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Problems Based on SHM with Spring Force - V

Q: The frequency of vibration \$'f'\$ of a mass \$'m'\$ suspended from a spring of spring constant \$K\$ is given by a relation of this type, \$f = Cm^{x}K^{y}\$; where \$C\$ is a dimensionless quantity. The value of \$x\$ and \$y\$ are

The frequency of oscillation spring mass system \$=\cfrac { 1 }{ 2\pi } \sqrt { \cfrac { k }{ m } } \$Where, \$k\$=spring constantand \$m\$=mass with springAnd given frequency, \$f=c{ m }^{ x }{ k }^{ y }\$Comparing above two frequency we get \$x=-\cfrac { 1 }{ 2 } ,y=\cfrac { 1 }{ 2 } \$

Q: Two masses \$m_1\$ and \$m_2\$ are suspended together by a massless spring of constant K as shown in figure. When the masses are in equilibrium, \$m_1\$ is removed without disturbing the system. Find the angular frequency and amplitude of oscillation of \$m_2\$.

If only mass \${ m }_{ 2 }\$ is present the extension of spring \$l\$ is given by \${ m }_{ 2 }g=kl\Rightarrow l=\cfrac { { m }_{ 2 }g }{ k } \$, where \$k\$=Spring constantNow with \${ m }_{ 1 }\$ and \${ m }_{ 2 }\$ the extension be \$l'\$ \$({ m }_{ 1 }+{ m }_{ 2 })g=kl'\$\$l'=\cfrac { { m }_{ 1 }+{ m }_{ 2 } }{ k } \$When \${ m }_{ 1 }\$ is removed, \$(l'-1)\$ acts as amplitude of oscillation\$A=(l'-l)=(\cfrac { { m }_{ 1 }+{ m }_{ 2 } }{ k } )g-\cfrac { { m }_{ 2 }g }{ k } \$\$\Rightarrow A=\cfrac { { m }_{ 1 }g }{ k } \$Now, if \${ m }_{ 2 }\$ is removed the angular frequency of oscillation \$\omega =\sqrt { \cfrac { k }{ { m }_{ 2 } } } \$(This is same as the case of \${ m }_{ 2 }\$ attach to spring)\$*\$N.B: When \${ m }_{ 1 }\$ is removed the spring mass system will oscillates with the same angular frequency as the case of spring mass system with only mass \${ m }_{ 2 }\$

Q: A spring having with a spring constant 1200 N m\$\displaystyle ^{-1}\$ is mounted on a horizontal table as shown in the Figure. A mass of 3 kg is attached to the free end of the spring. The mass is then pulled sideways to a distance of 2.0 cm and released.Determine (i) the frequency of oscillations, (ii) maximum acceleration of the mass, and (iii) the maximum speed of the mass.

Spring constant, \$k=1200\,m^{-1}\$Mass, \$m=3kg\$Displacement, \$A=2.0\,cm=0.02m\$(i) Frequency of oscillation v, is given by the relation:\$\displaystyle v=\frac{1}{T}=\frac{1}{2\pi}\sqrt{\frac{k}{m}}\$where, T is time period\$\therefore \displaystyle v=\frac{1}{2\times 3.14}\sqrt{\frac{1200}{3}} = 3.18 Hz\$Hence, the frequency of oscillations is 3.18 cycles per second.(ii) Maximum acceleration (a) is given by the relation:\$a=\omega^2\,A\$where,\$\omega =\$ Angular frequency \$=\sqrt{\frac{k}{m}}\$\$A\$ = maximum displacement\$\displaystyle \therefore a=\frac{k}{m}A=\frac{1200\times 0.02}{3}=8\, ms^{-2}\$Hence, the maximum acceleration of the mass is \$8.0\,m/s^2\$(iii) Maximum velocity, \$v_{max}=A\omega\$\$\displaystyle =A\sqrt{\frac{k}{m}}=0.02\times \sqrt{\frac{1200}{3}}=0.4\,m/s\$Hence, the maximum velocity of the mass is 0.4 m/s.

Q: Two blocks \$A\$ and \$B\$ of masses \$2m\$ and \$m\$, respectively are connected by a massless and inextensible string. The whole system is syspended by a massless spring as shown in the figure. The magnitude of acceleration of \$A\$ and \$B\$ immediately after the string is cut, are respectively:

Two blocks A and B of mass \$2m\$and \$m\$ respectively and they are connected by a mass less and in extensible string.The equilibrium condition FBD of \$2m\$\$Kx=T+2mg\$\$=mg+2mg=3mg\$After the string is cut\$T=0\$FBD of \$2m\$\$a_2=\cfrac{3mg-2mg}{2m}\\ \quad=\cfrac{g}{2}\$FBD of \$m\$Hence \$\cfrac{g}{2}\$ and \$g\$

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

Force constant, \$ K \propto \dfrac{1}{l}=\dfrac{c}{l}\$Let the spring cut into pieces of length \$x, 2x \$So, \$x+2x=l \Rightarrow x=\dfrac{l}{3}\$Force constant of long piece is: \$ k'=\dfrac{c}{2x}=\dfrac{Kl}{2l/3}=\dfrac{3K}{2}\$

Q: Two identical springs of constant are connected in series and parallel as shown in figure. A mass \$m\$ is suspended from them. The ratio of their frequencies of vertical oscillations will be

We know that, \$n=\dfrac{1}{2\pi}\sqrt{\dfrac{k}{m}}\$In series, \$k=\dfrac{k_1k_2}{k_1+k_2}=\dfrac{k^2}{2}\$In parallel, \$k=k_1+k_2=2k\$\$\therefore, n_1=\dfrac{1}{2\pi}\sqrt{\dfrac{k}{2m}}\$and \$n_2 = \dfrac{1}{2\pi}\sqrt{\dfrac{2k}{m}}\$hence, \$n_1:n_2=1:2\$

Q: A spring of spring constant 5 x \$10 ^ { 3 }\$ N/m is etched initially by 5 cm from the unstretched position the work required to stretch it further by another 5 cm is

Work done is change in potential energyThe spring is initially stretch to 5 cm and again further stretched by 5 cm and further stretched by 5 cm\$\begin{array}{l} w={ p_{ 10 } }-{ p_{ 5 } } \\ =\frac { 1 }{ 2 } \times 5\times { 10^{ 3 } }\left[ { I{ { \left( { \frac { { 10 } }{ { 100 } } } \right) }^{ 2 } }-{ { \left( { \dfrac { 5 }{ { 100 } } } \right) }^{ 2 } } } \right] \\ =\dfrac { 1 }{ 2 } \times 5\times { 10^{ 3 } }\left( { \dfrac { { 10 } }{ { 100 } } +\dfrac { 5 }{ { 100 } } } \right) \left( { \dfrac { { 10 } }{ { 100 } } -\dfrac { 5 }{ { 100 } } } \right) N-m \\ =\dfrac { 1 }{ 2 } \times 5\times { 10^{ 3 } }\times \dfrac { { 15 } }{ { 100 } } \times \dfrac { 5 }{ { 100 } } =2.5\times 15\times 5\times { 10^{ 3 } }\times { 10^{ -4 } } \\ =75\times 2.5\times { 10^{ -1 } }=7.5\times 2.5N-m \\ =18.75\, \, N-m \end{array}\$Option A.

Q: A spring balance has a scale that reads from \$0\$ to \$50 kg\$. The length of the scale is \$20 cm\$. A body suspended from this balance, when displaced and released, oscillates with period of \$0.6 s\$. What is the weight of the body ?

Maximum mass that the scale can read, \$M=50kg\$Maximum displacement of the spring = Length of the scale, \$l=20cm = 0.2 m\$Time period, \$T=0.6s\$Maximum force exerted on the spring, \$F=Mg\$where,\$g=\$ acceleration due to gravity \$=9.8\,m/s^2\$\$F=50\times 9.8=490\$\$\therefore\$ Spring constant, \$k=F/l=490/0.2=2450\,Nm^{-1}\$Mass m, is suspended from the balance.Times period, \$t=2\pi\sqrt{\dfrac{m}{k}}\$\$\therefore m=\left(\dfrac{T}{2\pi}\right)^2\times k=\left(\dfrac{0.6}{2\times 3.14}\right)^2\times 2450=22.36kg\$\$\therefore\$ Weight of the body \$=mg=22.36\times 9.8=219.16\,N\$Hence, the weight of the body is about 219 N.

Q: The time period of mass suspended from a spring is \$T\$. If the spring is cut into four equal parts and the same mass is suspended from one of the parts, then the new time period will be

\$T = 2\pi \sqrt {\dfrac{m}{k}}\$When the spring is cut into four equal parts, the spring constant of one part will becomes 4k, therefore the new time period will become\$T' = 2\pi \sqrt {\dfrac{m}{k'}}\$\$T = 2\pi \sqrt {\dfrac{m}{4k}}\$ \$T'=\dfrac{T}{2}\$

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Problems on SHM of Springs

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Problems on SHM of springs

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Important Questions

The frequency of vibration $^{'} f^{'}$ of a mass $^{'} m^{'}$ suspended from a spring of spring constant $K$ is given by a relation of this type, $f = C m^{x} K^{y}$ ; where $C$ is a dimensionless quantity. The value of $x$ and $y$ are

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Two masses $m_{1}$ and $m_{2}$ are suspended together by a massless spring of constant K as shown in figure. When the masses are in equilibrium, $m_{1}$ is removed without disturbing the system. Find the angular frequency and amplitude of oscillation of $m_{2}$ .

Medium

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A spring having with a spring constant 1200 N m $^{- 1}$ is mounted on a horizontal table as shown in the Figure. A mass of 3 kg is attached to the free end of the spring. The mass is then pulled sideways to a distance of 2.0 cm and released.
Determine (i) the frequency of oscillations, (ii) maximum acceleration of the mass, and (iii) the maximum speed of the mass.

Hard

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Two blocks $A$ and $B$ of masses $2 m$ and $m$ , respectively are connected by a massless and inextensible string. The whole system is syspended by a massless spring as shown in the figure. The magnitude of acceleration of $A$ and $B$ immediately after the string is cut, are respectively:

Medium

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

Medium

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Two masses of $1 k g$ and $2 k g$ respectively are connected by a massless spring as shown in figure. A force of $20 N$ acts on the $2 k g$ mass. At the instant when the $1 k g$ mass has an acceleration of $10 m s^{- 2}$ towards right, the acceleration of the $2 k g$ mass is :

Hard

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Two identical springs of constant are connected in series and parallel as shown in figure. A mass $m$ is suspended from them. The ratio of their frequencies of vertical oscillations will be

Medium

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A spring of spring constant 5 x $1 0^{3}$ N/m is etched initially by 5 cm from the unstretched position the work required to stretch it further by another 5 cm is

Medium

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A spring balance has a scale that reads from $0$ to $50 k g$ . The length of the scale is $20 c m$ . A body suspended from this balance, when displaced and released, oscillates with period of $0.6 s$ . What is the weight of the body ?

Medium

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The time period of mass suspended from a spring is $T$ . If the spring is cut into four equal parts and the same mass is suspended from one of the parts, then the new time period will be

Hard

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Problems Based on SHM with Spring Force - V

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Problems on SHM of Springs

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Problems on SHM of springs

The frequency of vibration ′f′ of a mass ′m′ suspended from a spring of spring constant K is given by a relation of this type, f=CmxKy; where C is a dimensionless quantity. The value of x and y are

Two masses m1​ and m2​ are suspended together by a massless spring of constant K as shown in figure. When the masses are in equilibrium, m1​ is removed without disturbing the system. Find the angular frequency and amplitude of oscillation of m2​.

Two blocks A and B of masses 2m and m, respectively are connected by a massless and inextensible string. The whole system is syspended by a massless spring as shown in the figure. The magnitude of acceleration of A and B immediately after the string is cut, are respectively:

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

Two masses of 1kg and 2kg respectively are connected by a massless spring as shown in figure. A force of 20N acts on the 2kg mass. At the instant when the 1kg mass has an acceleration of 10ms−2 towards right, the acceleration of the 2kg mass is :

Two identical springs of constant are connected in series and parallel as shown in figure. A mass m is suspended from them. The ratio of their frequencies of vertical oscillations will be

A spring of spring constant 5 x 103 N/m is etched initially by 5 cm from the unstretched position the work required to stretch it further by another 5 cm is

A spring balance has a scale that reads from 0 to 50kg. The length of the scale is 20cm. A body suspended from this balance, when displaced and released, oscillates with period of 0.6s. What is the weight of the body ?

The time period of mass suspended from a spring is T. If the spring is cut into four equal parts and the same mass is suspended from one of the parts, then the new time period will be

The frequency of vibration $^{'} f^{'}$ of a mass $^{'} m^{'}$ suspended from a spring of spring constant $K$ is given by a relation of this type, $f = C m^{x} K^{y}$ ; where $C$ is a dimensionless quantity. The value of $x$ and $y$ are

Two masses $m_{1}$ and $m_{2}$ are suspended together by a massless spring of constant K as shown in figure. When the masses are in equilibrium, $m_{1}$ is removed without disturbing the system. Find the angular frequency and amplitude of oscillation of $m_{2}$ .

Two blocks $A$ and $B$ of masses $2 m$ and $m$ , respectively are connected by a massless and inextensible string. The whole system is syspended by a massless spring as shown in the figure. The magnitude of acceleration of $A$ and $B$ immediately after the string is cut, are respectively:

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

Two masses of $1 k g$ and $2 k g$ respectively are connected by a massless spring as shown in figure. A force of $20 N$ acts on the $2 k g$ mass. At the instant when the $1 k g$ mass has an acceleration of $10 m s^{- 2}$ towards right, the acceleration of the $2 k g$ mass is :

Two identical springs of constant are connected in series and parallel as shown in figure. A mass $m$ is suspended from them. The ratio of their frequencies of vertical oscillations will be

A spring of spring constant 5 x $1 0^{3}$ N/m is etched initially by 5 cm from the unstretched position the work required to stretch it further by another 5 cm is

A spring balance has a scale that reads from $0$ to $50 k g$ . The length of the scale is $20 c m$ . A body suspended from this balance, when displaced and released, oscillates with period of $0.6 s$ . What is the weight of the body ?

The time period of mass suspended from a spring is $T$ . If the spring is cut into four equal parts and the same mass is suspended from one of the parts, then the new time period will be