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Physics

Impulse Moment Principle

Relates the angular impulse applied to an object to the change in its angular momentum.

τ Δ t = I (ω_{f} - ω_{i})

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Source & review

Difficulty

Intermediate

Read time

6 min

Prerequisites

Rotational Dynamics

Review statusNeeds review

Source

FormuLab initial formula library

Initial content draft pending verification against authoritative course or textbook sources.

Definition

Understanding the core concept

Just as a force applied over a period of time causes a change in linear motion (linear impulse-momentum), a torque applied over a period of time causes a change in rotational motion. This 'rotational push' or angular impulse directly impacts how an object spins, changing its angular velocity based on its resistance to rotation (moment of inertia).

Variables & Units

Understanding each component

Symbol	Meaning	Units
$τ$	Average Net Torque	$N \cdot m$
$Δ t$	Time Interval	$s$
$I$	Moment of Inertia	$kg \cdot m^{2}$
$ω_{f}$	Final Angular Velocity	$rad/s$
$ω_{i}$	Initial Angular Velocity	$rad/s$

Real-World Applications

Where this formula is used in practice

Spinning Up Industrial Machinery

Calculating the time needed for an electric motor to bring a heavy industrial flywheel or grinding wheel to its operating angular speed.

Vehicle Stability Control

Understanding how transient torques from steering or braking affect the rotational dynamics and stability of a vehicle, particularly in electronic stability control systems.

Robotics Joint Actuation

Designing robot arms where motors apply torque for a specific duration to achieve a desired angular velocity for a joint, considering the arm's moment of inertia.

Sports (e.g., Figure Skating)

Analyzing how a figure skater's push off the ice (applying torque) over a brief period changes their angular momentum, affecting their spin rate.

Worked Example

Step-by-step calculation with real numbers

Problem

A uniform solid disk with a moment of inertia of 0.8 kg·m² is initially rotating at 10 rad/s. A constant net torque of 4 N·m is applied to it for 5 seconds. What is the final angular velocity of the disk?

Given

I = 0.8 kg \cdot m^{2}

ω_{i} = 10 rad/s

τ = 4 N \cdot m

Δ t = 5 s

Solution

τ Δ t = I (ω_{f} - ω_{i})

(4 N \cdot m) (5 s) = (0.8 kg \cdot m^{2}) (ω_{f} - 10 rad/s)

20 N \cdot m \cdot s = (0.8 kg \cdot m^{2}) (ω_{f} - 10 rad/s)

\frac{20 N \cdot m \cdot s}{0.8 kg \cdot m ^{2}} = ω_{f} - 10 rad/s

25 rad/s = ω_{f} - 10 rad/s

ω_{f} = 25 rad/s + 10 rad/s

ω_{f} = 35 rad/s

Final Answer

ω_{f} = 35 rad/s

Formula Guide

Info

DifficultyIntermediate

Read Time6 min

PrerequisitesRotational Dynamics

Tools

Source & review

StatusNeeds review

Last reviewedPending

Source

FormuLab initial formula library

Initial content draft pending verification against authoritative course or textbook sources.

Assumptions

• Use SI units unless the formula or course context states otherwise.
• Check notation, sign conventions, and applicability against your course material.
• Confirm variables and assumptions before using the formula in graded or professional work.