(II) The 1.20-kg head of a hammer has a speed of 7.5 m/s just before it strikes a nail (Fig. 19–32) and is brought to rest. Estimate the temperature rise of a 14-g iron nail generated by eight such hammer blows done in quick succession. Assume the nail absorbs all the energy.

Verified step by step guidance

1. Determine the kinetic energy of the hammer head before it strikes the nail. Use the formula for kinetic energy: $E_k=\frac{1}{2}m{v}^2$, where $m$ is the mass of the hammer head and $v$ is its velocity.
2. Calculate the total energy transferred to the nail after eight hammer blows. Multiply the kinetic energy of a single blow by 8, since all the energy is assumed to be absorbed by the nail.
3. Relate the energy absorbed by the nail to the temperature rise using the formula: $Q=mc\mathrm{\Delta T}$, where $Q$ is the total energy absorbed, $m$ is the mass of the nail, $c$ is the specific heat capacity of iron, and $\mathrm{\Delta T}$ is the temperature rise.
4. Rearrange the formula to solve for the temperature rise: $\mathrm{\Delta T}=\frac{Q}{m}$. Substitute the total energy $Q$, the mass of the nail, and the specific heat capacity of iron (approximately 450 J/(kg·°C)) into the equation.
5. Perform the calculation to find the temperature rise of the nail after eight hammer blows. Ensure the units are consistent throughout the calculation (e.g., convert the mass of the nail to kilograms).

Key Concepts

Kinetic Energy

Kinetic energy is the energy possessed by an object due to its motion, calculated using the formula KE = 0.5 * m * v^2, where m is mass and v is velocity. In this scenario, the hammer's kinetic energy just before impact is converted into thermal energy upon striking the nail, leading to a temperature rise in the nail.

Energy Conservation

The principle of energy conservation states that energy cannot be created or destroyed, only transformed from one form to another. In this case, the kinetic energy of the hammer is transformed into thermal energy in the nail, allowing us to estimate the temperature increase based on the energy transferred during the hammer blows.

Specific Heat Capacity

Specific heat capacity is the amount of heat required to raise the temperature of a unit mass of a substance by one degree Celsius. For the iron nail, knowing its specific heat capacity allows us to calculate the temperature rise resulting from the absorbed energy from the hammer strikes, using the formula Q = mcΔT, where Q is the heat energy, m is mass, c is specific heat capacity, and ΔT is the change in temperature.