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Expression for Torsional Pendulum

Two blocks of...

Question

Two blocks of masses $m_{1}$ and $m_{2}$ ( $m_{1} > m_{2}$ ), are performing SHM together with the same amplitude and same time period as shown. Surface between $m_{1}$ and the ground is smooth, while between $m_{1}$ and $m_{2}$ , the coefficient of friction is $μ$ . Given that $\frac{k_{1}}{m_{1}} > \frac{k_{2}}{m_{2}}$ . Choose the correct option:

A

Time period of SHM is

2 π \sqrt{\frac{m_{1} m_{2}}{(m_{1} + m_{2}) (k_{1} + k_{2})}}

.

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B

Time period of SHM is

2 π \sqrt{\frac{(m_{1} + m_{2}) (k_{1} + k_{2})}{k_{1} k_{2}}}

.

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C

Time period of SHM is

2 π \sqrt{\frac{m_{1} + m_{2}}{k_{1} + k_{2}}}

.

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D

Maximum possible amplitude of this SHM is

\frac{2 μ m_{2} g (m_{1} + m_{2})}{k_{1} m_{2} - k_{2} m_{1}}

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Solution

The correct option is C Time period of SHM is $2 π \sqrt{\frac{m_{1} + m_{2}}{k_{1} + k_{2}}}$ .
Since the blocks perform SHM together (no slipping), we can take
Total mass, $m = m_{1} + m_{2}$ and $k_{e q} = k_{1} + k_{2}$
(springs in parallel)
Then, $ω = \sqrt{\frac{k_{1} + k_{2}}{m_{1} + m_{2}}}$
& $T = 2 π \sqrt{\frac{m_{1} + m_{2}}{k_{1} + k_{2}}}$

FBD of block $m_{2}$ at extreme position (amplitude $A$ ):

We know, maximum acceleration of block $m_{2}$
$a_{m a x} = ω^{2} A = (\frac{k_{1} + k_{2}}{m_{1} + m_{2}}) A$

Direction of Friction is towards mean position as $\frac{k_{1}}{m_{1}} > \frac{k_{2}}{m_{2}}$

From FBD,
$f_{m a x} + k_{2} A = m_{2} a_{m a x} = m_{2} (\frac{k_{1} + k_{2}}{m_{1} + m_{2}}) A$

For maximum possible amplitude, $f_{m a x} = μ m_{2} g$
$\Rightarrow μ m_{2} g = [m_{2} (\frac{k_{1} + k_{2}}{m_{1} + m_{2}}) - k_{2}] A$
$\Rightarrow A = \frac{μ m_{2} g (m_{1} + m_{2})}{k_{1} m_{2} - k_{2} m_{1}}$

Hence, only option (c) is correct.

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Similar questions

Two blocks of masses

m_{1}

m_{2} (m_{1} > m_{2})

are performing SHM together with same amplitude and same time period as shown. Surface between

m_{1}

and ground is smooth, while between

m_{1}

m_{2}

the coefficient of friction is

μ

\frac{k_{1}}{m_{1}} > \frac{k_{2}}{m_{2}},

choose the correct option(s)

Q. Two blocks of masses

M_{1}

M_{2}

are connected with a string passing over a pulley as shown in the figure. The block

M_{1}

lies on a horizontal surface. The coefficient of friction between the block

M_{1}

and the horizontal surface is

μ

. The system accelerates. What additional mass

m

should be placed on the block

M_{1}

so that the system does not accelerate?

Q. In the arrangement shown, both the springs are in their natural lengths. The coefficient of friction between

m_{2}

m_{1}

μ

. There is no friction between

m_{1}

and surface. If the blocks are displaced slightly, they perform SHM together.
then, the
(1) frequency of such oscillations

\frac{1}{2 π} \sqrt{\frac{k_{1} + k_{2}}{m_{1} + m_{2}}},

(2) condition if the frictional force on block

m_{2}

is to act in the direction of its displacement from the mean position is

\frac{m_{1}}{m_{2}} > \frac{k_{1}}{k_{2}}

(3) If the condition obtained in (b) is met, the maximum amplitude of their oscillations is

\frac{μ m_{2} g (m_{1} + m_{2})}{k_{2} m_{1} - k_{1} m_{2}}

Q. In the given figure, the coefficient of friction between

m_{1}

m_{2}

μ

m_{2}

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Q. Two blocks of masses

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M_{2}

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