• Motion of a mass suspended by two springs - class-XI
Back
0

Motion of a mass suspended by two springs - class-XI

Description: motion of a mass suspended by two springs
Number of Questions: 47
Created by:
Tags: oscillatory motion physics simple harmonic motion oscillations
Start the Quiz
Attempted 0/47 Correct 0 Score 0
‹
›

A spring of spring constant $k$ is cut into $3$ equal part find $k$ of each

physics oscillatory motion motion of a mass suspended by two springs example of simple harmonic motion oscillations due to a spring

  1. $3k$

  2. $\dfrac{k}{3}$

  3. $k$

  4. None of these

Show answer
Correct Option: A

A body of mass 'm' is suspended with an ideal spring of force constant 'k'. The expected change in the position of the body, due to an additional force 'F' acting vertically downwards is 

physics simple harmonic motion motion of a mass suspended by two springs example of simple harmonic motion oscillations due to a spring

  1. $\cfrac { 3F }{ 2K } $

  2. $\cfrac { 2F }{ K } $

  3. $\cfrac { 5F }{ 2K } $

  4. $\cfrac { 4F }{ K } $

Show answer
Correct Option: B

A block of mass m is suddenly released from the top of a string of stiffness constant k.
(i) The maximum compression in the spring will be
(ii) at equilibrium, the compression in the spring will be .......... 

physics oscillatory motion motion of a mass suspended by two springs example of simple harmonic motion oscillations due to a spring

  1. 2mg/k, mg/k

  2. mg/k, mg/k

  3. mg/k, 2mg/k

  4. 2mg/k, 2mg/k

Show answer
Correct Option: A

A body of $100 gm$ is attached to a spring balance suspended from the celling of an elevator. If the elevator cable breaks and itt falls freely down, the weight of the body as indicated by the spring balance would be $10gm$.

physics oscillatory motion motion of a mass suspended by two springs example of simple harmonic motion oscillations due to a spring

  1. $10gm$

  2. $0gm$

  3. $1gm$

  4. $None$

Show answer
Correct Option: A

A block of mass $m$ moving with speed v compresses a spring through distance $x$ before is halved. What is the value of spring constant?

physics oscillatory motion motion of a mass suspended by two springs example of simple harmonic motion oscillations due to a spring

  1. $\dfrac { 3 m v ^ { 2 } } { 4 x ^ { 2 } }$

  2. $\dfrac { m v ^ { 2 } } { 4 x ^ { 2 } }$

  3. $\dfrac { m v ^ { 2 } } { 2 x ^ { 2 } }$

  4. $\dfrac { 2 m v ^ { 2 } } { x ^ { 2 } }$

Show answer
Correct Option: A
Explanation:

Let the velocity at starting is $v$.

After compression change in velocity $ = \dfrac{v}{2}$
Here, Initial kinetic energy of a block $ = \left( {\dfrac{1}{2}} \right)m{v^2}$
After compression of spring,
Total energy at the point $x$= Kinetic energy of a block +Potential Energy which stored in the spring.
$\begin{array}{l} \left( { \dfrac { 1 }{ 2 }  } \right) m{ v^{ 2 } }=\dfrac { 1 }{ 2 } m{ \left( { \dfrac { v }{ 2 }  } \right) ^{ 2 } }+\dfrac { 1 }{ 2 } k{ v^{ 2 } } \ \dfrac { 1 }{ 2 } k{ v^{ 2 } }=\dfrac { 1 }{ 2 } m{ \left( { \dfrac { v }{ 2 }  } \right) ^{ 2 } }-\dfrac { 1 }{ 2 } \left( { m{ v^{ 2 } } } \right)  \ k{ x^{ 2 } }=m\left( { { v^{ 2 } }-\dfrac { { { v^{ 2 } } } }{ 4 }  } \right)  \ k{ x^{ 2 } }=\dfrac { { 3m{ v^{ ^{ 2 } } } } }{ 4 }  \ \therefore k=\dfrac { { 3m{ v^{ ^{ 2 } } } } }{ { 4{ x^{ 2 } } } }  \end{array}$

A hollow pipe of length $0.8\ m$ is closed at one end. At its open end, a $0.5\ m$ long uniform string is vibrating in its second harmonic and it resonates with the fundamental frequency of the pipe. If the tension in the wire is $50\ N$ and the speed of sound is $320\ ms^{-1}$, the mass of the string is

physics oscillatory motion motion of a mass suspended by two springs example of simple harmonic motion oscillations due to a spring

  1. $5\ grams$

  2. $10\ grams$

  3. $20\ grams$

  4. $40\ grams$

Show answer
Correct Option: B
Explanation:

Velocity of sound $= c$

$\dfrac{c}{4L} = \dfrac{2v}{l}$

$\Rightarrow \dfrac{320}{4\times 0.8} = \dfrac{1}{5} \sqrt{\dfrac{T}{\mu}}$

$\Rightarrow \mu = 0.02\space kgm^{-1}$

$\Rightarrow m = \mu l = 10\space g $

 

Two blocks are connected to an ideal spring (K = 200 N/m) and placed on a smooth surface. Initially spring is in its natural lenght and blocks are projected as shown. The maximum extension in the spring will be

physics oscillatory motion motion of a mass suspended by two springs example of simple harmonic motion oscillations due to a spring

  1. 30 cm

  2. 25 cm

  3. 20 cm

  4. 15 cm

Show answer
Correct Option: A

Will it make any difference in the extension of the spring, if 3 springs of spring constant k are joined in series to life a load W as compared to one string of spring constant k to lift the same load

physics oscillatory motion motion of a mass suspended by two springs example of simple harmonic motion oscillations due to a spring

  1. Extension in long spring < extension in shorter spring

  2. Extension in long spring > extension in shorter spring

  3. Extension in both the springs are same

  4. None of the above

Show answer
Correct Option: A
Explanation:

If three springs are joined together, their effective spring constant will be k/3. Since load is W, we can write $W=(k/3)x _1$.

If these strings are replaced by a long spring of spring constant k, let the extension of the load be W, we can still write $W=kx _2$

Comparing these two equations, we get, $x _2=x _1/3$ or the extension in the long spring is less than the shorter springs

How many identical springs of spring constant k should be joined in series, so the effective spring constant is k/2

physics oscillatory motion motion of a mass suspended by two springs example of simple harmonic motion oscillations due to a spring

  1. 8

  2. 4

  3. 1

  4. 2

Show answer
Correct Option: D
Explanation:

effective spring constant will be $K _{eff}=(k)(k)/(k+k)=k/2$

The correct answer is option(d)

If two springs of spring constants $k _1$ and $k _2$ whose extensions upon applying a force F are $x _1$ and $x _2$ respectively are joined together in a series configuration, the net extension will be 

physics oscillatory motion motion of a mass suspended by two springs example of simple harmonic motion oscillations due to a spring

  1. $x= F(1/k _1+1/k _2)$

  2. $x= F(1/k _1-1/k _2)$

  3. $x= F(k _1+k _2)$

  4. $x= F(k _1-k _2)$

Show answer
Correct Option: A
Explanation:

$x _1 = F/k _1$ and $x _2=F/k _2$

Upon joining both the springs together, the net extension will be $x =x _1+x _2$

Substituting, we get, $x= F(1/k _1+1/k _2)$

The correct option is (a)


A spring of force constant k is cut into 4 equal parts. The spring constant of each piece become_______ times and time period will become______ times.

physics oscillatory motion motion of a mass suspended by two springs example of simple harmonic motion oscillations due to a spring

  1. [5, 1/2]

  2. [4, 1/2]

  3. [7, 1/2]

  4. [4, 1/3]

Show answer
Correct Option: B

When two blocks connected by a spring move towards each other under mutual interaction:

physics oscillatory motion motion of a mass suspended by two springs example of simple harmonic motion oscillations due to a spring

  1. Their velocities are equal and opposite

  2. Their accelerations are equal and opposite

  3. The forces acting on them are equal and opposite

  4. Their momenta are equal and opposite.

Show answer
Correct Option: C
Explanation:

If we take the two blocks plus spring as the system there is no external force acting on this system.
The accelerations will be equal and opposite if masses are equal. Since the forces are internal, they will be equal and opposite.

Two springs have their force constants ${ K } _ { 1 }$ and ${ K } _ { 2 }.$ Both are stretched till their elastic energies are equal. Then,ratio of stretching forces ${ K } _ { 1 } / { K } _ { 2 }$ is equal to:

physics oscillatory motion motion of a mass suspended by two springs example of simple harmonic motion oscillations due to a spring

  1. $K _ { 1 } / K _ { 2 }$

  2. $\mathbf { K } _ { 2 } : \mathbf { K } _ { 1 }$

  3. $\sqrt { K _ { 1 } } : \sqrt { K _ { 2 } }$

  4. $\mathbf { K } _ { 2 } ^ { 2 } : \mathbf { K } _ { 2 } ^ { 2 }$

Show answer
Correct Option: C

A mass of 2 kg falls from a height of 40 cm, on a spring with a force constant of 1960 N/m. The spring is compressed by ? (Take $g=9.8m/s^2$)

physics simple harmonic motion motion of a mass suspended by two springs example of simple harmonic motion oscillations due to a spring

  1. 9 cm

  2. 1.0 cm

  3. 20 cm

  4. 5 cm

Show answer
Correct Option: A

A string fixed at both ends vibrates in a resonant mode with separation of $6.0$cm between the consecutive nodes. For the next to next higher resonant frequency this separation is reduced by $2.0$cm. The length of the spring is 

physics simple harmonic motion motion of a mass suspended by two springs example of simple harmonic motion oscillations due to a spring

  1. 8 cm

  2. 16 cm

  3. 24 cm

  4. 32 cm

Show answer
Correct Option: A

One end of a light spring of force constant K is fixed to ceiling the other end is fixed to block of mass M initially the spring is relaxed the work done by the external agent to lower the Hanging body of mass M slowly till it comes to equilibrium is

physics oscillatory motion motion of a mass suspended by two springs example of simple harmonic motion oscillations due to a spring

  1. $3 m^2 g^2/ 2k$

  2. $m^2 g^2/ 2k$

  3. $-3 m^2 g^2/ 2k$

  4. $- m^2 g^2/ 2k$

Show answer
Correct Option: A

A spring oscillates with frequency $1$ cycle per second. What approximate length must a simple pendulum have to oscillate with that same frequency?

physics oscillatory motion motion of a mass suspended by two springs example of simple harmonic motion oscillations due to a spring

  1. 25 cm

  2. 50 cm

  3. 67 cm

  4. 90 cm

Show answer
Correct Option: B

Two identical springs are fixed at one end and masses $1$ $kg$ and $4$ $kg$ are suspended at their other ends. They are both stretched down from their mean position and let go simultaneously. If they are in the same phase after every $4$ seconds then the springs constant $k$ is 

physics oscillatory motion motion of a mass suspended by two springs example of simple harmonic motion oscillations due to a spring

  1. $\pi \dfrac { N }{ m } $

  2. ${ \pi }^{ 2 }\dfrac { N }{ m } $

  3. $2\pi \dfrac { N }{ m } $

  4. $given$ $data$ $is$ $insufficient$

Show answer
Correct Option: C

A body is attached to the lower end of a vertical spiral spring and it is gradually lowered to its equilibrium position.This stretches the spring by a length d.If the same body attached to the same spring is allowed to fall suddenly, what would be the maximum stretching in this case?

physics oscillatory motion motion of a mass suspended by two springs example of simple harmonic motion oscillations due to a spring

  1. d

  2. 2d

  3. 3d

  4. 1/2d

Show answer
Correct Option: B

A spring $40\ mm$ long is stretched by the application of a force. If $10\ N$ force required to stretch the spring through $1\ mm$, then work done in stretching the spring through $40\ mm$ is:

physics oscillatory motion motion of a mass suspended by two springs example of simple harmonic motion oscillations due to a spring

  1. 84 J

  2. 68 J

  3. 23 J

  4. 8 J

Show answer
Correct Option: D

A mass of $0.98kg$ suspended using a spring of constant $K=300Nm^{-1}$ is hit by a bullet of 20gm moving with a velocity $3.0m/s$ vertically. The bullet gets embedded and oscillates with the mass .  The amplitude of oscillation will be-

physics oscillatory motion motion of a mass suspended by two springs example of simple harmonic motion oscillations due to a spring

  1. $0.15cm$

  2. $0.12cm$

  3. $1.2cm$

  4. $12m$

Show answer
Correct Option: A

A spring of force constant K is cut into two pieces such that one piece is double the length of the other Then the long piece will have a force constant of

physics oscillatory motion motion of a mass suspended by two springs example of simple harmonic motion oscillations due to a spring

  1. 2 k/3

  2. 3 k/2

  3. 3 k

  4. 6 k

Show answer
Correct Option: B
Explanation:
Length of the spring $= L$
Force constant of spring $= K$
Ratio in which spring is cut $= 1 : 2$
Length of larger piece $= 2L / (2 + 1) = 2L/3$
Force constant of larger piece $= K’$
Force constant ∝ 1 / Length of the spring
$K / K’ = (2L / 3) / L$
$K / K’ = 2 / 3$
$K’ = 3K / 2$
$K’ = 1.5 K$
Force constant of larger piece is $1.5 K$

The potential energy of a particle executing  $S.H.M$ is $2.5 J$.

When its displacement is half of amplitude the total energy of the particle  be

physics oscillatory motion motion of a mass suspended by two springs example of simple harmonic motion oscillations due to a spring

  1. 18 J

  2. 15 J

  3. 10 J

  4. 12 J

Show answer
Correct Option: C
Explanation:

$\begin{array}{l} We\, \, know, \ \dfrac { { potential\, \, energy\, \left( U \right)  } }{ { Total\, \, energy\left( E \right)  } } =\dfrac { { \dfrac { 1 }{ 2 } m{ \omega ^{ 2 } }{ y^{ 2 } } } }{ { \dfrac { 1 }{ 2 } m{ \omega ^{ 2 } }{ a^{ 2 } } } } =\dfrac { { { y^{ 2 } } } }{ { { a^{ 2 } } } }  \ So, \ \dfrac { { 2.5 } }{ E } =\dfrac { { { { \left( { \dfrac { a }{ 2 }  } \right)  }^{ 2 } } } }{ { { a^{ 2 } } } }  \ E=10\, \, J \end{array}$

A force of 6.4 N stretches a vertical spring by 0.1 m. The mass that must be suspended from the spring so that it oscillates with a period of ($\pi/4$) sec is:  

physics oscillatory motion motion of a mass suspended by two springs example of simple harmonic motion oscillations due to a spring

  1. $(\pi/4)$ kg

  2. 1 kg

  3. $(1 / \pi)$

  4. 10 kg

Show answer
Correct Option: B
Explanation:

$\begin{array}{l} k=\frac { f }{ x } =\frac { { 6.4 } }{ { 0.1 } } =64 \ T=2\pi \sqrt { \frac { m }{ k }  }  \ \frac { \pi  }{ 4 } =2\pi \sqrt { \frac { m }{ { 64 } }  }  \ m=1\, kg \end{array}$

Hence,
option $(B)$ is correct answer.

A spring of force constant $800 Nm^{-1}$ has an extension of 5 cm . The work done in extending it from 5 cm to 15 cm is

physics oscillatory motion motion of a mass suspended by two springs example of simple harmonic motion oscillations due to a spring

  1. 16 J

  2. 8 J

  3. 32 J

  4. 24 J

Show answer
Correct Option: B

A spring of spring constant ($k$) is attached to a block of mass ($m$). During free fall its time period of oscillations will be

physics oscillatory motion motion of a mass suspended by two springs example of simple harmonic motion oscillations due to a spring

  1. Zero

  2. Infinite

  3. $2\pi \sqrt{\cfrac{m}{k}}$

  4. $\pi \sqrt{\cfrac{m}{k}}$

Show answer
Correct Option: C

A man weighing 60 kg stands on the horizontal platform of a spring balance. The platform starts executing simple harmonic motion of amplitude 0.1 m and frequency $2/ \pi$ Hz. Which of the following statements is correct?

physics simple harmonic motion motion of a mass suspended by two springs example of simple harmonic motion oscillations due to a spring

  1. The spring balance reads the weight of man as 60kg

  2. The spring balance reading fluctuates between 60 kg. and 70 kg

  3. The spring balance reading fluctuates between 50 kg and 60 kg

  4. The spring balance reading fluctuates between 50 kg and 70 kg

Show answer
Correct Option: A
Explanation:

Here,

The option $A$ is the correct answer because

as you know spring balance observe normal reaction between contacting surface. it is effected only when lift accelerated or decelerated . 
if lift is moving upward with acceleration $a $
then, observation of spring balance will be $= m(g + a)$ , where m is mass of man 
when lift is moving downward with acceleration a then, observation of spring balance will be $= m(g - a) .$
but when lift is moving upward or donward with constant velocity then, observation will be remain same.
hence, observation of man's weight is $60kg$ on spring balance.

Two identical springs are attached to a mass and the system is made to oscillate. ${ T } _{ 1 }$ is the time period when springs are joined in parallel and ${ T } _{ 2 }$ is the time period when they are joined in series then

physics oscillatory motion motion of a mass suspended by two springs example of simple harmonic motion oscillations due to a spring

  1. ${ T } _{ 1 }=2{ T } _{ 2 }$

  2. ${ T } _{ 1 }=\sqrt { 2 } { T } _{ 2 }$

  3. ${ T } _{ 2 }=2{ T } _{ 1 }$

  4. ${ T } _{ 2 }=\sqrt { 2 } { T } _{ 1 }$

Show answer
Correct Option: C

A loaded spring gun. Initially at rest on a horizontal frictioneles surface fires a marble of  mass m in at an angle of elevation ${ 0 }^{ o }$. The mass of the gun is M that of the marble is m and its muzzle velocity of the marble is ${ V } _{ 0 }$ then Velocity of the gem just after the firing is 

physics simple harmonic motion motion of a mass suspended by two springs example of simple harmonic motion oscillations due to a spring

  1. $\dfrac { m{ v } _{ 0 } }{ M } $

  2. $\dfrac { m{ v } _{ 0 }\cos { \theta } }{ M } $

  3. $\dfrac { m{ v } _{ 0 }\cos { \theta } }{ M+m } $

  4. $\dfrac { m{ v } _{ 0 }\cos { 2\theta } }{ M+m } $

Show answer
Correct Option: A

A block tied between two springs is in equilibrium. If upper spring is cut then the acceleration of the block just after cut is 6 ${ m/s }^{ 2 }$ downwards. Now, if instead of upper spring, lower spring is cut then the magnitude of acceleration of the block just after the cut will be : (Take g = 10 ${ m/s }^{ 2 }$)

physics oscillatory motion motion of a mass suspended by two springs example of simple harmonic motion oscillations due to a spring

  1. 16 ${ m/s }^{ 2 }$

  2. 4 ${ m/s }^{ 2 }$

  3. Cannot be determined

  4. None of these

Show answer
Correct Option: B

A light spring of length 20 cm and force constant 2 N/cm is placed vertically on a table. A small block of mass 1 kg falls on it. The length h from the surface of the table at which the block will have the maximum velocity is  

physics oscillatory motion motion of a mass suspended by two springs example of simple harmonic motion oscillations due to a spring

  1. 20 cm

  2. 15 cm

  3. 10 cm

  4. 5 cm

Show answer
Correct Option: D

Two dissimilar spring fixed at one end are stretched by 10cm and 20cm respectively, when masses ${ m } _{ 1 }$ and ${ m } _{ 2 }$ are suspended at their lower ends. When displaced slightly from their mean positions and released, they will oscillate with period in the ratio

physics oscillatory motion motion of a mass suspended by two springs example of simple harmonic motion oscillations due to a spring

  1. 1 : 2

  2. 2 : 1

  3. 1 : 1.41

  4. 1.41 :4

Show answer
Correct Option: A

A bob of mass  $\mathrm { M }$  is hung using a string of length  $\mathrm { l }.$  A mass  $m$  moving with a velocity  $u$  pierces through the bob and emerges out with velocity  $\dfrac { u } { 3 } ,$  The frequency of oscillation of the bob considering as amplitude  $A$ is

physics simple harmonic motion motion of a mass suspended by two springs example of simple harmonic motion oscillations due to a spring

  1. $2 \pi \sqrt { \dfrac { 3 m u } { 2 M A } }$

  2. $\dfrac { 1 } { 2 \pi } \sqrt { \dfrac { 2 m } { 3 M A } }$

  3. $\dfrac { 1 } { 2 \pi } \left( \dfrac { 2 m u } { 3 M A } \right)$

  4. cannot be found

Show answer
Correct Option: A

A  body of mass 0.98 Kg is suspended from a spring of spring constant K = 2N/m. Then the period is. 

physics simple harmonic motion motion of a mass suspended by two springs example of simple harmonic motion oscillations due to a spring

  1. 4.9s

  2. 4.4s

  3. 5.2s

  4. None

Show answer
Correct Option: B

Two particles  $A$  and  $B$  of equal masses are suspended from two massless springs of spring constants  $k _ { 1 }$  and  $k _ { 2 }$  respectively. If the maximum velocities during oscillations are equal, the ratio of the amplitudes of  $A$  and  $B$  is

physics simple harmonic motion motion of a mass suspended by two springs example of simple harmonic motion oscillations due to a spring

  1. $\sqrt { k _ { 1 } / k _ { 2 } }$

  2. $k _ { 1 } / k _ { 2 }$

  3. $\sqrt { k _ { 2 } / k _ { 1 } }$

  4. $k _ { 2 } / k _ { 1 }$

Show answer
Correct Option: C

A body of mass $4\, kg$ hangs from a spring and oscillates with a period $0.5$ second. On the removed of the body, the spring is shortened by

physics oscillatory motion motion of a mass suspended by two springs example of simple harmonic motion oscillations due to a spring

  1. $6.4\, cm$

  2. $6.2\, cm$

  3. $6.8\, cm$

  4. $7.1\, cm$

Show answer
Correct Option: B

A mass m is suspended from the two coupled springs connected in series. The force constant for springs are $ K _1 and K _2 $. The time period of the suspended mass will be-

physics oscillatory motion motion of a mass suspended by two springs example of simple harmonic motion oscillations due to a spring

  1. $ T = 2 \pi \sqrt { \left( \dfrac { m }{ k _ 1-k _ 2 } \right) } $

  2. $ T = 2 \pi \sqrt { \left( \dfrac { m }{ k _ 1+k _ 2 } \right) } $

  3. $ T = 2 \pi \sqrt { \left( \dfrac { m\left( k _ 1+k _ 2 \right) }{ k _{ 1 }k _{ 2 } } \right) } $

  4. $ T = 2 \pi \sqrt { \left( \dfrac { mk _ 1k _ 2 }{ k _{ 1 }+k _{ 2 } } \right) } $

Show answer
Correct Option: C

A block of mass m is suspended separately by two different spring have time period $ t _1 and t _2 $ . if same mass is connected to parallel combination of both springs , then its time period is given by

physics simple harmonic motion motion of a mass suspended by two springs example of simple harmonic motion oscillations due to a spring

  1. $ \dfrac {t _1t _2}{t _1 +t _2} $

  2. $ \dfrac {t _1t _2}{\sqrt {t^2 _1+ t^2 _1} } $

  3. $ \sqrt {\dfrac { t _1t _2}{ t _1 +t _2}} $

  4. $\sqrt {(t _1)^2 + (t _2)^2} $

Show answer
Correct Option: D

Two massless springs of force constants ${ k } _{ 1 }$ and ${ k } _{ 2 }$ are joined end to end. The resultant force constant $k$ of the system is

physics oscillatory motion motion of a mass suspended by two springs example of simple harmonic motion oscillations due to a spring

  1. $k=\dfrac { { k } _{ 1 }+{ k } _{ 2 } }{ { k } _{ 1 }{ k } _{ 2 } } $

  2. $k=\dfrac { { k } _{ 1 }-{ k } _{ 2 } }{ { k } _{ 1 }{ k } _{ 2 } } $

  3. $k=\dfrac { { k } _{ 1 }{ k } _{ 2 } }{ { k } _{ 1 }+{ k } _{ 2 } } $

  4. $k=\dfrac { { k } _{ 1 }{ k } _{ 2 } }{ { k } _{ 1 }-{ k } _{ 2 } } $

Show answer
Correct Option: C
Explanation:

In series, resultant force constant is given as
  $\dfrac { 1 }{ { k } _{ eq } } =\dfrac { 1 }{ { k } _{ 1 } } +\dfrac { 1 }{ { k } _{ 2 } } $
$\Rightarrow { k } _{ eq }=\dfrac { { k } _{ 1 }{ k } _{ 2 } }{ { k } _{ 1 }+{ k } _{ 2 } } $

One end of a long metallic wire of length $L$ area of cross-section $A$ and Young's modulus $Y$ is tied to the ceiling. The other end is tied to a massless spring of force constant $k$. A mass $m$ hangs freely from the free end of the spring. It is slightly pulled down and released. Its time period is given by-

physics oscillatory motion motion of a mass suspended by two springs example of simple harmonic motion oscillations due to a spring

  1. $\displaystyle 2\pi \sqrt{\frac{m}{k}}$

  2. $\displaystyle 2\pi \sqrt{\frac{mYA}{kL}}$

  3. $\displaystyle 2\pi \sqrt{\frac{mk}{YA}}$

  4. $\displaystyle 2\pi \sqrt{\frac{m(kL+YA)}{kYA}}$

Show answer
Correct Option: D
Explanation:
$F = \dfrac{YA\Delta l}{L} = k _2 \Delta l$
we can consider the system as two springs in series hence 
$\dfrac{1}{k _{eq}} = \dfrac{1}{k _1} +\dfrac{1}{k _2}$
$=\dfrac{1}{k} + \dfrac{L}{YA} = \dfrac{YAk +kL}{YAk}$

$ T = 2\pi \sqrt{\dfrac{m}{k _{eq}}} = 2\pi \sqrt{\dfrac{m(YAk + kL)}{YAk}}$

The frequency $f$ of vibrations of a mass $m$ suspended from a spring of spring constant $k$ is given by $f = Cm^xk^y$, where $C$ is a dimensionless constant. The values of $x$ and $y$ are respectively:

physics oscillatory motion motion of a mass suspended by two springs example of simple harmonic motion oscillations due to a spring

  1. $\dfrac{1}{2}, \dfrac{1}{2}$

  2. $-\dfrac{1}{2}, -\dfrac{1}{2}$

  3. $\dfrac{1}{2}, -\dfrac{1}{2}$

  4. $-\dfrac{1}{2}, \dfrac{1}{2}$

Show answer
Correct Option: D
Explanation:
We know $F=-KK\Rightarrow dim\left( K \right) =\left[ { MLT }^{ -2 } \right] \left[ { L }^{ -1 } \right] ={ ML }^{ 0 }{ T }^{ -2 }$
$dim\left( M \right) ={ ML }^{ 0 }{ T }^{ 0 }$    $dim\left( f \right) =\left[ { M }^{ 0 }{ L }^{ 0 }{ T }^{ -1 } \right] $
$f={ Cm }^{ x }{ K }^{ y }\Rightarrow { M }^{ 0 }{ L }^{ 0 }{ T }^{ -1 }={ \left[ { ML }^{ 0 }{ T }^{ 0 } \right]  }^{ k }{ \left[ { ML }^{ 0 }{ T }^{ -2 } \right]  }^{ y }$
$\Rightarrow$  Comparing powers of $M,L$ and $T$ gives,
$x+y=0\quad \quad -2y=-1\quad \Rightarrow \quad y=\dfrac { 1 }{ 2 } $
and $x=-1/2$

Frequency of a block in spring-mass system is $\displaystyle \upsilon $, if it is taken in a lift slowly accelerating upward, then frequency will 

physics simple harmonic motion motion of a mass suspended by two springs example of simple harmonic motion oscillations due to a spring

  1. decrease

  2. increase

  3. remain constant

  4. none

Show answer
Correct Option: C
Explanation:

$\omega=2\pi\sqrt{\dfrac{K}{M}}$

Frequency is independent of gravity
Hence it will remain constant

A uniform spring has certain mass suspended from it and it's period of vertical oscillations is ${t} _{1}$. The spring is now cut in $2$ parts having lengths in ratio $1:2$  and these springs are now connected in series and then in parallel. find out the ratio of the time period of these two ossillation?

physics oscillatory motion motion of a mass suspended by two springs example of simple harmonic motion oscillations due to a spring

  1. $1$

  2. $\sin \theta$

  3. $\sqrt {\dfrac {2}{9}}$

  4. $\sqrt {\dfrac {9}{2}}$

Show answer
Correct Option: C
Explanation:
Let $k$ be initial force constant of spring,${k} _{1}$ and ${k} _{2}$ be the force constant of neew springs
We can derive,
$ kl= constant $
$\Rightarrow \dfrac{{x} _{1}}{{x} _{2}}=1/2$
$\Rightarrow \dfrac{{k} _{1}}{{k} _{2}}=2$     ........$(1)$
$so k _1=3k, k _2=3k/2 $
As initially these lengths were in series:
$\dfrac{1}{k}=\dfrac{1}{{k} _{1}}+\dfrac{1}{{k} _{2}}$
$\Rightarrow \dfrac{1}{k' _1}=\dfrac{1}{3{k} _{1}}+\dfrac{2}{3{k} _{1}}$
$\Rightarrow \dfrac{1}{k' _1}=\dfrac{1}{{k} _{1}}$
$\Rightarrow {k'} _{1}= k\ $
When these two stringd are connected in parallel,
${k _2}^{\prime}={k} _{1}+{k} _{2}$

${k _2}^{\prime}=\dfrac{3k}{2}+3k$

${k _2}^{\prime}=\dfrac{9k}{2}$

Time period is 
$\dfrac{{T _1}^{\prime}}{T' _2}=\sqrt{\dfrac{k' _1}{{k' _2}^{\prime}}}$
$\Rightarrow \dfrac{{T _1}^{\prime}}{T' _2}=\sqrt{\dfrac{2}{9}}$

A $1.5$ kg block at rest on a tabletop is attached to a horizontal spring having a spring constant of $19.6$ N/m. The spring is initially unstretched. A constant $20.0$ N horizontal force is applied to the object causing the spring to stretch.Determine the speed of the block after it has moved $0.30$ m from equilibrium if the surface between the block and the tabletop is frictionless.

physics simple harmonic motion motion of a mass suspended by two springs example of simple harmonic motion oscillations due to a spring

  1. $2.61\ m/s$

  2. $3.61\ m/s$

  3. $7.61\ m/s$

  4. $8.1\ m/s$

Show answer
Correct Option: B
Explanation:

This system will exhibits S.H.M with angular frequency $\omega =\sqrt { \cfrac { k }{ m }  } =\sqrt { \cfrac { 19.6 }{ 1.5 }  } $

$k$= spring constant 
$m$= mass of the body
The string stretched by the maximum A restoring force at maximum stretch= force acting on body
$\Rightarrow kA=F$
$\Rightarrow A=\cfrac { F }{ k } =\cfrac { 20 }{ 19.6 } $
$F=20N$
$K=19.6\quad N/m$
Now if x is displacement from mean position, the velocity is given by:
$v=\omega =\sqrt { ({ A }^{ 2 }-{ x }^{ 2 }) } $
$v=\omega =\sqrt { \cfrac { 19.6 }{ 1.5 } ({ \cfrac { 20 }{ 19.6 }  }^{ 2 }-{ 0.3 }^{ 2 }) } $
$=3.523m/s$

An infinite number of springs having force constants as K, 2K, 4K, 8K, .......$\displaystyle \infty $ respectively are connected in series; then equivalent spring constant is 

physics oscillatory motion motion of a mass suspended by two springs example of simple harmonic motion oscillations due to a spring

  1. K

  2. 2K

  3. $\displaystyle \frac{K}{2}$

  4. $\displaystyle \infty $

Show answer
Correct Option: C
Explanation:

For the springs connected in series

$\dfrac{1}{K _{eq}}=\dfrac{1}{K}+\dfrac{1}{2K}+\dfrac{1}{4K}+\dfrac{1}{8K}+......$
$\dfrac{1}{K _{eq}}=\dfrac{1}{K}(1+\dfrac{1}{2}+\dfrac{1}{4}+.....)$
$\dfrac{1}{K _{eq}}=\dfrac{1}{K}(\dfrac{1}{1-\dfrac{1}{2}})$
$\dfrac{1}{K _{eq}}=\dfrac{1}{K}(2)$
$K _{eq}=\dfrac{K}{2}$

A body of mass $m$ is suspended from a spring of spring constant $k$. A damping force proportional to the velocity exerts itself on the mass. An appropriate representation of the motion is 

physics simple harmonic motion motion of a mass suspended by two springs example of simple harmonic motion oscillations due to a spring

  1. $ m \dfrac{d^2x}{dt^2} - c \dfrac{dx}{dt} + kx = 0$

  2. $ m \dfrac{d^2x}{dt^2} + c \dfrac{dx}{dt} - kx = 0$

  3. $ m \dfrac{d^2x}{dt^2} - c \dfrac{dx}{dt} - kx = 0$

  4. $ m \dfrac{d^2x}{dt^2} + c \dfrac{dx}{dt} + kx = 0$

Show answer
Correct Option: D
Explanation:
The forces on the mass $m$ are due to spring and damper.

Suppose the mass moves by a distance $x$ from equilibrium and travels with a velocity $\displaystyle \frac{dx}{dt}$, then

Force due to spring is $-kx$ and force due to damper is $\displaystyle -c\frac{dx}{dt}$

By Newton's Second Law of motion, we have $m\displaystyle \frac{d^2x}{dt^2} = -kx-c\frac{dx}{dt}$

Thus, the equation of motion is $\displaystyle m\frac{d^2x}{dt^2}+c\frac{dx}{dt}+kx=0$

A body of mass $m$ attached to the spring experiences a drag force proportional to its velocity and an external force $F(t) = F _o \cos \omega _ot$. The position of the mass at any point in time can be given by:

physics simple harmonic motion motion of a mass suspended by two springs example of simple harmonic motion oscillations due to a spring

  1. $x(t) = c _1 \sin (\omega t + \phi) + (\dfrac{F _o}{\omega ^2 - \omega _o ^2}) \cos \omega _o t$

  2. $x(t) = c _1 \cos (\omega t + \phi) + (\dfrac{F _o}{\omega ^2 - \omega _o ^2}) \cos \omega _o t$

  3. $x(t) = c _1 \sin (\omega t) + (\dfrac{F _o}{\omega ^2 - \omega _o ^2}) \cos \omega _o t$

  4. $x(t) = c _1 \cos (\omega t) + (\dfrac{F _o}{\omega ^2 - \omega _o ^2}) \cos \omega _o t$

Show answer
Correct Option: A,B,C,D
Explanation:

Initially, we already know that the displacement at any moment (Instant) of time for spring mass system is:

$x=a\sin { (\omega t+\phi ) } $ or,
$x=a\cos { (\omega t+\phi ) } $
And here an external force is experienced thus, all the four options give position at any time.

- Hide questions
ADVERTISEMENT

Find more quizzes from top tags

general knowledge
3615 Quizzes
119858 Questions
 technology
307 Quizzes
36705 Questions
 sports
963 Quizzes
19148 Questions
 history
1187 Quizzes
13475 Questions
 physics
598 Quizzes
13441 Questions
 biology
1160 Quizzes
12174 Questions
 science & technology
551 Quizzes
11015 Questions
 chemistry
490 Quizzes
9892 Questions
 maths
518 Quizzes
9324 Questions
 softskills
0 Quizzes
7131 Questions