Water at 15^ ^ ^ C is contained inside a closed, thermally...

According to the first law of thermodynamics, the change in internal energy is given by $\Delta U = Q + W$, where $Q$ is the heat transferred to the system and $W$ is the work done on the system.

The problem states that the copper vessel is thermally insulated. This means that no heat can flow into or out of the system from the surroundings. Therefore, the heat transferred to the liquid-vessel system is $Q = 0$.

The increase in temperature from 15^{}^{{}^{\circ}} \text{C} to 17^{}^{{}^{\circ}} \text{C} is entirely due to the mechanical work done on the system by vigorous shaking. We can calculate the increase in internal energy (which equals the work done) as follows:

$\Delta U = (m_w c_w + m_c c_c) \Delta T$

$\Delta U = (0.2 \times 4200 + 0.1 \times 420) \times (17 - 15)$

$\Delta U = (840 + 42) \times 2 = 1764 \text{ J}$

Although the internal energy increases by $1764 \text{ J}$, this energy enters the system as work, not as heat. Thus, the heat transferred is zero.