Explanation:
1. Identify the System and Initial/Final States:
The system consists of the 1 kg block ($m_1$), the 2 kg plank ($m_2$), and the spring.
- Initial State: The system is at rest. The spring is compressed, storing potential energy ($U_i$).
- Mass of block, $m_1 = 1$ kg
- Mass of plank, $m_2 = 2$ kg
- Initial velocity of block, $u_1 = 0$ m/s
- Initial velocity of plank, $u_2 = 0$ m/s
- Final State: The thread is burnt, and the spring expands to its natural length. The potential energy in the spring becomes zero. Both the block and the plank are in motion.
- Final velocity of block, $v_1 = 6$ m/s
- Let the final velocity of the plank be $v_2$.
2. Apply Conservation of Momentum:
Since friction is negligible and there are no external horizontal forces acting on the system, the total momentum of the system is conserved.
Initial momentum ($P_i$) = Final momentum ($P_f$)
$P_i = m_1 u_1 + m_2 u_2$
$P_i = (1 \text{ kg})(0 \text{ m/s}) + (2 \text{ kg})(0 \text{ m/s}) = 0$
$P_f = m_1 v_1 + m_2 v_2$
$0 = (1 \text{ kg})(6 \text{ m/s}) + (2 \text{ kg})v_2$
$0 = 6 + 2v_2$
$2v_2 = -6$
$v_2 = -3 \text{ m/s}$
The negative sign indicates that the plank moves in the opposite direction to the block.
3. Apply Conservation of Energy:
Since friction is negligible, the total mechanical energy of the system is conserved. The initial potential energy stored in the spring is converted into the kinetic energy of the block and the plank.
Initial total energy ($E_i$) = Final total energy ($E_f$)
$E_i = U_i + \frac{1}{2}m_1 u_1^2 + \frac{1}{2}m_2 u_2^2$
Since $u_1 = 0$ and $u_2 = 0$,
$E_i = U_i$
$E_f = \frac{1}{2}m_1 v_1^2 + \frac{1}{2}m_2 v_2^2$ (The potential energy of the spring is zero at its natural length)
Equating initial and final energies:
$U_i = \frac{1}{2}m_1 v_1^2 + \frac{1}{2}m_2 v_2^2$
Substitute the values:
$U_i = \frac{1}{2}(1 \text{ kg})(6 \text{ m/s})^2 + \frac{1}{2}(2 \text{ kg})(-3 \text{ m/s})^2$
$U_i = \frac{1}{2}(1)(36) + \frac{1}{2}(2)(9)$
$U_i = 18 + 9$
$U_i = 27 \text{ J}$
Answer:
The potential energy originally stored in the spring is $27 \text{ J}$.
