Show that the oscillations due to a spring are simple harmonic oscillations and obtain the expression of periodic time.
Show that the oscillations due to a spring are simple harmonic oscillations and obtain the expression of periodic time.
According to figure a block of mass $m$ fixed to a spring, which in turn is fixed to a rigid wall.
The block is placed on a friction less horizontal surface.
If the block is pulled on one side and is released it then executes a to and fro motion about a mean position.
Let $x=0$, indicate the position of the centre of the block when the spring is in equilibrium. The positions - A and + A indicate the maximum displacements to the left and the right of the mean position.
For spring Robert Hooke law, "Spring when deformed, is subject to a restoring force, the magnitude of which is proportional to the deformation or the displacement and acts is opposite direction."
Let any time $t$, if the displacement of the block from its mean position is $x$, the restoring force $\mathrm{F}$ acting on the block it
$\mathrm{F}(x)=-k x \quad \ldots$ $(1)$
Where $k$ is constant of proportionality and is called the spring constant or spring force constant.
Equation $(1)$ is same as the force law for $SHM$ and therefore the system executes a simple harmonic motion.
Similar Questions
What will be the force constant of the spring system shown in the figure
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The time period of simple harmonic motion of mass $\mathrm{M}$ in the given figure is $\pi \sqrt{\frac{\alpha M}{5 K}}$, where the value of $\alpha$ is____.
The time period of simple harmonic motion of mass $\mathrm{M}$ in the given figure is $\pi \sqrt{\frac{\alpha M}{5 K}}$, where the value of $\alpha$ is____.
- [JEE MAIN 2024]
A mass $m$ is attached to two springs of same force constant $K$, as shown in following four arrangements. If $T_1, T_2, T_3$ and $T_4$ respectively be the time periods of oscillation in the following arrangements, in which case time period is maximum?
A mass $m$ is attached to two springs of same force constant $K$, as shown in following four arrangements. If $T_1, T_2, T_3$ and $T_4$ respectively be the time periods of oscillation in the following arrangements, in which case time period is maximum?
The angular frequency of a spring block system is $\omega _0.$ This system is suspended from the ceiling of an elevator moving downwards with a constant speed $v_0.$ The block is at rest relative to the elevator. Lift is suddenly stopped. Assuming the downwards as a positive direction, choose the wrong statement :
The angular frequency of a spring block system is $\omega _0.$ This system is suspended from the ceiling of an elevator moving downwards with a constant speed $v_0.$ The block is at rest relative to the elevator. Lift is suddenly stopped. Assuming the downwards as a positive direction, choose the wrong statement :
When a particle of mass $m$ is attached to a vertical spring of spring constant $k$ and released, its motion is described by $y ( t )= y _{0} \sin ^{2} \omega t ,$ where $'y'$ is measured from the lower end of unstretched spring. Then $\omega$ is
When a particle of mass $m$ is attached to a vertical spring of spring constant $k$ and released, its motion is described by $y ( t )= y _{0} \sin ^{2} \omega t ,$ where $'y'$ is measured from the lower end of unstretched spring. Then $\omega$ is
- [JEE MAIN 2020]