Rotational Kinematics Equations (College Board AP® Physics 1: Algebra-Based)

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Physics

Rotational kinematics equations

Rotating systems in a state of constant angular acceleration can be described by three rotational kinematic equations
These are analogous to the linear kinematic equations
- Each of the three rotational kinematics equations are listed on the equation sheet and will be provided in the exams
For all of these equations, the following conditions apply:
- angular acceleration is constant
- motion is relative to an axis of rotation
For all these equations:
- Time interval, $∆ t = t$ (i.e. the timer is assumed to start from zero, $t_{0} = 0$ )
- Change in angular velocity, $∆ ω = ω - ω_{0}$
- Angular displacement, $∆ θ = θ - θ_{0}$

Rotational kinematic equation 1

This equation is used when angular displacement is not required

$ω = ω_{0} + α t$

Where:
- $ω$ = final angular velocity, in $rad / s$
- $ω_{0}$ = initial angular velocity, in $rad / s$
- $α$ = angular acceleration, in $rad / s^{2}$
- $t$ = time interval, in $s$

Rotational kinematic equation 2

This equation is used when final angular velocity is not required

$θ = θ_{0} + ω_{0} t + \frac{1}{2} α t^{2}$

Where:
- $θ$ = final angular position, in $rad$
- $θ_{0}$ = initial angular position, in $rad$
- $ω_{0}$ = initial angular velocity, in $rad / s$
- $α$ = angular acceleration, in $rad / s^{2}$
- $t$ = time interval, measured in $s$

Rotational kinematic equation 3

This equation is used when time is not required

$ω^{2} = ω_{0}^{2} + 2 α (θ - θ_{0})$

Where:
- $ω$ = final angular velocity, in $rad / s$
- $ω_{0}$ = initial angular velocity, in $rad / s$
- $α$ = angular acceleration, in $rad / s^{2}$
- $θ - θ_{0} = ∆ θ$ = angular displacement, in $rad$

Other helpful equations in rotational kinematics

Angular displacement can be calculated using angular velocity and time when angular acceleration is not required

$∆ θ = \frac{1}{2} (ω_{0} + ω) t$

Where:
- $∆ θ$ = angular displacement, in $rad$
- $ω_{0}$ = initial angular velocity, in $rad / s$
- $ω$ = final angular velocity, in $rad / s$
- $t$ = time interval, in $s$
Final angular position can be calculated when initial angular velocity is not required using the following equation:

$θ = θ_{0} + ω t - \frac{1}{2} α t^{2}$

Where:
- $θ$ = final angular position, in $rad$
- $θ_{0}$ = initial angular position, in $rad$
- $ω$ = final angular velocity, in $rad / s$
- $α$ = angular acceleration, in $rad / s^{2}$
- $t$ = time interval, in $s$

Table of rotational kinematics equations

Linear equation	Rotational equation	Quantity not required
$v_{x} = v_{x 0} + a_{x} t$	$ω = ω_{0} + α t$	$∆ θ$
$x = x_{0} + v_{x 0} t + \frac{1}{2} a_{x} t^{2}$	$θ = θ_{0} + ω_{0} t + \frac{1}{2} α t^{2}$	$ω$
$v_{x}^{2} = v_{x 0}^{2} + 2 a_{x} (x - x_{0})$	$ω^{2} = ω_{0}^{2} + 2 α (θ - θ_{0})$	$t$
$∆ x = \frac{1}{2} (v_{x 0} + v_{x}) t$	$∆ θ = \frac{1}{2} (ω_{0} + ω) t$	$α$
$x = x_{0} + v_{x} t - \frac{1}{2} a_{x} t^{2}$	$θ = θ_{0} + ω t - \frac{1}{2} α t^{2}$	$ω_{0}$

Worked Example

The turntable of a record player spins at an angular velocity of $45 rpm$ just before it is turned off. Its rotation then decelerates at a rate of $0.8 rad / s^{2}$ .

Determine the number of rotations the turntable completes before it comes to rest.

Answer:

Step 1: List the known quantities

Taking the initial direction of rotation as positive
Final angular velocity, $ω = 0$
Initial angular velocity, $ω_{0} = 45 rpm$
Angular acceleration, $α = - 0.8 rad / s^{2}$

Step 2: Convert the angular velocity from rpm to rad/s

One revolution corresponds to a rotation angle of $2 π$ radians
Therefore, the initial angular velocity is:

$ω_{0} = 45 rpm \times \frac{2 π}{60} = \frac{3 π}{2} rad / s$

Step 3: Choose the relevant rotational kinematic equation

The question asks for the number of rotations completed, which is equal to the ratio

$\frac{angular displacement of the turntable}{angular displacement of one rotation}$

Therefore, the quantity we need to calculate is $∆ θ$
The quantities we know are $ω$ , $ω_{0}$ and $α$
The quantity not required in this calculation is $t$

$ω^{2} = ω_{0}^{2} + 2 α (θ - θ_{0})$

Step 4: Rearrange the equation

$0 = ω_{0}^{2} + 2 α (θ - θ_{0})$

$- 2 α (θ - θ_{0}) = ω_{0}^{2}$

$θ - θ_{0} = - \frac{ω_{0}^{2}}{2 α}$

Since angular displacement is $∆ θ = θ - θ_{0}$

$∆ θ = - \frac{ω_{0}^{2}}{2 α}$

Step 5: Substitute the known values and calculate the angular displacement

$∆ θ = - \frac{{(\frac{3 π}{2})}^{2}}{2 \times (- 0.8)} = 13.88 rad$

Step 6: Determine the number of rotations completed

There are $2 π$ radians in one rotation
Therefore, the number of rotations completed is

$\frac{angular displacement of the turntable}{angular displacement of one rotation} = \frac{13.88}{2 π} = 2.2$

This means the turntable spins 2.2 times before coming to rest

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