Chapter 14, Problem 18P

To determine

The temperature rises of a nail when struck by a hammer.

Answer to Problem 18P

Solution:

The temperature rises of a nail when struck by a hammer is 43 degrees

Explanation of Solution

Formula Used:

The formula used is the relation between temperature and specific heat from thermodynamics

$Q=mcp(Tf-Ti)$

Where,

$Q$ is the heat transferred

$cp$ is the specific heat

$m$ is the mass

$Tf$ is the final temperature

$Ti$ is the initial temperature

The other formula used is the kinetic energy:

$T=\frac{m}{2}{v}^2$

Where,

$T$ is the kinetic energy

$v$ is the velocity

$m$ is the mass

Given:

$cp$ of iron is 460 J/kgK

$mn$ is mass of nail which is 0.014 kg

$Tf-Ti$ is the temperature rise of the nail

$v$ is the velocity of hammer which is 7.5 m/s

$mh$ is the mass which is 1.2 kg

Calculation:

When striking one the kinetic energy of the hammer is converted to heat energy in the nail.

The kinetic energy of the hammer is $T=\frac{mh}{2}{v}^2$

Substituting,

$v$ is the velocity of hammer which is 7.5 m/s

$mh$ is the mass which is 1.2 kg

The kinetic energy is now,

$T=\frac{(1.2)}{2}{(7.5)}^2$

$T=33.75$ J

This kinetic energy is transferred as heat energy. Since the hammer is struck 8 times the total kinetic energy $Ttot$ is

$Ttot=8(33.75)=270$ J

Since this is transferred to the nail, the energy of the nail is

$Ttot=Q=270$ J

Inserting the given values in the heat capacity formula: $Q=mcp(Tf-Ti)$

$270=(0.014)(460)(Tf-Ti)$

$(Tf-Ti)=43$ C