3.$ True. Complex systems (like coupled oscillators or vibrating strings) can have multiple natural frequencies (normal modes).
$4.$ False. The periodic time $T = 2\pi \sqrt{m/k}$ of an ideal $SHM$ is independent of amplitude,energy,and phase constant.
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Similar Questions
The displacement-time graph of a particle executing $S.H.M.$ is given in the figure: (sketch is schematic and not to scale). Which of the following statements is/are true for this motion?
$(A)$ The force is zero at $t = \frac{3T}{4}$
$(B)$ The acceleration is maximum at $t = T$
$(C)$ The speed is maximum at $t = \frac{T}{4}$
$(D)$ The $P.E.$ is equal to $K.E.$ of the oscillation at $t = \frac{T}{2}$
$(A)$ The force is zero at $t = \frac{3T}{4}$
$(B)$ The acceleration is maximum at $t = T$
$(C)$ The speed is maximum at $t = \frac{T}{4}$
$(D)$ The $P.E.$ is equal to $K.E.$ of the oscillation at $t = \frac{T}{2}$
DifficultJEE MAIN 2020
View Solution$A$ graph of the square of the velocity against the square of the acceleration of a given simple harmonic motion is
Difficult
View SolutionThe amplitude of a particle executing $SHM$ about $O$ is $10 \, cm.$ Then:
Column $I$ gives a list of possible sets of parameters measured in some experiments. The variations of the parameters in the form of graphs are shown in Column $II$. Match the set of parameters given in Column $I$ with the graph given in Column $II$.
Column $I$ Column $II$ $(A)$ Potential energy of a simple pendulum ($y$-axis) as a function of displacement ($x$-axis) $(p)$ Parabolic curve opening upwards $(B)$ Displacement ($y$-axis) as a function of time ($x$-axis) for a one-dimensional motion at zero or constant acceleration $(q)$ Linear graph passing through origin $(C)$ Range of a projectile ($y$-axis) as a function of its velocity ($x$-axis) when projected at a fixed angle $(r)$ Linear graph with non-zero intercept $(D)$ The square of the time period ($y$-axis) of a simple pendulum as a function of its length ($x$-axis) $(s)$ Parabolic curve opening upwards (starting from origin)
| Column $I$ | Column $II$ |
|---|---|
| $(A)$ Potential energy of a simple pendulum ($y$-axis) as a function of displacement ($x$-axis) | $(p)$ Parabolic curve opening upwards |
| $(B)$ Displacement ($y$-axis) as a function of time ($x$-axis) for a one-dimensional motion at zero or constant acceleration | $(q)$ Linear graph passing through origin |
| $(C)$ Range of a projectile ($y$-axis) as a function of its velocity ($x$-axis) when projected at a fixed angle | $(r)$ Linear graph with non-zero intercept |
| $(D)$ The square of the time period ($y$-axis) of a simple pendulum as a function of its length ($x$-axis) | $(s)$ Parabolic curve opening upwards (starting from origin) |
The energy of a particle executing simple harmonic motion is given by $E = Ax^2 + Bv^2$,where $x$ is the displacement from mean position $x = 0$ and $v$ is the velocity of the particle at $x$. Choose the incorrect statement.
Difficult
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