Impulse–Momentum Theorem in Rotational Form (College Board AP® Physics 1: Algebra-Based)

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Physics

Rate of change of angular momentum

In terms of angular momentum, Newton's second law states:

The rate of change of angular momentum is equal to the net external torque exerted on an object (or system)

This can be written as:

$τ = \frac{∆ L}{∆ t}$

Where:
- $τ$ = torque exerted on the system, measured in $N \cdot m$
- $∆ L$ = change in angular momentum, measured in $kg \cdot m^{2} / s$
- $∆ t$ = time interval, measured in $s$
The magnitude of the change in angular momentum is the difference between an object or system's final and initial angular momenta:

change in angular momentum = final angular momentum − initial angular momentum

$∆ L = L - L_{0}$

The equation above can be used in situations where the rotational inertia of the body is not constant
When the rotational inertia is constant:

$τ_{n e t} = \frac{∆ L}{∆ t} = \frac{L - L_{0}}{∆ t} = \frac{I ω - I ω_{0}}{∆ t} = \frac{I (ω - ω_{0})}{∆ t} = \frac{I ∆ ω}{∆ t}$

Since angular acceleration $α$ is equal to the rate of change of angular velocity, the equation becomes:

$τ_{n e t} = I α$

Where:
- $I$ = rotational inertia of the body, measured in $kg \cdot m^{2}$
- $∆ ω$ = change in angular velocity, measured in $rad / s$
- $ω$ = final angular velocity, measured in $rad / s$
- $ω_{0}$ = initial angular velocity, measured in $rad / s$
- $α$ = angular acceleration, measured in $rad / s^{2}$
More about this equation can also be found in the study guide for Newton’s second law in rotational form

Impulse–momentum theorem in rotational form

In terms of angular momentum, Newton’s first law states:

The angular momentum of an object (or system) remains constant unless an external net torque acts upon it

An angular impulse is exerted when an external torque is applied for a time
Therefore, it must also change the angular momentum of the system
- This is known as the rotational form of the impulse-momentum theorem
The rotational form of the impulse-momentum theorem states that the angular impulse exerted on an object or rigid system is equal to the change in angular momentum

$angular impulse = ∆ L = τ ∆ t$

Where:
- angular impulse is measured in $N \cdot m \cdot s$
- $∆ L$ = change in angular momentum, measured in $kg \cdot m^{2} / s$
- $τ$ = torque exerted on the system, measured in $N \cdot m$
- $∆ t$ = time interval, measured in $s$
The equation will appear in this form on your equation sheet

In calculations, a more useful form of this equation is

$∆ L = τ ∆ t = I (ω - ω_{0})$

Where:
- $I$ = rotational inertia of the body, measured in $kg \cdot m^{2}$
- $ω$ = final angular velocity, measured in $rad / s$
- $ω_{0}$ = initial angular velocity, measured in $rad / s$
This form can be derived from the equation for angular momentum $(L = I ω)$
The rotational form of Newton’s second law is a direct result of the rotational form of the impulse-momentum theorem applied to systems with constant rotational inertia

$τ_{n e t} = \frac{∆ L}{∆ t} = \frac{I ∆ ω}{∆ t} = I α$

Where:
- $∆ ω$ = change in angular velocity, measured in $rad / s$
- $α$ = angular acceleration, measured in $rad / s^{2}$
The rotational form of the impulse-momentum theorem tells us
- for a given change in angular momentum, a small torque acting over a long time has the same effect as a large torque acting over a short time
- for a constant torque, applying the torque over a longer time will lead to a greater change in angular momentum
- for a specified time, a greater torque will lead to a greater change in angular momentum

Examiner Tip

The following equation does not appear on the equation sheet:

$τ_{n e t} = \frac{∆ L}{∆ t} = \frac{I ∆ ω}{∆ t} = I α$

Each part of this equation can be easily derived, however, using the following equations from the equation sheet:

$Δ L = τ Δ t$

$L = I ω$

$α_{s y s} = \frac{τ_{n e t}}{I_{s y s}}$

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