Equation For Force And Torque On A Rigid Body

Force and Torque Equations:

\[ F = m \cdot a_{cm} \] \[ \tau = I \cdot \alpha \]

Mass (m):

Acceleration (a_cm):

m/s²

Moment of Inertia (I):

kg m²

Angular Acceleration (α):

rad/s²

Force (F):

Torque (τ):

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1. What are Force and Torque Equations for a Rigid Body?

Definition: These equations describe the relationship between force, mass, acceleration, torque, moment of inertia, and angular acceleration in rigid body dynamics.

Purpose: They are fundamental to understanding and calculating the motion of rigid bodies in physics and engineering applications.

2. How Do These Equations Work?

The calculator uses two fundamental equations:

\[ F = m \cdot a_{cm} \] \[ \tau = I \cdot \alpha \]

Where:

\( F \) — Net force acting on the body (Newtons, N)
\( m \) — Mass of the body (kilograms, kg)
\( a_{cm} \) — Acceleration of the center of mass (m/s²)
\( \tau \) — Net torque acting on the body (Newton-meters, N m)
\( I \) — Moment of inertia about the axis of rotation (kg m²)
\( \alpha \) — Angular acceleration (radians per second squared, rad/s²)

Explanation: The first equation relates linear force to linear acceleration, while the second relates torque to angular acceleration.

3. Importance of These Equations

Details: These equations are essential for analyzing the motion of objects in engineering, physics, and mechanical design, from simple machines to complex mechanical systems.

4. Using the Calculator

Tips: Enter the mass, linear acceleration, moment of inertia, and angular acceleration. All values must be positive numbers.

5. Frequently Asked Questions (FAQ)

Q1: What is a rigid body in physics?
A: A rigid body is an idealized solid where deformation is negligible, and the distance between any two points remains constant.

Q2: How is moment of inertia different from mass?
A: While mass measures resistance to linear acceleration, moment of inertia measures resistance to angular acceleration.

Q3: What are typical units for these quantities?
A: Mass (kg), acceleration (m/s²), force (N), moment of inertia (kg m²), angular acceleration (rad/s²), torque (N m).

Q4: Can these equations be used for non-rigid bodies?
A: They provide approximate solutions, but exact analysis requires more complex equations accounting for deformation.

Q5: How do I find moment of inertia for specific shapes?
A: Standard formulas exist for common shapes (e.g., I = ½MR² for a solid cylinder about its axis).