A spring whose unstretched length is ell has a force constant k . spring is cut into two pieces of u

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13.Oscillations

medium

A spring whose unstretched length is $\ell $ has a force constant $k$. The spring is cut into two pieces of unstretched lengths $\ell_1$ and $\ell_2$ where, $\ell_1 = n\ell_2$ and $n$ is an integer. The ratio $k_1/k_2$ of the corresponding force constants, $k_1$ and $k_2$ will be

$n$

$\frac{1}{n^2}$

$n^2$

$\frac{1}{n}$

(JEE MAIN-2019)

Solution

$\begin{array}{l}
{k_1} = \frac{C}{{{\ell _1}}}\\
{k_2} = \frac{C}{{{\ell _2}}}\\
\frac{{{k_1}}}{{{k_2}}} = \frac{{C{\ell _2}}}{{{\ell _1}C}} = \frac{{{\ell _2}}}{{n{\ell _2}}} = \frac{1}{n}
\end{array}$

Standard 11

Physics

Two masses ${m_1}$ and ${m_2}$ are suspended together by a massless spring of constant k. When the masses are in equilibrium, ${m_1}$ is removed without disturbing the system. Then the angular frequency of oscillation of ${m_2}$ is

easy

The period of oscillation of a mass $M$ suspended from a spring of negligible mass is $T$. If along with it another mass $M$ is also suspended, the period of oscillation will now be

medium

A block with mass $M$ is connected by a massless spring with stiffiess constant $k$ to a rigid wall and moves without friction on a horizontal surface. The block oscillates with small amplitude $A$ about an equilibrium position $x_0$. Consider two cases: ($i$) when the block is at $x_0$; and ($ii$) when the block is at $x=x_0+A$. In both the cases, a perticle with mass $m$ is placed on the mass $M$ ?

($A$) The amplitude of oscillation in the first case changes by a factor of $\sqrt{\frac{M}{m+M}}$, whereas in the second case it remains unchanged

($B$) The final time period of oscillation in both the cases is same

($C$) The total energy decreases in both the cases

($D$) The instantaneous speed at $x_0$ of the combined masses decreases in both the cases

normal

(IIT-2016)

Two springs of constant ${k_1}$and ${k_2}$are joined in series. The effective spring constant of the combination is given by

easy

(AIPMT-2004)

Two bodies $M$ and $N $ of equal masses are suspended from two separate massless springs of force constants $k_1$ and $k_2$ respectively. If the two bodies oscillate vertically such that their maximum velocities are equal, the ratio of the amplitude $M$ to that of $N$ is

medium

(IIT-1988)

A spring whose unstretched length is $\ell $ has a force constant $k$. The spring is cut into two pieces of unstretched lengths $\ell_1$ and $\ell_2$ where, $\ell_1 = n\ell_2$ and $n$ is an integer. The ratio $k_1/k_2$ of the corresponding force constants, $k_1$ and $k_2$ will be

Solution

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