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 . Volume has dimensions . Thus, kinetic energy per unit volume has dimensions .
(2) Force per unit area: Force has dimensions . Area has dimensions . Thus, force per unit area has dimensions .
(3) Potential energy per unit volume: Potential energy also has dimensions of energy, which is . Volume has dimensions . Thus, potential energy per unit volume has dimensions .
(4) Pressure: Pressure is defined as force per unit area. As calculated in (2), it has dimensions .
(5) Angular momentum: Angular momentum is defined as where is moment of inertia and is angular velocity. Moment of inertia has dimensions , and angular velocity has dimensions . Thus, angular momentum has dimensions .
Comparing the dimensions, we can see that options (1), (2), (3), and (4) all have the same dimensions . Option (5) has dimensions . Therefore, angular momentum has different dimensions.
3. Final Answer
Angular momentum