Question

Two blocks, of masses $M$ and $2M$, are connected to a light spring of spring constant $K$ that has one end fixed, as shown in figure. The horizontal surface and the pulley are frictionless. The blocks are released from rest when the spring is non deformed. The string is light.

(I) Maximum extension in the spring is $\frac{4Mg}{K}$.

(II) Maximum kinetic energy of the system is $\frac{2M^2g^2}{K}$.

(III) Maximum extension in the spring is $\frac{2Mg}{K}$.

(IV) Maximum kinetic energy of the system is $\frac{4M^2g^2}{K}$.

Then which of the following option is correct?

• A. I and IV
• B. I and II
• C. III and IV
• D. None of these

Answer 1

Answer: (B)

(B)

Answer 2

The problem involves a system of two blocks, a spring, and a pulley. We need to find the maximum extension of the spring and the maximum kinetic energy of the system. The system is released from rest with the spring non-deformed. Since the surfaces and pulley are frictionless, mechanical energy is conserved.

Let $M_1 = M$ be the mass of the block on the horizontal surface, and $M_2 = 2M$ be the mass of the hanging block. Let $K$ be the spring constant.

1. Calculate the equilibrium extension ($x_{eq}$):

The equilibrium position is where the net force on the system is zero. Let $x$ be the extension of the spring. The force exerted by the spring on $M_1$ is $Kx$ to the left. The tension in the string pulls $M_1$ to the right and $M_2$ upwards. For $M_1$ (horizontal motion): $T - Kx = 0 \implies T = Kx$. For $M_2$ (vertical motion): $2Mg - T = 0 \implies T = 2Mg$. Equating the tensions: $Kx_{eq} = 2Mg$ $x_{eq} = \frac{2Mg}{K}$

This is the extension at which the system would remain at rest if placed there.

2. Calculate the maximum extension in the spring ($x_{max}$):

The system starts from rest at $x=0$ (relaxed spring). It moves downwards, accelerating until it reaches the equilibrium position, and then continues to move until it momentarily comes to rest at the maximum extension. We can use the conservation of mechanical energy. Let the initial position of $M_2$ be the reference for gravitational potential energy ($U_g = 0$). Initial state (released from rest): $KE_i = 0$ $U_{s,i} = 0$ (spring relaxed) $U_{g,i} = 0$ Total initial energy $E_i = 0$.

Final state (maximum extension $x_{max}$): At maximum extension, the system momentarily comes to rest, so $KE_f = 0$. The spring is extended by $x_{max}$, so $U_{s,f} = \frac{1}{2}Kx_{max}^2$. The block $M_2$ has descended by $x_{max}$, so $U_{g,f} = -2Mgx_{max}$. Total final energy $E_f = 0 + \frac{1}{2}Kx_{max}^2 - 2Mgx_{max}$.

By conservation of mechanical energy, $E_i = E_f$: $0 = \frac{1}{2}Kx_{max}^2 - 2Mgx_{max}$ $\frac{1}{2}Kx_{max}^2 = 2Mgx_{max}$ Since $x_{max} \neq 0$, we can divide by $x_{max}$: $\frac{1}{2}Kx_{max} = 2Mg$ $x_{max} = \frac{4Mg}{K}$

This matches statement (I).

3. Calculate the maximum kinetic energy of the system ($KE_{max}$):

The maximum kinetic energy occurs at the equilibrium position, where $x = x_{eq} = \frac{2Mg}{K}$. Let $v_{max}$ be the speed of the blocks at this point. Using conservation of mechanical energy between the initial state and the equilibrium state: Initial energy $E_i = 0$. Energy at equilibrium $E_{eq}$: $KE_{eq} = \frac{1}{2}M_1v_{max}^2 + \frac{1}{2}M_2v_{max}^2 = \frac{1}{2}(M + 2M)v_{max}^2 = \frac{1}{2}(3M)v_{max}^2$. $U_{s,eq} = \frac{1}{2}Kx_{eq}^2$. $U_{g,eq} = -2Mgx_{eq}$. So, $E_{eq} = KE_{max} + \frac{1}{2}Kx_{eq}^2 - 2Mgx_{eq}$.

By conservation of mechanical energy, $E_i = E_{eq}$: $0 = KE_{max} + \frac{1}{2}Kx_{eq}^2 - 2Mgx_{eq}$ $KE_{max} = 2Mgx_{eq} - \frac{1}{2}Kx_{eq}^2$ Substitute $x_{eq} = \frac{2Mg}{K}$: $KE_{max} = 2Mg\left(\frac{2Mg}{K}\right) - \frac{1}{2}K\left(\frac{2Mg}{K}\right)^2$ $KE_{max} = \frac{4M^2g^2}{K} - \frac{1}{2}K\frac{4M^2g^2}{K^2}$ $KE_{max} = \frac{4M^2g^2}{K} - \frac{2M^2g^2}{K}$ $KE_{max} = \frac{2M^2g^2}{K}$

This matches statement (II).

Conclusion:

Statement (I) is correct: Maximum extension in the spring is $\frac{4Mg}{K}$. Statement (II) is correct: Maximum kinetic energy of the system is $\frac{2M^2g^2}{K}$. Statement (III) is incorrect. Statement (IV) is incorrect.

Therefore, the correct option is (B) I and II.