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Chapter 15: Problem 28

Show that pressure has the units of energy density.

Short Answer

Expert verified
The SI unit for pressure is \( kg/(m.s^2) \) and the SI unit for energy density is also \( kg/(m.s^2) \), confirming that pressure indeed has the units of energy density.

Step by step solution

01

Determining the unit of pressure

Pressure is defined as force per unit area. The SI unit for force is the Newton (N) and for area is square meter (\(m^2\)), so the unit for pressure is \(N/m^2\). A newton is defined as the force necessary to accelerate 1 kilogram of mass at 1 meter per second per second. So, we can rewrite the newton as kilogram meter per second squared (\(kg.m/s^2\)). This means the SI unit for pressure can also be expressed as \(kg/(m.s^2)\).
02

Determining the unit of energy density

Energy density is defined as energy per unit volume. The SI unit for energy is the Joule (J), and for volume is cubic meter (\(m^3\)), therefore the unit for energy density is \(J/m^3\). A Joule is defined as the work done by a force of one newton over a displacement of one meter, so we can rewrite the Joule as \(N.m\). Recall that we've derived the Newton as \(kg.m/s^2\), so substituting this we get \(J = kg.m^2/s^2\). Therefore, the SI unit for energy density is \(kg.m^2/(s^2.m^3) = kg/(m.s^2)\).
03

Comparing the units of pressure and energy density

Comparing the SI units for pressure (Step 1), which is \(kg/(m.s^2)\), with that of energy density (Step 2), which is also \(kg/(m.s^2)\) demonstrates that they are indeed the same. Thus, it can be concluded that pressure has the units of energy density.

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