The problem asks to identify which of the given quantities has different dimensions compared to the others. The options are: (1) Kinetic energy per unit volume (2) Force per unit area (3) Potential energy per unit volume (4) Pressure (5) Angular momentum

Applied MathematicsDimensional AnalysisPhysicsKinetic EnergyPotential EnergyPressureAngular Momentum
2025/6/26

1. Problem Description

The problem asks to identify which of the given quantities has different dimensions compared to the others. The options are:
(1) Kinetic energy per unit volume
(2) Force per unit area
(3) Potential energy per unit volume
(4) Pressure
(5) Angular momentum

2. Solution Steps

We need to find the dimensions of each quantity.
(1) Kinetic energy per unit volume: Kinetic energy has dimensions of energy, which is ML2T2ML^2T^{-2}. Volume has dimensions L3L^3. Thus, kinetic energy per unit volume has dimensions ML2T2L3=ML1T2\frac{ML^2T^{-2}}{L^3} = ML^{-1}T^{-2}.
(2) Force per unit area: Force has dimensions MLT2MLT^{-2}. Area has dimensions L2L^2. Thus, force per unit area has dimensions MLT2L2=ML1T2\frac{MLT^{-2}}{L^2} = ML^{-1}T^{-2}.
(3) Potential energy per unit volume: Potential energy also has dimensions of energy, which is ML2T2ML^2T^{-2}. Volume has dimensions L3L^3. Thus, potential energy per unit volume has dimensions ML2T2L3=ML1T2\frac{ML^2T^{-2}}{L^3} = ML^{-1}T^{-2}.
(4) Pressure: Pressure is defined as force per unit area. As calculated in (2), it has dimensions ML1T2ML^{-1}T^{-2}.
(5) Angular momentum: Angular momentum is defined as IωI\omega where II is moment of inertia and ω\omega is angular velocity. Moment of inertia has dimensions ML2ML^2, and angular velocity has dimensions T1T^{-1}. Thus, angular momentum has dimensions ML2T1ML^2T^{-1}.
Comparing the dimensions, we can see that options (1), (2), (3), and (4) all have the same dimensions ML1T2ML^{-1}T^{-2}. Option (5) has dimensions ML2T1ML^2T^{-1}. Therefore, angular momentum has different dimensions.

3. Final Answer

Angular momentum

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