Combination of Springs

Q. A spring of force constant $k$ is cut into lengths of ratio $1:2:3$. They are connected in series and the new force constant is $k^{\prime }$. Then they are connected in parallel and force constant is $k^{\prime \prime }$. Then $k^{\prime }:k^{\prime \prime }$ is

1. 1 : 9
2. 1 : 11
3. 1 : 14
4. 1 : 16

Q. Figure shows a system consisting of a massless pulley, a spring of force constant $k$ and a block of mass $m$. If the block is slightly displaced vertically down from its equilibrium position and then released, the period of its vertical oscillation is

1. $2\pi \sqrt{\frac{m}{k}}$
2. $\pi \sqrt{\frac{m}{4k}}$
3. $\pi \sqrt{\frac{m}{k}}$
4. $4\pi \sqrt{\frac{m}{k}}$

Q. A block of mass 10 kg is suspended through two light springs which are balanced as shown in the figure. Then

1. both the scales will read 10 kg.
2. both the scales will read 5 kg.
3. the readings may be of any value, but their sum will be 10 kg.
4. the upper scale will read 10 kg and the lower one zero.

Q. The system shown is in equilibrium. Find the acceleration of the block A at the instant just after the spring between the ceiling and A is cut. All blocks are of equal masses ‘$m$’ each and the springs are of equal stiffness. (Assume springs to be ideal).

1. Zero
2. $2g$
3. $g$
4. $\frac{3g}{2}$

Q. Two blocks of equal mass $m$ each are connected by an unstretched spring and the system is kept at rest on a frictionless horizontal surface. A constant force $F$ is applied on one of the blocks pulling it away from the other as shown in figure. If the extension of the spring is $x_0$ at time $t$, The displacement of the left and right blocks at this instant is $x_1$ and $x_2$ respectively, then

1. $x_1=\frac{1}{2}(\frac{F{t}^2}{2m}+x_0),x_2=\frac{1}{2}(\frac{F{t}^2}{2m}-x_0)$
2. $x_1=\frac{1}{2}(\frac{F{t}^2}{2m}-x_0),x_2=\frac{1}{2}(\frac{F{t}^2}{2m}+x_0)$
3. $x_1=\frac{1}{2}(\frac{F{t}^2}{2m}-x_0),x_2=\frac{1}{2}(\frac{F{t}^2}{2m}-x_0)$
4. $x_1=\frac{1}{2}(\frac{F{t}^2}{2m}+x_0),x_2=\frac{1}{2}(\frac{F{t}^2}{2m}+x_0)$

Q. A spring of force constant $K$ is cut into lengths of ratio 1 : 2 : 3. They are connected in series and the new force constant is $K^{{}^{\prime }}$. Then, they are connected in parallel and the force constant is $K^{\prime \prime }$. Then, $\frac{K^{{}^{\prime }}}{K^{\prime \prime }}$ is

1. $1:11$
2. $1:14$
3. $1:6$
4. $1:9$

Q. The system shown is in equilibrium. Find the accelerations of the blocks A, B and C just after the spring between B and C is cut. All blocks are of equal masses $‘m^{\prime }$ each and springs are of equal stiffness. (Assume springs to be ideal and take downward acceleration to be positive).

1. $+g,+g,-g$
2. $-\frac{g}{2},-\frac{g}{2},+g$
3. $+g,-\frac{g}{2},-g$
4. $0,-\frac{g}{2},-g$

Q. Find the $K_{equivalent}$.

1. $K_{equivalent}=K_1$.
2. $K_{equivalent}=K_2$.
3. $K_{equivalent}=K_1+K_2$.
4. $K_{equivalent}=\frac{K_1{K}_2}{K_1+K_2}$.

Q. Spring of force constant $k$ is cut into springs of length ratio $1:2:3$. They are then connected in series and the new force constant is $k^{\prime }.$ If they are connected in parallel, then the force constant is $k^{\prime \prime }.$ Find $k^{\prime }:k^{\prime \prime }$

Q. A block of mass $m$ placed on a smooth floor is connected to a fixed support with the help of a spring of stiffness $k$. It is pulled by a rope as shown in the figure. Tension force $T$ of the rope is increased gradually without changing its direction until the block losses contact from the floor. The increase in rope tension $T$ is so gradual that acceleration in the block can be neglected. What is the extension in the spring, when the block losses contact from the floor?

1. $\frac{mg\cos \theta }{k}$
2. $\frac{mg\sin \theta }{k}$
3. $\frac{mg\tan \theta }{k}$
4. $\frac{mg\cot \theta }{k}$

Q. The system shown is in equilibrium. Find the acceleration of the block C at the instant just after the spring between ceiling and A is cut. All blocks are of equal masses ‘$m$’ each and springs are of equal stiffness. (Assume springs to be ideal).

1. $2g$
2. $g$
3. Zero
4. $\frac{3g}{2}$

Q. A spring of force constant $K$ is cut into lengths of ratio 1 : 2 : 3. They are connected in series and the new force constant is $K^{{}^{\prime }}$. Then, they are connected in parallel and the force constant is $K^{\prime \prime }$. Then, $\frac{K^{{}^{\prime }}}{K^{\prime \prime }}$ is

1. $1:9$
2. $1:11$
3. $1:14$
4. $1:6$

Q. A spring of force constant k is cut into lengths of ratio They are connected in series and the new force constant is k then they are connected in parallel and then the force constants is:

Q.

On applying a force F the point 'P' is displaced vertically downwards by y units from equilibrium position. Find the force F in terms of the force constant k of the spring and displacement y, for the cases (A) and (B), as shown in figure.

1. Case A : F = ky

   Case B : F = ky

2. Case A : F = ky

   Case B : F = 2ky

3. Case A : F = ky

   Case B : F = 4ky

4. Case A : F = 2ky

   Case B : F = ky

Q. The system shown in the figure is in equilibrium. Find the accelerations of the blocks A, B and C just after the string between A and B is cut. All blocks are of equal masses ‘$m$’ each and springs are of equal stiffness. (Assume springs to be ideal and take downward acceleration to be positive).

1. $0,+2g,+g$
2. $0,+g,+g$
3. $-2g,+2g,0$
4. $-2g,+2g,+g$

Q. The degree of static indeterminacy of truss given below is.

1. 3

Q. When a mass $M$ hangs from a spring of length $L$, it stretches the spring by a distance $x$. Now the spring is cut in two parts of length $\frac{L}{3}$ and $\frac{2L}{3}$, and the two springs thus formed are connected to a straight rod of mass $M$ which is horizontal in the configuration as shown in the figure. Find the stretch in each of the spring.

1. $x_1=\frac{x}{6},x_2=\frac{x}{2}$
2. $x_1=\frac{x}{3},x_2=\frac{x}{6}$
3. $x_1=\frac{x}{2},x_2=\frac{x}{2}$
4. $x_1=\frac{x}{6},x_2=\frac{x}{3}$

Q. In the given figure, the block displaces down to the equilibrium position by an amount (K, 2K and 3K represents stiffness of the respective springs)

1. $\frac{9}{4}\frac{mg}{K}$
2. $\frac{3}{7}\frac{mg}{K}$
3. $\frac{5}{8}\frac{mg}{K}$
4. $\frac{4}{11}\frac{mg}{K}$

Q. What is the equivalent spring constant for the system shown below?

1. $5k$
2. $3.5k$
3. $1.5k$
4. $0.5k$

Q. The system shown is in equilibrium. Find the acceleration of the block C at the instant just after the spring between ceiling and A is cut. All blocks are of equal masses ‘$m$’ each and springs are of equal stiffness. (Assume springs to be ideal).

1. $2g$
2. $\frac{3g}{2}$
3. $g$
4. Zero

Q. If you cut a spring into two equal springs, what will the value of their respective spring constants be, given, that of the original one is K?

1. 2K
2. Incalculable
3. K
4. K/2

Q. For the spring block arrangement as shown in figure:


Given that $k_1=2k_2$. If force applied by the first spring on the block is $F_1$ and force applied by the second spring on the block is $F_2$, choose the correct option if the block is in equilibrium with the action of an external force $30\text{N}$ on the block.

1. $F_1=F_2=30\text{N}$
2. $2x_1=x_2$
3. $x_1=2x_2$
4. $F_1=20\text{N},F_2=10\text{N}$

Q. A spring of force constant $k$ is cut into lengths of ratio $1:2:3$. They are connected in series and the new force constant is $k^{\prime }$. Then they are connected in parallel and force constant is $k^{\prime \prime }$. Then $k^{\prime }:k^{\prime \prime }$ is

1. 1 : 9
2. 1 : 11
3. 1 : 14
4. 1 : 16

Q. A particle moving in a straight line with initial velocity $u$ and uniform acceleration $a$. If the sum of the distance travelled in $t^{th}$ and ${(t+1)}^{th}$ seconds is $100cm$, then its velocity after $t$ seconds in $cm/s$ is

1. $20$
2. $30$
3. $50$
4. $80$

Q. The system shown is in equilibrium. Find the accelerations of the blocks A, B and C just after the spring between B and C is cut. All blocks are of equal masses $‘m^{\prime }$ each and springs are of equal stiffness. (Assume springs to be ideal and take downward acceleration to be positive).

1. $+g,+g,-g$
2. $-\frac{g}{2},-\frac{g}{2},+g$
3. $+g,-\frac{g}{2},-g$
4. $0,-\frac{g}{2},-g$

Q. Consider the truss shown in figure below:



The degree of static indeterminacy of the above shown truss is

1. 0

Q. The elevator shown in figure is descending with an acceleration of $2m/s^2$. The mass of the block $A=0.5kg$. The force exerted by the block $A$ on block $B$ is :

1. $6N$
2. $2N$
3. $4N$
4. $8N$

Q. The time period of a simple pendulum is T. When the length is increased by 10 cm, the period is $T_1$.When the length is decreased by 10 cm, its periodis T${}_2$. Then relation between T, T${}_1$, T${}_2$ is

1. $2T^2=T_1^2-T_2^2$
2. $2T^2=T_1^2+T_2^2$
3. $2/T_2=1/T_1^2+1/T_2^2$
4. $2/T_2=1/T_1^2-1/T_2^2$

Q. For the spring block arrangement as shown in figure:


Given that $k_1=2k_2$. If force applied by the first spring on the block is $F_1$ and force applied by the second spring on the block is $F_2$, choose the correct option(s) if the block is in equilibrium with the action of an external force $30\text{N}$ on the block.

1. $x_1=2x_2$
2. $x_1=x_2$
3. $F_1=F_2=30\text{N}$
4. $F_1=20\text{N},F_2=10\text{N}$

Q. For the given figure List - I gives acceleration of blocks of masses $m_1,m_2,m_3\&m_4$ and List - II gives their magnitudes. Here $m_1=m_2=2m$ and $m_3=m_4=m$

$\begin{array}{cc}\text{List - I}& \text{List - II}\\ \text{(I)}{a}_1& \text{(P)}2g\\ \text{(II)}{a}_2& \text{(Q)}g\\ \text{(III)}{a}_3& \text{(R)}\frac{g}{2}\\ \text{(IV)}{a}_4& \text{(S)}\frac{g}{4}\\ & \text{(T)}0\\ & \text{(U)}\text{None}\end{array}$
Just after cutting string connecting $m_1$ and $m_2$ correct match for accelerations of blocks is

1. $I\to Q;II\to Q;III\to P;IV\to Q$
2. $I\to Q;II\to P;III\to R;IV\to T$
3. $I\to Q;II\to Q;III\to T;IV\to T$
4. $I\to P;II\to Q;III\to R;IV\to U$