Spring-Mass Systems and Oscillations
Comprehensive physics quiz on spring mechanics, covering spring constants in series and parallel, oscillation periods, energy in springs, and mass-spring dynamics
Questions
A spring of spring constant $k$ is cut into $3$ equal part find $k$ of each
- $3k$
- $\dfrac{k}{3}$
- $k$
- None of these
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
- $\cfrac { 3F }{ 2K } $
- $\cfrac { 2F }{ K } $
- $\cfrac { 5F }{ 2K } $
- $\cfrac { 4F }{ K } $
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 ..........
- 2mg/k, mg/k
- mg/k, mg/k
- mg/k, 2mg/k
- 2mg/k, 2mg/k
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 $10gm$.
- $10gm$
- $0gm$
- $1gm$
- $None$
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?
- $\dfrac { 3 m v ^ { 2 } } { 4 x ^ { 2 } }$
- $\dfrac { m v ^ { 2 } } { 4 x ^ { 2 } }$
- $\dfrac { m v ^ { 2 } } { 2 x ^ { 2 } }$
- $\dfrac { 2 m v ^ { 2 } } { x ^ { 2 } }$
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
- 30 cm
- 25 cm
- 20 cm
- 15 cm
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
- Extension in long spring < extension in shorter spring
- Extension in long spring > extension in shorter spring
- Extension in both the springs are same
- None of the above
How many identical springs of spring constant k should be joined in series, so the effective spring constant is k/2
- 8
- 4
- 1
- 2
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
- $x= F(1/k _1+1/k _2)$
- $x= F(1/k _1-1/k _2)$
- $x= F(k _1+k _2)$
- $x= F(k _1-k _2)$
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.
- [5, 1/2]
- [4, 1/2]
- [7, 1/2]
- [4, 1/3]
When two blocks connected by a spring move towards each other under mutual interaction:
- Their velocities are equal and opposite
- Their accelerations are equal and opposite
- The forces acting on them are equal and opposite
- Their momenta are 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:
- $K _ { 1 } / K _ { 2 }$
- $\mathbf { K } _ { 2 } : \mathbf { K } _ { 1 }$
- $\sqrt { K _ { 1 } } : \sqrt { K _ { 2 } }$
- $\mathbf { K } _ { 2 } ^ { 2 } : \mathbf { K } _ { 2 } ^ { 2 }$
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 $g=9.8m/s^2$)
- 9 cm
- 1.0 cm
- 20 cm
- 5 cm
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
- $3 m^2 g^2/ 2k$
- $m^2 g^2/ 2k$
- $-3 m^2 g^2/ 2k$
- $- m^2 g^2/ 2k$
A spring oscillates with frequency $1$ cycle per second. What approximate length must a simple pendulum have to oscillate with that same frequency?
- 25 cm
- 50 cm
- 67 cm
- 90 cm
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
- $\pi \dfrac { N }{ m } $
- ${ \pi }^{ 2 }\dfrac { N }{ m } $
- $2\pi \dfrac { N }{ m } $
- $given$ $data$ $is$ $insufficient$
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?
- d
- 2d
- 3d
- 1/2d
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:
- 84 J
- 68 J
- 23 J
- 8 J
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-
- $0.15cm$
- $0.12cm$
- $1.2cm$
- $12m$
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
- 2 k/3
- 3 k/2
- 3 k
- 6 k
The potential energy of a particle executing $S.H.M$ is $2.5 J$.
- 18 J
- 15 J
- 10 J
- 12 J
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:
- $(\pi/4)$ kg
- 1 kg
- $(1 / \pi)$
- 10 kg
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
- 16 J
- 8 J
- 32 J
- 24 J
A spring of spring constant ($k$) is attached to a block of mass ($m$). During free fall its time period of oscillations will be
- Zero
- Infinite
- $2\pi \sqrt{\cfrac{m}{k}}$
- $\pi \sqrt{\cfrac{m}{k}}$
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?
- The spring balance reads the weight of man as 60kg
- The spring balance reading fluctuates between 60 kg. and 70 kg
- The spring balance reading fluctuates between 50 kg and 60 kg
- The spring balance reading fluctuates between 50 kg and 70 kg
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
- ${ T } _{ 1 }=2{ T } _{ 2 }$
- ${ T } _{ 1 }=\sqrt { 2 } { T } _{ 2 }$
- ${ T } _{ 2 }=2{ T } _{ 1 }$
- ${ T } _{ 2 }=\sqrt { 2 } { T } _{ 1 }$
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
- $\dfrac { m{ v } _{ 0 } }{ M } $
- $\dfrac { m{ v } _{ 0 }\cos { \theta } }{ M } $
- $\dfrac { m{ v } _{ 0 }\cos { \theta } }{ M+m } $
- $\dfrac { m{ v } _{ 0 }\cos { 2\theta } }{ M+m } $
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 ${ m/s }^{ 2 }$)
- 16 ${ m/s }^{ 2 }$
- 4 ${ m/s }^{ 2 }$
- Cannot be determined
- None of these
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
- 20 cm
- 15 cm
- 10 cm
- 5 cm
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
- 1 : 2
- 2 : 1
- 1 : 1.41
- 1.41 :4
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
- $2 \pi \sqrt { \dfrac { 3 m u } { 2 M A } }$
- $\dfrac { 1 } { 2 \pi } \sqrt { \dfrac { 2 m } { 3 M A } }$
- $\dfrac { 1 } { 2 \pi } \left( \dfrac { 2 m u } { 3 M A } \right)$
- cannot be found
A body of mass 0.98 Kg is suspended from a spring of spring constant K = 2N/m. Then the period is.
- 4.9s
- 4.4s
- 5.2s
- None
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
- $\sqrt { k _ { 1 } / k _ { 2 } }$
- $k _ { 1 } / k _ { 2 }$
- $\sqrt { k _ { 2 } / k _ { 1 } }$
- $k _ { 2 } / k _ { 1 }$
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
- $6.4\, cm$
- $6.2\, cm$
- $6.8\, cm$
- $7.1\, cm$
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-
- $ T = 2 \pi \sqrt { \left( \dfrac { m }{ k _ 1-k _ 2 } \right) } $
- $ T = 2 \pi \sqrt { \left( \dfrac { m }{ k _ 1+k _ 2 } \right) } $
- $ T = 2 \pi \sqrt { \left( \dfrac { m\left( k _ 1+k _ 2 \right) }{ k _{ 1 }k _{ 2 } } \right) } $
- $ T = 2 \pi \sqrt { \left( \dfrac { mk _ 1k _ 2 }{ k _{ 1 }+k _{ 2 } } \right) } $
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
- $ \dfrac {t _1t _2}{t _1 +t _2} $
- $ \dfrac {t _1t _2}{\sqrt {t^2 _1+ t^2 _1} } $
- $ \sqrt {\dfrac { t _1t _2}{ t _1 +t _2}} $
- $\sqrt {(t _1)^2 + (t _2)^2} $
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
- $k=\dfrac { { k } _{ 1 }+{ k } _{ 2 } }{ { k } _{ 1 }{ k } _{ 2 } } $
- $k=\dfrac { { k } _{ 1 }-{ k } _{ 2 } }{ { k } _{ 1 }{ k } _{ 2 } } $
- $k=\dfrac { { k } _{ 1 }{ k } _{ 2 } }{ { k } _{ 1 }+{ k } _{ 2 } } $
- $k=\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-
- $\displaystyle 2\pi \sqrt{\frac{m}{k}}$
- $\displaystyle 2\pi \sqrt{\frac{mYA}{kL}}$
- $\displaystyle 2\pi \sqrt{\frac{mk}{YA}}$
- $\displaystyle 2\pi \sqrt{\frac{m(kL+YA)}{kYA}}$
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:
- $\dfrac{1}{2}, \dfrac{1}{2}$
- $-\dfrac{1}{2}, -\dfrac{1}{2}$
- $\dfrac{1}{2}, -\dfrac{1}{2}$
- $-\dfrac{1}{2}, \dfrac{1}{2}$
Frequency of a block in spring-mass system is $\displaystyle \upsilon $, if it is taken in a lift slowly accelerating upward, then frequency will
- decrease
- increase
- remain constant
- none
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?
- $1$
- $\sin \theta$
- $\sqrt {\dfrac {2}{9}}$
- $\sqrt {\dfrac {9}{2}}$
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.
- $2.61\ m/s$
- $3.61\ m/s$
- $7.61\ m/s$
- $8.1\ m/s$
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
- K
- 2K
- $\displaystyle \frac{K}{2}$
- $\displaystyle \infty $
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
- $ m \dfrac{d^2x}{dt^2} - c \dfrac{dx}{dt} + kx = 0$
- $ m \dfrac{d^2x}{dt^2} + c \dfrac{dx}{dt} - kx = 0$
- $ m \dfrac{d^2x}{dt^2} - c \dfrac{dx}{dt} - kx = 0$
- $ m \dfrac{d^2x}{dt^2} + c \dfrac{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:
- $x(t) = c _1 \sin (\omega t + \phi) + (\dfrac{F _o}{\omega ^2 - \omega _o ^2}) \cos \omega _o t$
- $x(t) = c _1 \cos (\omega t + \phi) + (\dfrac{F _o}{\omega ^2 - \omega _o ^2}) \cos \omega _o t$
- $x(t) = c _1 \sin (\omega t) + (\dfrac{F _o}{\omega ^2 - \omega _o ^2}) \cos \omega _o t$
- $x(t) = c _1 \cos (\omega t) + (\dfrac{F _o}{\omega ^2 - \omega _o ^2}) \cos \omega _o t$