MyQuestionIcon

7

You visited us 7 times! Enjoying our articles? Unlock Full Access!

Velocity Displacement Relationship

In the arrang...

Question

In the arrangement shown, both the springs are in their natural lengths. The coefficient of friction between $m_{2}$ and $m_{1}$ is $μ$ . 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 $\frac{μ m_{2} g (m_{1} + m_{2})}{k_{2} m_{1} - k_{1} m_{2}}$

A

1, 2 are correct

No worries! We‘ve got your back. Try BYJU‘S free classes today!

B

2, 3 are correct

No worries! We‘ve got your back. Try BYJU‘S free classes today!

C

all three are correct

Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses

D

1 is only correct

No worries! We‘ve got your back. Try BYJU‘S free classes today!

Open in App

Solution

The correct option is C all three are correct
(a) If the blocks oscillate together, the equivalent spring constant is $(k_{1} + k_{2})$
$∴$ Frequency = $\frac{1}{2 π} \sqrt{\frac{k_{1} + k_{2}}{m_{1} + m_{2}}}$
(b) Assuming that the blocks $(m_{1} + m_{2})$ are displaced towards right, the various forces (except friction) on $m_{2}$ in the reference frame of m1 are as shown. where a = $(\frac{k_{1} + k_{2}}{m_{1} + m_{2}}) x$
For friction to act on $m_{2}$ in the direction of its displacement,
$k_{2} x > m_{2} (\frac{k_{1} + k_{2}}{m_{1} + m_{2}}) x$
$\Rightarrow m_{1} k_{2} + m_{2} k_{2} - m_{2} k_{1} - m_{2} k_{2} > 0$
$\Rightarrow m_{1} k_{2} - m_{2} k_{1} > 0$
$\Rightarrow \frac{m_{1}}{m_{2}} > \frac{k_{1}}{k_{2}}$ This is the desired condition.
(c) Assuming , $\frac{m_{1}}{m_{2}} > \frac{k_{1}}{k_{2}}$ the free body diagram of $m_{2}$ with $m_{1}$ and $m_{2}$ together displaced towards right is as shown.
If A is amplitude of oscillation of $m_{2}, k_{2} A = f + m_{2} ω^{2} A$
$A = \frac{μ m_{2} g}{k_{2} - m_{2} ω^{2}} = \frac{μ m_{2} g}{k_{2} - m_{2} (\frac{k_{1} + k_{2}}{m_{1} + m_{2}})}$
$= \frac{μ m_{2} g (m_{1} + m_{2})}{k_{2} m_{1} - k_{1} m_{2}}$

Suggest Corrections

thumbs-up

1

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}

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}

m_{2}

, the coefficient of friction is

μ

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

. Choose the correct option:

Q. The condition, if the frictional force on block

m_{2}

is to act in the direction of its displacement from the mean position is met, what can be maximum amplitude of their oscillations?

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. Two blocks of masses

m_{1}

m_{2}

are connected by spring constant

K

. Initially the spring is at its natural length. The coefficient of friction between the bars and the surface is

μ

. What minimum constant force has to be applied in the horizontal direction on the block of mass

m_{1}

, in order to shift the other block?

View More

Join BYJU'S Learning Program

similar_icon

Related Videos

lock

Aftermath of SHM

PHYSICS

Watch in App

explore_more_icon

Explore more

Velocity Displacement Relationship

Standard XII Physics

Join BYJU'S Learning Program

CrossIcon