Physics

Spring Mass Systems

172 Questions

A spring mass system is a key physics concept used to study oscillations and simple harmonic motion. It involves understanding spring constants, damping, and series or parallel combinations. These principles are frequently tested in engineering entrance examinations.

Series and parallel springsSpring constant calculationsDamped oscillationsSpring compression energyElevator systems

Spring Mass Systems Questions

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

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?

  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. $\dfrac { 2 m v ^ { 2 } } { x ^ { 2 } }$
Reveal answer Fill a bubble to check yourself
A Correct answer
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.
$\begin{array}{l} \left( { \dfrac { 1 }{ 2 }  } \right) m{ v^{ 2 } }=\dfrac { 1 }{ 2 } m{ \left( { \dfrac { v }{ 2 }  } \right) ^{ 2 } }+\dfrac { 1 }{ 2 } k{ v^{ 2 } } \ \dfrac { 1 }{ 2 } k{ v^{ 2 } }=\dfrac { 1 }{ 2 } m{ \left( { \dfrac { v }{ 2 }  } \right) ^{ 2 } }-\dfrac { 1 }{ 2 } \left( { m{ v^{ 2 } } } \right)  \ k{ x^{ 2 } }=m\left( { { v^{ 2 } }-\dfrac { { { v^{ 2 } } } }{ 4 }  } \right)  \ k{ x^{ 2 } }=\dfrac { { 3m{ v^{ ^{ 2 } } } } }{ 4 }  \ \therefore k=\dfrac { { 3m{ v^{ ^{ 2 } } } } }{ { 4{ x^{ 2 } } } }  \end{array}$

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

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

  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
Reveal answer Fill a bubble to check yourself
A Correct answer
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

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

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 

  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. $x= F(k _1-k _2)$
Reveal answer Fill a bubble to check yourself
A Correct answer
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)


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

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.

  1. [5, 1/2]

  2. [4, 1/2]

  3. [7, 1/2]

  4. [4, 1/3]
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Cutting a spring into 4 equal parts makes the spring constant of each piece 4k. The time period T = 2 * pi * sqrt(m/k). Since k becomes 4k, the new time period T' = 2 * pi * sqrt(m/4k) = T/2.

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

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

  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.
Reveal answer Fill a bubble to check yourself
C Correct answer
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.

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

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:

  1. $K _ { 1 } / K _ { 2 }$
  2. $\mathbf { K } _ { 2 } : \mathbf { K } _ { 1 }$
  3. $\sqrt { K _ { 1 } } : \sqrt { K _ { 2 } }$
  4. $\mathbf { K } _ { 2 } ^ { 2 } : \mathbf { K } _ { 2 } ^ { 2 }$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Elastic energy U = F^2 / (2k). If U1 = U2, then F1^2 / (2k1) = F2^2 / (2k2). Rearranging gives (F1/F2)^2 = k1/k2, so F1/F2 = sqrt(k1)/sqrt(k2).

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

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 $g=9.8m/s^2$)

  1. 9 cm

  2. 1.0 cm

  3. 20 cm

  4. 5 cm
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Using conservation of energy: m * g * (h + x) = 1/2 * k * x^2. Plugging in m=2, g=9.8, h=0.4, k=1960 results in a quadratic equation for compression x. Solving this yields x = 0.09m or 9cm.

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

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

  1. $3 m^2 g^2/ 2k$
  2. $m^2 g^2/ 2k$
  3. $-3 m^2 g^2/ 2k$
  4. $- m^2 g^2/ 2k$
Reveal answer Fill a bubble to check yourself
A Correct answer
Multiple choice physics oscillatory motion motion of a mass suspended by two springs example of simple harmonic motion oscillations due to a spring

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 

  1. $\pi \dfrac { N }{ m } $
  2. ${ \pi }^{ 2 }\dfrac { N }{ m } $
  3. $2\pi \dfrac { N }{ m } $
  4. $given$ $data$ $is$ $insufficient$
Reveal answer Fill a bubble to check yourself
C Correct answer
Multiple choice physics oscillatory motion motion of a mass suspended by two springs example of simple harmonic motion oscillations due to a spring

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?

  1. d

  2. 2d

  3. 3d

  4. 1/2d
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Gradual lowering reaches equilibrium at d = mg/k. Sudden release results in maximum extension at 2d because the potential energy lost by the mass (mg * 2d) equals the energy stored in the spring (1/2 * k * (2d)^2).

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

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:

  1. 84 J

  2. 68 J

  3. 23 J

  4. 8 J
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Force constant k = F/x = 10N / 1mm = 10,000 N/m. Work done W = 1/2 * k * x^2 = 0.5 * 10,000 * (0.04m)^2 = 5,000 * 0.0016 = 8 J.

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

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

  1. 2 k/3

  2. 3 k/2

  3. 3 k

  4. 6 k
Reveal answer Fill a bubble to check yourself
B Correct answer
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 $1.5 K$
Multiple choice physics oscillatory motion motion of a mass suspended by two springs example of simple harmonic motion oscillations due to a spring

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:  

  1. $(\pi/4)$ kg

  2. 1 kg
  3. $(1 / \pi)$

  4. 10 kg
Reveal answer Fill a bubble to check yourself
B Correct answer
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.