2025-12-13 13:06:49 +01:00
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---
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tags:
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- Level - Intermediate
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- System - ISQ
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- System - SI
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- System - Strong Angular
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- Feature - Quantity Concepts
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- Feature - Unit Conversions
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- Domain - Mechanics
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---
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# Angular Quantities and Torque Calculations
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[Try it live on Compiler Explorer](https://godbolt.org/z/8G4KPsW8a){ .md-button }
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## Overview
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Angular mechanics requires careful handling of _angles_, _angular positions_, and
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angular-dependent quantities like _torque_. This example demonstrates using the
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dedicated ISQ angular system to calculate _torque_ from _force,_ lever arm _position_,
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and _angle_ with proper dimensional analysis.
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## Key Concepts
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### The ISQ Angular System
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The library provides `isq_angle`, an alternative ISQ system that treats angles as
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dimensional quantities rather than dimensionless:
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```cpp
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--8<-- "example/strong_angular_quantities.cpp:42:50"
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```
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In this system:
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- `isq_angle::position_vector` represents a _position_ with angular dependence
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- `isq_angle::force` represents _force_ with angular considerations
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- `isq_angle::angular_measure` explicitly represents _angles_ (not dimensionless)
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- `isq_angle::torque` has dimensions that include angular components
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### Dimensional Torque Calculation
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The torque formula **τ = r × F × sin(θ)** is implemented with full dimensional checking:
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```cpp
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--8<-- "example/strong_angular_quantities.cpp:50:50"
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```
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The division by `cotes_angle` normalizes the angular measure to work correctly with
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the dimensional analysis system, ensuring that the resulting torque has the proper
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dimensions.
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### Working with Angles as Quantities
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Unlike treating _angles_ as dimensionless (radians as pure numbers), this approach makes
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_angles_ first-class quantities:
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```cpp
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const quantity angle = isq_angle::angular_measure(90. * deg);
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```
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This enables:
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- Compile-time checking that _angles_ are used correctly
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- Explicit conversion between _angle_ units (degrees, radians, etc.)
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- Prevention of accidentally treating _angles_ as scalars
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### Output Formatting
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The result is displayed with explicit angular units:
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```cpp
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--8<-- "example/strong_angular_quantities.cpp:51:52"
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```
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**Sample Output:**
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```text
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Applying a perpendicular force of 500 N to a 20 cm long lever results in 100 m N/rad of torque.
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```
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## Why This Matters
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- **Dimensional Correctness**: Angles are strong quantities, not just numbers — this
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system enforces that
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- **Rotational Mechanics**: Essential for robotics, mechanical engineering, aerospace
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applications
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- **Type Safety**: Prevents mixing angular and linear quantities inappropriately
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- **ISQ Compliance**: Follows international standards for quantity systems
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## Alternative Approach
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For simpler cases where angles can be treated as dimensionless, the standard `isq` system
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can be used. The `isq_angle` system is valuable when you need stronger type checking for
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angular quantities or when working with standards that treat angles dimensionally.
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## Related Concepts
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- _Angular velocity and acceleration_
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- _Moment of inertia_
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- _Angular momentum_
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- _Rotational kinetic energy_
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All of these benefit from treating angles as dimensional quantities rather than pure scalars.
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For more details on the theoretical background and design decisions, see the
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2025-12-15 15:39:44 +01:00
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[Strong Angular System](../users_guide/systems/strong_angular_system.md) chapter
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2025-12-13 13:06:49 +01:00
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in the user's guide.
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