7.1 Rotational kinematics

Achievable AP Physics 1

7. Torque and rotational mechanics

Our AP Physics 1 course is currently in development and is a work-in-progress.

Rotational kinematics

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In Chapter 5, we analyzed motion along a straight line by tracking an object’s position $x (t)$ , velocity $v = d x / d t$ , and acceleration $a = d v / d t$ . We used kinematic equations, work-energy relations, and Newton’s laws to predict and explain linear motion.

Many physical systems, however, involve rotation about an axis rather than translation along a line. Think of a spinning wheel, a planet orbiting its star, or a windmill turning in the breeze. Just as we describe linear motion using displacement, velocity, and acceleration, we can describe rotational motion using three analogous quantities:

$angular displacement θ, angular velocity ω, angular acceleration α .$

This section develops the tools of rotational kinematics: the rotational analogues of the familiar linear relationships.

$v = \frac{d x}{d t}, a = \frac{d v}{d t}, x = x_{0} + v_{0} t + \frac{1}{2} a t^{2},$

We’ll also connect linear quantities like arc length $s$ , tangential speed $v$ , and tangential acceleration $a_{t}$ to their angular counterparts using the radius $r$ .

$s = r θ, v = r ω, a_{t} = r α .$

Once these parallels are clear, you can apply the same problem-solving style you used in linear motion to rotational versions of Newton’s laws, energy methods, and momentum conservation.

Definitions

Angular displacement $θ$ : the angle through which a point or line has been rotated in a specified sense about a specified axis. Units: radians.
Angular velocity $ω$ : the rate of change of angular displacement.

$ω = \frac{d θ}{d t}, [ω] = rad/s .$
Angular acceleration $α$ : the rate of change of angular velocity.

$α = \frac{d ω}{d t}, [α] = rad/ s^{2} .$

Relation to linear kinematics

For a point moving in a circle of radius $r$ , angular and linear quantities are linked by the geometry of a circle:

$arc length: s = r θ, tangential speed: v = r ω, tangential acceleration: a_{t} = r α .$

Relating rotational and linear kinematics

Constant- $α$ kinematic equations

When $α$ is constant, the rotational kinematic equations match the structure of the constant-acceleration equations for linear motion:

$ω_{f} Δ θ ω_{f}^{2} = ω_{i} + α t, = ω_{i} t + \frac{1}{2} α t^{2}, = ω_{i}^{2} + 2 α Δ θ .$

Example problem 1

A turntable is spinning at $ω_{i} = 10.0$ rad/s and slows with constant angular acceleration $α = - 2.0$ rad/s $^{2}$ . How long until it stops, and through what angle does it turn?

Solution.

Time to stop (use $ω_{f} = ω_{i} + α t$ with $ω_{f} = 0$ ):

$0 = 10.0 + (- 2.0) t ⟹ t = 5.0 s .$

Angle turned (use $Δ θ = ω_{i} t + \frac{1}{2} α t^{2}$ ):

$Δ θ = ω_{i} t + \frac{1}{2} α t^{2} = (10.0) (5.0) + \frac{1}{2} (- 2.0) (5.0)^{2} = 50.0 - 25.0 = 25.0 rad .$

Example problem 2

A hoop of radius $0.50$ m rolls without slipping so that its center moves at $v = 2.0$ m/s. Find its angular speed $ω$ and the time to complete one rotation.

Solution.

No slipping means the tangential speed at the rim matches the center’s speed, so $v = r ω$ :

$ω = \frac{v}{r} = \frac{2.0}{0.50} = 4.0 rad/s .$

One rotation corresponds to $Δ θ = 2 π$ rad, so the time is:

$t = \frac{Δ θ}{ω} = \frac{2 π}{4.0} = \frac{π}{2} \approx 1.57 s .$

Example problem 3

A fan blade of length $0.30$ m accelerates from rest to $1200$ rpm in $6.0$ s with constant angular acceleration.

Convert $1200$ rpm to rad/s.
Find $α$ .
How many revolutions does it make?

Solution.

Convert to rad/s:

$ω_{f} = 1200 rpm = 1200 \frac{rev}{min} \times \frac{2 π rad}{rev} \times \frac{1 min}{60 s} = 40 π rad/s .$
Find $α$ using $ω_{f} = ω_{i} + α t$ with $ω_{i} = 0$ and $t = 6.0$ s:

$40 π = α (6.0) ⟹ α = \frac{40 π}{6.0} \approx 20.94 rad/ s^{2} .$
Find the number of revolutions.

First compute total angular displacement using $Δ θ = ω_{i} t + \frac{1}{2} α t^{2}$ with $ω_{i} = 0$ :

$Δ θ = ω_{i} t + \frac{1}{2} α t^{2} = \frac{1}{2} (20.94) (6.0)^{2} = 376.92 rad .$

Then convert radians to revolutions:

$N = \frac{Δ θ}{2 π} \approx \frac{376.92}{2 π} = 60.0 revolutions .$

Example problem 4

A CD spins down from $2000$ rpm to $0$ in $40$ s under constant $α$ . How far (in meters) does a point on the rim travel? (Use $r = 0.06$ m.)

Solution.

Convert the initial angular speed to rad/s:

$ω_{i} = 2000 rpm = 2000 \frac{2 π}{60} \approx 209.44 rad/s .$
Find angular acceleration from $ω_{f} = ω_{i} + α t$ with $ω_{f} = 0$ :

$0 = ω_{i} + α t ⟹ α = - \frac{ω _{i}}{t} = - \frac{209.44}{40} = - 5.236 rad/ s^{2} .$
Find total angular displacement:

$Δ θ = ω_{i} t + \frac{1}{2} α t^{2} = (209.44) (40) + \frac{1}{2} (- 5.236) (40)^{2} = 4187.6 rad .$
Convert angular displacement to arc length using $s = r Δ θ$ :

$s = r Δ θ = 0.06 \times 4187.6 = 251.26 m .$

Example problem 5

A wheel’s angular position is given by $θ (t) = 3.0 t + 0.50 t^{2}$ (rad), where $t$ is in seconds.

Find $ω (t)$ and $α (t)$ .
Is $α$ constant?
Find $ω$ at $t = 4.0$ s.

Solution.

$ω (t) = \frac{d θ}{d t} = 3.0 + 1.0 t, α (t) = \frac{d ω}{d t} = 1.0 rad/ s^{2} .$

Yes - $α (t) = 1.0$ is constant.
Evaluate $ω (t)$ at $t = 4.0$ s:

$ω (4.0) = 3.0 + 1.0 \times 4.0 = 7.0 rad/s .$

Example problem 6

A record player spins at $33 \frac{1}{3}$ rpm. You start it from rest and reach playing speed in $4.0$ s at constant $α$ .

What is $α$ ?
How many turns does it take to get up to speed?

Solution.

Convert the final speed to rad/s, then use $α = (ω_{f} - ω_{i}) / t$ :

$ω_{f} = 33 \frac{1}{3} rpm = \frac{100}{3} \times \frac{2 π}{60} = \frac{10 π}{9} \approx 3.49 rad/s, α = \frac{ω _{f} - 0}{4.0} = \frac{3.49}{4.0} = 0.873 rad/ s^{2} .$
Use $Δ θ = \frac{1}{2} α t^{2}$ (since $ω_{i} = 0$ ), then convert radians to revolutions:

$Δ θ = \frac{1}{2} α t^{2} = \frac{1}{2} (0.873) (4.0)^{2} = 6.984 rad, N = \frac{6.984}{2 π} \approx 1.11 revolutions .$

Rotational Kinematics: Fundamental Quantities\

Angular displacement ( $θ$ ): angle rotated, units = radians\
Angular velocity ( $ω$ ): rate of change of $θ$ , $ω = d θ / d t$ , units = rad/s\
Angular acceleration ( $α$ ): rate of change of $ω$ , $α = d ω / d t$ , units = rad/s $^{2}$

Relation to Linear Kinematics\

Arc length: $s = r θ$ \
Tangential speed: $v = r ω$ \
Tangential acceleration: $a_{t} = r α$

Constant Angular Acceleration (Constant- $α$ ) Equations\

$ω_{f} = ω_{i} + α t$ \
$Δ θ = ω_{i} t + \frac{1}{2} α t^{2}$ \
$ω_{f}^{2} = ω_{i}^{2} + 2 α Δ θ$

Problem-Solving Habits\

Always use radians for angular quantities\
Maintain consistent sign convention (clockwise vs counterclockwise)\
Explicitly relate linear and angular variables via radius when needed

Key Example Takeaways\

Convert rpm to rad/s: multiply by $2 π /60$ \
For rolling without slipping: $v = r ω$ \
Total angle turned: use kinematic equations for $Δ θ$ \
Arc length traveled: $s = r Δ θ$ \
Number of revolutions: $N = Δ θ /2 π$

Preparation for Further Topics\

These concepts underpin torque, rotational inertia, and rotational energy\
Approach rotational problems using analogous strategies to linear motion

Next | 7.2 Torque and rotational inertia

Rotational kinematics

$angular displacement θ, angular velocity ω, angular acceleration α .$

This section develops the tools of rotational kinematics: the rotational analogues of the familiar linear relationships.

$v = \frac{d x}{d t}, a = \frac{d v}{d t}, x = x_{0} + v_{0} t + \frac{1}{2} a t^{2},$

We’ll also connect linear quantities like arc length $s$ , tangential speed $v$ , and tangential acceleration $a_{t}$ to their angular counterparts using the radius $r$ .

$s = r θ, v = r ω, a_{t} = r α .$

Once these parallels are clear, you can apply the same problem-solving style you used in linear motion to rotational versions of Newton’s laws, energy methods, and momentum conservation.

Definitions

Angular displacement $θ$ : the angle through which a point or line has been rotated in a specified sense about a specified axis. Units: radians.
Angular velocity $ω$ : the rate of change of angular displacement.

$ω = \frac{d θ}{d t}, [ω] = rad/s .$
Angular acceleration $α$ : the rate of change of angular velocity.

$α = \frac{d ω}{d t}, [α] = rad/ s^{2} .$

Relation to linear kinematics

For a point moving in a circle of radius $r$ , angular and linear quantities are linked by the geometry of a circle:

$arc length: s = r θ, tangential speed: v = r ω, tangential acceleration: a_{t} = r α .$

Constant- $α$ kinematic equations

When $α$ is constant, the rotational kinematic equations match the structure of the constant-acceleration equations for linear motion:

$ω_{f} Δ θ ω_{f}^{2} = ω_{i} + α t, = ω_{i} t + \frac{1}{2} α t^{2}, = ω_{i}^{2} + 2 α Δ θ .$

Example problem 1

A turntable is spinning at $ω_{i} = 10.0$ rad/s and slows with constant angular acceleration $α = - 2.0$ rad/s $^{2}$ . How long until it stops, and through what angle does it turn?

Solution.

Time to stop (use $ω_{f} = ω_{i} + α t$ with $ω_{f} = 0$ ):

$0 = 10.0 + (- 2.0) t ⟹ t = 5.0 s .$

Angle turned (use $Δ θ = ω_{i} t + \frac{1}{2} α t^{2}$ ):

$Δ θ = ω_{i} t + \frac{1}{2} α t^{2} = (10.0) (5.0) + \frac{1}{2} (- 2.0) (5.0)^{2} = 50.0 - 25.0 = 25.0 rad .$

Example problem 2

A hoop of radius $0.50$ m rolls without slipping so that its center moves at $v = 2.0$ m/s. Find its angular speed $ω$ and the time to complete one rotation.

Solution.

No slipping means the tangential speed at the rim matches the center’s speed, so $v = r ω$ :

$ω = \frac{v}{r} = \frac{2.0}{0.50} = 4.0 rad/s .$

One rotation corresponds to $Δ θ = 2 π$ rad, so the time is:

$t = \frac{Δ θ}{ω} = \frac{2 π}{4.0} = \frac{π}{2} \approx 1.57 s .$

Example problem 3

A fan blade of length $0.30$ m accelerates from rest to $1200$ rpm in $6.0$ s with constant angular acceleration.

Convert $1200$ rpm to rad/s.
Find $α$ .
How many revolutions does it make?

Solution.

Convert to rad/s:

$ω_{f} = 1200 rpm = 1200 \frac{rev}{min} \times \frac{2 π rad}{rev} \times \frac{1 min}{60 s} = 40 π rad/s .$
Find $α$ using $ω_{f} = ω_{i} + α t$ with $ω_{i} = 0$ and $t = 6.0$ s:

$40 π = α (6.0) ⟹ α = \frac{40 π}{6.0} \approx 20.94 rad/ s^{2} .$
Find the number of revolutions.

First compute total angular displacement using $Δ θ = ω_{i} t + \frac{1}{2} α t^{2}$ with $ω_{i} = 0$ :

$Δ θ = ω_{i} t + \frac{1}{2} α t^{2} = \frac{1}{2} (20.94) (6.0)^{2} = 376.92 rad .$

Then convert radians to revolutions:

$N = \frac{Δ θ}{2 π} \approx \frac{376.92}{2 π} = 60.0 revolutions .$

Example problem 4

A CD spins down from $2000$ rpm to $0$ in $40$ s under constant $α$ . How far (in meters) does a point on the rim travel? (Use $r = 0.06$ m.)

Solution.

Convert the initial angular speed to rad/s:

$ω_{i} = 2000 rpm = 2000 \frac{2 π}{60} \approx 209.44 rad/s .$
Find angular acceleration from $ω_{f} = ω_{i} + α t$ with $ω_{f} = 0$ :

$0 = ω_{i} + α t ⟹ α = - \frac{ω _{i}}{t} = - \frac{209.44}{40} = - 5.236 rad/ s^{2} .$
Find total angular displacement:

$Δ θ = ω_{i} t + \frac{1}{2} α t^{2} = (209.44) (40) + \frac{1}{2} (- 5.236) (40)^{2} = 4187.6 rad .$
Convert angular displacement to arc length using $s = r Δ θ$ :

$s = r Δ θ = 0.06 \times 4187.6 = 251.26 m .$

Example problem 5

A wheel’s angular position is given by $θ (t) = 3.0 t + 0.50 t^{2}$ (rad), where $t$ is in seconds.

Find $ω (t)$ and $α (t)$ .
Is $α$ constant?
Find $ω$ at $t = 4.0$ s.

Solution.

$ω (t) = \frac{d θ}{d t} = 3.0 + 1.0 t, α (t) = \frac{d ω}{d t} = 1.0 rad/ s^{2} .$

Yes - $α (t) = 1.0$ is constant.
Evaluate $ω (t)$ at $t = 4.0$ s:

$ω (4.0) = 3.0 + 1.0 \times 4.0 = 7.0 rad/s .$

Example problem 6

A record player spins at $33 \frac{1}{3}$ rpm. You start it from rest and reach playing speed in $4.0$ s at constant $α$ .

What is $α$ ?
How many turns does it take to get up to speed?

Solution.

Convert the final speed to rad/s, then use $α = (ω_{f} - ω_{i}) / t$ :

$ω_{f} = 33 \frac{1}{3} rpm = \frac{100}{3} \times \frac{2 π}{60} = \frac{10 π}{9} \approx 3.49 rad/s, α = \frac{ω _{f} - 0}{4.0} = \frac{3.49}{4.0} = 0.873 rad/ s^{2} .$
Use $Δ θ = \frac{1}{2} α t^{2}$ (since $ω_{i} = 0$ ), then convert radians to revolutions:

$Δ θ = \frac{1}{2} α t^{2} = \frac{1}{2} (0.873) (4.0)^{2} = 6.984 rad, N = \frac{6.984}{2 π} \approx 1.11 revolutions .$

Achievable AP Physics 1

7. Torque and rotational mechanics

Our AP Physics 1 course is currently in development and is a work-in-progress.