Question

For each of the following systems and time intervals, write the appropriate version of Equation 8.2, the conservation of energy equation. (a) the heating coils in your toaster during the first five seconds after you turn the toaster on (b) your automobile from just before you fill it with gasoline until you pull away from the gas station at speed v (c) your body while you sit quietly and eat a peanut butter and jelly sandwich for lunch (d) your home during five minutes of a sunny afternoon while the temperature in the home remains fixed.

Short answer

(a) The appropriate version of Equation 8.2 for heating coil is $\Delta E_{int}=Q+Q\text{'}+T_{ET}$.

(b) The appropriate version of Equation 8.2 for automobile before filling gasoline to move from gas station with speed v is $\Delta K+\Delta E_{int}=W+Q+T_{MT}+T_{MW}$.

(c) The appropriate version of Equation 8.2 for sitting quietly and eating peanut and sandwich is $\Delta U=\text{Q}+{\text{T}}_{\text{MT}}$.

(d) The appropriate version of Equation 8.2 for fixed temperature home at sunny afternoon is $0=Q+T_{\text{ET}}+T_{\text{ER}}$.

Step-by-step solution

General equation of energy conservation

The general equation of energy conservation is given as

$\mathbf{\Delta K}\mathbf{+}\mathbf{\Delta U}\mathbf{+}{\mathbf{\Delta E}}_{\mathbf{int}}\mathbf{=}\mathbf{W}\mathbf{+}\mathbf{Q}\mathbf{+}{\mathbf{T}}_{\mathbf{MW}}\mathbf{+}{\mathbf{T}}_{\mathbf{MT}}\mathbf{+}{\mathbf{T}}_{\mathbf{ET}}\mathbf{+}{\mathbf{T}}_{\mathbf{ER}}$

Here, $\mathbf{\Delta K}$is the change in kinetic energy, $\mathbf{\Delta U}$is the change in potential energy, ${\mathbf{\Delta E}}_{\mathbf{int}}$ is the change in internal energy of system, $\mathbf{W}$ is the work done by system, $\mathbf{Q}$is the heat transfer from system, ${\mathbf{T}}_{\mathbf{MW}}$is the energy transfer from mechanical waves, ${\mathbf{T}}_{\mathbf{MT}}$ is the energy due to mass transfer, ${\mathbf{T}}_{\mathbf{ET}}$is the energy transfer due to electrical transmission, ${\mathbf{T}}_{\mathbf{ER}}$is the energy transfer by electromagnetic radiation.

Determination of appropriate version of equation 8.2 for heating coil of toaster

(a)

The energy interaction for a heating coil of toaster during the first five seconds involves only energy transfer from electricity to the heating coil $\left(T_{ET}\right)$, increase in internal energy of the coil $\left(\Delta E_{int}\right)$ heat transfer from coil to medium$\left(Q_1\right)$ and some energy of heating coil transferred to the surrounding medium as electromagnetic radiation $\left(Q\text{'}\right)$. The remaining terms in general equation of energy conservation are zero because no change occurs in them. So, the law of conservation of energy can be written as

$\Delta E_{int}=Q+Q\text{'}+T_{ET}$

Determination of appropriate version of equation 8.2 for automobile before filling gasoline to move from gas station with speed v

(b)

The mass transfer occurs in filling gas $\left(T_{MT}\right)$, kinetic energy changes $\left(\Delta K\right)$ due to movement, internal energy changes due to fuel burning $\left(\Delta E_{int}\right)$, work $\left(W\right)$ and heat transferoccurs from automobile and energy transfers from mechanical waves from exhaust of automobile $\left(T_{MW}\right)$. So, the law of conservation of energy can be written as

$\Delta K+\Delta E_{int}=W+Q+T_{MT}+T_{MW}$

Determination of appropriate version of equation 8.2 for sitting quietly and eating peanut and sandwich

(c)

The change in potential energy of the body$\left(\Delta P\right)$, energy due to mass transfer by peanut and sandwich$\left(T_{MT}\right)$, conversion of energy into heat$\left(Q\right)$ are the only energy interaction for this situation. So, the law of conservation of energy can be written as

$\Delta U=\text{Q}+{\text{T}}_{\text{MT}}$

Determination of appropriate version of equation 8.2 for fixed temperature home at sunny afternoon

(d)

The home is stationary and at fixed temperature so changes in kinetic, potential and internal energy are zero. The heat transfer$\left(Q\right)$ and electromagnetic radiation$\left(T_{ER}\right)$ occurs from sun, and electricity is used to fix temperature of home so electrical transmission energy$\left(T_{ET}\right)$ also occurs. So, the law of conservation of energy can be written as

$0=Q+T_{\text{ET}}+T_{\text{ER}}$

Thus, the law of conservation of energy for the four cases given are $\Delta E_{int}=Q+Q\text{'}+T_{ET}$, $\Delta K+\Delta E_{int}=W+Q+T_{MT}+T_{MW}$, $\Delta U=\text{Q}+{\text{T}}_{\text{MT}}$, and $0=Q+T_{\text{ET}}+T_{\text{ER}}$ respectively.