Rotational Motion | AP - Wyatt's Notes

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Rotational Kinematics (CED Unit 7)

Angular Quantities

Linear Quantity	Angular Quantity	Relation
Displacement $x$	Angle $\theta$	$x = r\theta$
Velocity $v$	Angular velocity $\omega$	$v = r\omega$
Acceleration $a$	Angular acceleration $\alpha$	$a_t = r\alpha$

Angular Velocity and Acceleration

\omega = \frac{d\theta}{dt}, \qquad \alpha = \frac{d\omega}{dt}

Constant Angular Acceleration Equations

\omega = \omega_0 + \alpha t

\theta = \theta_0 + \omega_0 t + \frac{1}{2}\alpha t^2

\omega^2 = \omega_0^2 + 2\alpha(\theta - \theta_0)

Tangential vs Centripetal Acceleration

For circular motion, the total acceleration has two components:

A_t = r\alpha \quad \mathrm{(tangential, changes speed)

A_c = r\omega^2 = \frac{v^2}{r} \quad \mathrm{(centripetal, changes direction)

A_{\mathrm{total} = \sqrt{a_t^2 + a_c^2}

The Analogy Between Linear and Rotational Kinematics

Every linear kinematic equation has a direct rotational analogue. Replace $x$ with $\theta$ , $v$ With $\omega$ And $a$ with $\alpha$ . This is not a coincidence: it reflects the fact that rotation Is a one-dimensional motion in the angular coordinate. The mathematics is identical.

Example A wheel starts from rest and accelerates at $2.0 \mathrm{ rad/s^2$ for $5.0 \mathrm{ s$ . Find the Angular velocity, total angle rotated, and the tangential speed of a point 0.3 m from the center.

\omega = 0 + (2.0)(5.0) = 10 \mathrm{ rad/s

\theta = 0 + 0 + \frac{1}{2}(2.0)(5.0)^2 = 25 \mathrm{ rad

V = r\omega = (0.3)(10) = 3.0 \mathrm{ m/s

Moment of Inertia (CED Unit 7)

The moment of inertia is the rotational analog of mass. It measures an object”s resistance to Angular acceleration:

I = \sum m_i r_i^2

For a continuous body:

I = \int r^2\, dm

Why the Moment of Inertia Depends on the Axis

Mass is a scalar, but moment of inertia depends on how the mass is distributed relative to the axis Of rotation. A rod rotated about its center has $I = \frac{1}{12}ML^2$ But the same rod rotated About one end has $I = \frac{1}{3}ML^2$ — four times larger. The same physical object can have Different moments of inertia depending on the axis. This has no linear analogue: mass is mass, Regardless of the direction of motion.

Common Moments of Inertia

Object	Axis	Moment of Inertia
Thin rod, length $L$ Mass $M$	Center, perpendicular to rod	$\frac{1}{12}ML^2$
Thin rod, length $L$ Mass $M$	End, perpendicular to rod	$\frac{1}{3}ML^2$
Solid cylinder/disk, radius $R$	Central axis	$\frac{1}{2}MR^2$
Hollow cylinder, radius $R$	Central axis	$MR^2$
Solid sphere, radius $R$	Diameter	$\frac{2}{5}MR^2$
Hollow sphere, radius $R$	Diameter	$\frac{2}{3}MR^2$
Point mass $m$ at distance $r$	—	$mr^2$

Derivation: Moment of Inertia of a Solid Cylinder

Consider a solid cylinder of mass $M$ Radius $R$ And length $L$ Rotating about its central axis. Divide the cylinder into thin cylindrical shells of radius $r$ and thickness $dr$ .

Dm = \frac{M}{\pi R^2 L} \cdot 2\pi r L\, dr = \frac{2M}{R^2} r\, dr

I = \int_0^R r^2\, dm = \int_0^R r^2 \frac{2M}{R^2} r\, dr = \frac{2M}{R^2} \int_0^R r^3\, dr = \frac{2M}{R^2} \cdot \frac{R^4}{4} = \frac{1}{2}MR^2

Parallel Axis Theorem

I = I_{\mathrm{cm} + Md^2

Where $d$ is the distance from the center of mass to the new axis.

Proof of the Parallel Axis Theorem

Consider a body rotating about an axis parallel to an axis through the center of mass, at distance $d$ . Let $r_i$ be the distance from the CM axis to mass element $m_i$ And $R_i$ be the distance From the new axis. Then $R_i^2 = r_i^2 + d^2 - 2r_i d\cos\phi_i$ .

I = \sum m_i R_i^2 = \sum m_i r_i^2 + \sum m_i d^2 - 2d \sum m_i r_i \cos\phi_i

The last term is zero because the CM is at the origin: $\sum m_i r_i \cos\phi_i = 0$ .

I = I_{\mathrm{cm} + Md^2

$\blacksquare$

Example Find the moment of inertia of a solid sphere of mass $M$ and radius $R$ about an axis tangent to its Surface.

I = I_{\mathrm{cm} + Md^2 = \frac{2}{5}MR^2 + MR^2 = \frac{7}{5}MR^2

Torque (CED Unit 7)

Torque is the rotational analog of force:

\vec{\tau} = \vec{r} \times \vec{F}

The magnitude is:

\tau = rF\sin\theta = Fd

Where $d = r\sin\theta$ is the moment arm (perpendicular distance from the axis to the line of Action of the force).

Sign Convention

Counterclockwise (CCW) torque: positive
Clockwise (CW) torque: negative

Newton’s Second Law for Rotation (CED Unit 7)

\sum \tau = I\alpha

This is the rotational analog of $\sum F = ma$ .

Example A solid disk of mass $5.0 \mathrm{ kg$ and radius $0.3 \mathrm{ m$ is mounted on a frictionless Axle. A $20 \mathrm{ N$ force is applied tangentially at the rim. Find the angular acceleration.

I = \frac{1}{2}MR^2 = \frac{1}{2}(5.0)(0.3)^2 = 0.225 \mathrm{ kg \cdot \mathrm{m^2

\tau = Fr = (20)(0.3) = 6.0 \mathrm{ N \cdot \mathrm{m

\alpha = \frac{\tau}{I} = \frac{6.0}{0.225} = 26.7 \mathrm{ rad/s^2

Rotational Energy (CED Unit 7)

The kinetic energy of a rotating object:

K_{\mathrm{rot} = \frac{1}{2}I\omega^2

For an object that is both translating and rotating (like a rolling ball):

K_{\mathrm{total} = \frac{1}{2}mv_{\mathrm{cm}^2 + \frac{1}{2}I_{\mathrm{cm}\omega^2

Rolling Without Slipping

When an object rolls without slipping: $v_{\mathrm{cm} = R\omega$ .

Conservation of Energy with Rotation

Mgh_i + \frac{1}{2}mv_i^2 + \frac{1}{2}I\omega_i^2 = mgh_f + \frac{1}{2}mv_f^2 + \frac{1}{2}I\omega_f^2

Why Hollow Objects Roll Slower Than Solid Ones

A hollow cylinder has $I = MR^2$ So its kinetic energy is split as $K = \frac{1}{2}Mv^2 + \frac{1}{2}(MR^2)(v/R)^2 = Mv^2$ . Half the energy goes to translation and half To rotation. A solid cylinder has $I = \frac{1}{2}MR^2$ So $K = \frac{1}{2}Mv^2 + \frac{1}{4}Mv^2 = \frac{3}{4}Mv^2$ . More energy goes to translation, so the Solid cylinder moves faster.

Example A solid sphere of mass $M$ and radius $R$ rolls without slipping from rest down an incline of height $h$ and angle $\theta$ . Find its speed at the bottom.

Using conservation of energy (starting from rest, ending at the bottom):

Mgh = \frac{1}{2}Mv^2 + \frac{1}{2}\!\left(\frac{2}{5}MR^2\right)\!\left(\frac{v}{R}\right)^2

Mgh = \frac{1}{2}Mv^2 + \frac{1}{5}Mv^2 = \frac{7}{10}Mv^2

V = \sqrt{\frac{10gh}{7}}

Note: the answer is independent of both the mass and the radius. For a hollow sphere, the factor Would be $\frac{5}{6}$ instead of $\frac{7}{10}$ So the solid sphere is always faster.

Angular Momentum (CED Unit 7)

The angular momentum of a particle:

\vec{L} = \vec{r} \times \vec{p} = \vec{r} \times m\vec{v}

For a rigid body rotating about a fixed axis:

L = I\omega

Conservation of Angular Momentum

If the net external torque on a system is zero:

\vec{L}_{\mathrm{initial} = \vec{L}_{\mathrm{final} \implies I_1\omega_1 = I_2\omega_2

Rotational Impulse

\int \tau\, dt = \Delta L = I\omega_f - I\omega_i

Example A figure skater with moment of inertia $4.0 \mathrm{ kg \cdot \mathrm{m^2$ is spinning at $3.0 \mathrm{ rad/s$ . She pulls in her arms, reducing her moment of inertia to $1.5 \mathrm{ kg \cdot \mathrm{m^2$ . Find her new angular speed.

I_1\omega_1 = I_2\omega_2

(4.0)(3.0) = (1.5)\omega_2 \implies \omega_2 = \frac{12.0}{1.5} = 8.0 \mathrm{ rad/s

Why Angular Momentum Is a Vector

Angular momentum is a vector pointing along the axis of rotation, with direction given by the Right-hand rule. Conservation of angular momentum applies to each component separately. This is why A spinning top precesses: gravity exerts a torque that changes the direction (but not the magnitude) Of the angular momentum vector.

Torque and the Cross Product (AP Physics C)

The torque vector is:

\vec{\tau} = \vec{r} \times \vec{F}

In component form:

\vec{\tau} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ r_x & r_y & r_z \\ F_x & F_y & F_z \end{vmatrix}

The magnitude: $\tau = rF\sin\theta$ Where $\theta$ is the angle from $\vec{r}$ to $\vec{F}$ .

The direction is determined by the right-hand rule.

Static Equilibrium

An object is in static equilibrium when:

$\sum \vec{F} = 0$ (translational equilibrium)
$\sum \vec{\tau} = 0$ (rotational equilibrium)

Choosing the Pivot Point

The choice of pivot point is arbitrary when the system is in equilibrium. Choosing the pivot at an Unknown force eliminates that force from the torque equation.

flowchart TD
    A[5_Rotational Motion] --> B[Key Concepts]
    A --> C[Core Principles]
    A --> D[Practical Applications]
    B --> E[Fundamental definitions]
    C --> F[Design patterns]
    D --> G[Real-world usage]

Intuition

Rotational motion is the mirror image of linear motion — every linear concept has a rotational analogue. Replace $x$ with $\theta$ , $v$ with $\omega$ , $a$ with $\alpha$ , $m$ with $I$ , and $F$ with $\tau$ . The equations are identical in form. This isn’t a coincidence — it reflects the fact that rotation is one-dimensional motion in the angular coordinate.

Moment of inertia intuition: Just as mass resists linear acceleration, moment of inertia ( $I$ ) resists angular acceleration. But unlike mass, $I$ depends on how the mass is distributed relative to the axis. A hollow cylinder has more $I$ than a solid one of the same mass because its mass is farther from the axis. This is why a figure skater spins faster when pulling in her arms — she reduces $I$ , and angular momentum conservation forces $\omega$ to increase.

Rolling intuition: When a ball rolls without slipping, its kinetic energy splits between translation ( $\frac{1}{2}mv^2$ ) and rotation ( $\frac{1}{2}I\omega^2$ ). Objects with more mass near the rim (hollow cylinder) put more energy into rotation and less into translation, so they roll slower down an incline than solid objects.

Angular momentum conservation: When no external torque acts, $L = I\omega$ is constant. This is why gyroscopes precess, why planets move faster when closer to the Sun (Kepler’s second law), and why neutron stars spin incredibly fast.

Common Pitfalls

Using the wrong moment of inertia. Always check which axis the object rotates about. Use the parallel axis theorem when the axis does not pass through the center of mass.
Confusing torque and force. Torque depends on both the force and the moment arm (distance from the axis).
Forgetting that angular momentum is a vector. Direction matters and is determined by the right-hand rule.
Incorrectly applying the rolling condition. Rolling without slipping means $v = R\omega$ Not $v = R\alpha$ .
Not accounting for rotational KE in energy problems. Rolling objects have both translational and rotational kinetic energy.
Sign errors with torque. Be consistent with the sign convention (CCW positive, CW negative).
Assuming angular momentum is conserved when external torques act. Conservation requires zero net external torque.
Confusing moment of inertia with mass. Mass is a scalar; moment of inertia depends on the axis.
Using the wrong radius in $v = R\omega$ . $R$ is the radius of the rolling object, not the radius of the incline or the path.

Practice Questions

A disk of mass $8.0 \mathrm{ kg$ and radius $0.5 \mathrm{ m$ is mounted on a frictionless axle. A block of mass $2.0 \mathrm{ kg$ is attached to a string wrapped around the disk. Find the acceleration of the block.
A solid cylinder and a hollow cylinder of the same mass and radius are released from rest at the top of an incline. Which reaches the bottom first? Prove your answer.
A uniform rod of length $L$ and mass $M$ is pivoted at one end and held horizontally, then released. Find its angular speed as it passes through the vertical position.
A merry-go-round of radius $3.0 \mathrm{ m$ and moment of inertia $600 \mathrm{ kg \cdot \mathrm{m^2$ is rotating at $0.50 \mathrm{ rad/s$ . A child of mass $30 \mathrm{ kg$ jumps on at the edge. Find the new angular speed.
A $4.0 \mathrm{ m$ uniform beam of mass $50 \mathrm{ kg$ is supported by two vertical cables, one at each end. A $200 \mathrm{ kg$ mass hangs $1.0 \mathrm{ m$ from the left end. Find the tension in each cable.
Calculate the moment of inertia of a uniform solid sphere about its diameter by integration.
A $0.50 \mathrm{ kg$ ball of radius $0.05 \mathrm{ m$ rolls without slipping down a $30^\circ$ incline from a height of $2.0 \mathrm{ m$ . Find the translational and rotational speeds at the bottom.
Derive the parallel axis theorem from the definition of moment of inertia.
A solid sphere rolls without slipping up a $20^\circ$ incline. If its initial speed is $5.0 \mathrm{ m/s$ How far up the incline does it travel before stopping and rolling back?
A flywheel of moment of inertia $50 \mathrm{ kg \cdot \mathrm{m^2$ rotating at $300 \mathrm{ rpm$ is brought to rest by a constant frictional torque of $10 \mathrm{ N \cdot \mathrm{m$ in $785 \mathrm{ s$ . Verify this using the rotational impulse equation, and calculate the angle through which the flywheel rotates while stopping.

11. Moment of Inertia: Extended Derivations

Derivation: Solid Sphere About Its Diameter

Divide a solid sphere of mass $M$ and radius $R$ into thin disks of radius $r$ and thickness $dz$ at Height $z$ from the centre.

$r^2 = R^2 - z^2$

The volume of each disk is $dV = \pi r^2 dz = \pi(R^2 - z^2)dz$ .

$dm = \frac{M}{\frac{4}{3}\pi R^3} \cdot \pi(R^2 - z^2)dz = \frac{3M}{4R^3}(R^2 - z^2)dz$

$I = \int_{-R}^{R} \frac{1}{2}r^2\, dm = \int_{-R}^{R} \frac{1}{2}(R^2 - z^2) \cdot \frac{3M}{4R^3}(R^2 - z^2)dz$

$= \frac{3M}{8R^3}\int_{-R}^{R} (R^2 - z^2)^2 dz = \frac{3M}{8R^3}\int_{-R}^{R} (R^4 - 2R^2z^2 + z^4)dz$

$= \frac{3M}{8R^3}\left[R^4 \cdot 2R - 2R^2 \cdot \frac{2R^3}{3} + \frac{2R^5}{5}\right] = \frac{3M}{8R^3}\left[2R^5 - \frac{4R^5}{3} + \frac{2R^5}{5}\right]$

$= \frac{3M}{8R^3} \cdot \frac{2R^5}{15}(15 - 10 + 3) = \frac{3M}{8R^3} \cdot \frac{16R^5}{15} = \frac{2MR^2}{5}$

Derivation: Thin Rod About One End

A thin uniform rod of mass $M$ and length $L$ Pivoted at one end.

$I = \int_0^L x^2 \frac{M}{L}\, dx = \frac{M}{L}\left[\frac{x^3}{3}\right]_0^L = \frac{ML^2}{3}$

Using the parallel axis theorem: $I_{\mathrm{end} = I_{\mathrm{cm} + Md^2 = \frac{ML^2}{12} + M\left(\frac{L}{2}\right)^2 = \frac{ML^2}{12} + \frac{ML^2}{4} = \frac{ML^2}{3}$ . Confirmed.

12. Rolling Without Slipping: Extended Analysis

Worked Example: Race Down an Incline

A solid sphere, a solid cylinder, a hollow sphere, and a hollow cylinder, all of mass $M$ and radius $R$ Are released from rest at the top of an incline of height $h$ . Rank them by their speed at the Bottom.

Object	$I_{\mathrm{cm}$	$v$ at bottom	Fraction as KE of translation
Solid sphere	$\frac{2}{5}MR^2$	$\sqrt{10gh/7}$	$5/7 = 71\%$
Solid cylinder	$\frac{1}{2}MR^2$	$\sqrt{4gh/3}$	$2/3 = 67\%$
Hollow sphere	$\frac{2}{3}MR^2$	$\sqrt{6gh/5}$	$3/5 = 60\%$
Hollow cylinder	$MR^2$	$\sqrt{gh}$	$1/2 = 50\%$

Ranking (fastest to slowest): solid sphere, solid cylinder, hollow sphere, hollow cylinder. Objects With more mass concentrated near the rim (larger $I$ ) have more rotational KE and less translational KE, so they move more slowly.

Worked Example: Rolling Up an Incline

A solid sphere rolls without slipping up a $20^{\circ}$ incline with initial speed $5 \mathrm{ m/s$ . How far up does it travel?

$Mgh = \frac{1}{2}Mv^2 + \frac{1}{2}I\omega^2 = \frac{1}{2}Mv^2 + \frac{1}{2}\cdot\frac{2}{5}MR^2\cdot\frac{v^2}{R^2} = \frac{7}{10}Mv^2$

$h = \frac{7v^2}{10g} = \frac{7 \times 25}{10 \times 9.8} = \frac{175}{98} = 1.786 \mathrm{ m$

$d = \frac{h}{\sin 20^{\circ}} = \frac{1.786}{0.342} = 5.22 \mathrm{ m$

13. Angular Momentum: Extended Examples

Worked Example: Angular Momentum of a Collision

A $0.05 \mathrm{ kg$ ball moving at $8 \mathrm{ m/s$ strikes a rod of mass $2 \mathrm{ kg$ and Length $1 \mathrm{ m$ that is free to rotate about one end. The ball strikes the rod at its free End, perpendicular to the rod. Find the angular velocity of the rod after the collision.

Conservation of angular momentum about the pivot (the external forces at the pivot produce no Torque about the pivot):

$L_{\mathrm{initial} = L_{\mathrm{final}$

$mv \cdot L = I_{\mathrm{rod}\omega + mv' \cdot L$

Since the ball sticks to the end of the rod (perfectly inelastic collision):

$mvr = (I_{\mathrm{rod} + mr^2)\omega = \left(\frac{1}{3}Mr^2 + mr^2\right)\omega$

$\omega = \frac{mvr}{r^2\left(\frac{M}{3} + m\right)} = \frac{mv}{r\left(\frac{M}{3} + m\right)} = \frac{0.05 \times 8}{1 \times \left(\frac{2}{3} + 0.05\right)}$

$= \frac{0.4}{0.717} = 0.558 \mathrm{ rad/s$

Worked Example: Person on a Rotating Platform

A merry-go-round of radius $3 \mathrm{ m$ and moment of inertia $600 \mathrm{ kg\cdot\mathrm{m^2$ Rotates at $0.5 \mathrm{ rad/s$ . A $60 \mathrm{ kg$ person standing at the edge walks to the Centre. Find the new angular speed.

Initially: $I_i = 600 + 60 \times 3^2 = 600 + 540 = 1140 \mathrm{ kg\cdot\mathrm{m^2$ .

Finally: $I_f = 600 + 60 \times 0^2 = 600 \mathrm{ kg\cdot\mathrm{m^2$ .

$I_i\omega_i = I_f\omega_f$

$1140 \times 0.5 = 600 \times \omega_f$

$\omega_f = \frac{570}{600} = 0.95 \mathrm{ rad/s$

The angular speed nearly doubles. This is the same principle behind figure skaters pulling in their Arms to spin faster.

14. Static Equilibrium: Extended Examples

Worked Example: Ladder Against a Wall

A uniform ladder of mass $15 \mathrm{ kg$ and length $4 \mathrm{ m$ leans against a smooth (frictionless) wall at $65^{\circ}$ to the horizontal. The floor is rough with $\mu_s = 0.4$ . Will The ladder slip?

Take torques about the base of the ladder (eliminates the friction and normal force at the base):

Forces on the ladder:

Weight $Mg$ at the centre ( $2 \mathrm{ m$ from base)
Normal force from wall $N_w$ at the top ( $4 \mathrm{ m$ from base, horizontal)
Normal force from floor $N_f$ at the base (vertical)
Friction $f$ at the base (horizontal, towards wall)

Torque equation (about base):

Clockwise (positive): $Mg \times 2\cos 65^{\circ} = 15 \times 9.8 \times 2 \times 0.4226 = 124.2 \mathrm{ N\cdot\mathrm{m$

Anticlockwise (negative): $N_w \times 4\sin 65^{\circ} = N_w \times 3.625$

$N_w \times 3.625 = 124.2 \implies N_w = 34.3 \mathrm{ N$

Horizontal equilibrium: $f = N_w = 34.3 \mathrm{ N$

Vertical equilibrium: $N_f = Mg = 147 \mathrm{ N$

Check slipping: $f_{\max} = \mu_s N_f = 0.4 \times 147 = 58.8 \mathrm{ N$

Since $f = 34.3 \mathrm{ N \lt 58.8 \mathrm{ N = f_{\max}$ The ladder does not slip.

15. Summary Table: Linear vs Rotational Quantities

Linear	Rotational	Relationship
$x$	$\theta$	$x = r\theta$
$v$	$\omega$	$v = r\omega$
$a$	$\alpha$	$a_t = r\alpha$
$m$	$I$	$I = \int r^2 dm$
$F$	$\tau$	$\tau = rF\sin\theta$
$p = mv$	$L = I\omega$	$L = r \times p$
$K = \frac{1}{2}mv^2$	$K = \frac{1}{2}I\omega^2$	$K_{\mathrm{total} = \frac{1}{2}mv^2 + \frac{1}{2}I\omega^2$
$F = ma$	$\tau = I\alpha$	Newton’s second law for rotation
$W = Fd$	$W = \tau\theta$	Work-energy theorem for rotation

16. Practice Questions (Additional)

A uniform rod of mass $4 \mathrm{ kg$ and length $2 \mathrm{ m$ is pivoted at one end and held horizontally. It is released from rest. Find (a) the initial angular acceleration, (b) the angular speed as it passes through the vertical, and (c) the angular speed when it makes an angle of $45^{\circ}$ with the vertical.
A solid cylinder of mass $10 \mathrm{ kg$ and radius $0.2 \mathrm{ m$ rolls without slipping down an incline of $30^{\circ}$ from a height of $2 \mathrm{ m$ . Find the linear speed at the bottom and the time taken.
A disk of moment of inertia $0.5 \mathrm{ kg\cdot\mathrm{m^2$ rotating at $20 \mathrm{ rad/s$ has a braking torque of $0.2 \mathrm{ N\cdot\mathrm{m$ applied. Find (a) the angular deceleration, (b) the time to stop, and (c) the number of revolutions made while stopping.
A $3 \mathrm{ m$ uniform beam of mass $40 \mathrm{ kg$ is hinged at a wall and supported by a cable at $30^{\circ}$ to the beam at the far end. A $100 \mathrm{ kg$ mass hangs $1 \mathrm{ m$ from the hinge. Find the tension in the cable and the force at the hinge.
Calculate the moment of inertia of a hollow cylinder (thin-walled) of mass $M$ and radius $R$ about its central axis by integration. Verify the result using the parallel axis theorem if necessary.

Extended Worked Examples

Example 16: Gyroscopic Precession

A spinning bicycle wheel of mass $2 \mathrm{ kg$ and radius $0.35 \mathrm{ m$ is supported on one End of its axle. The wheel spins at $50 \mathrm{ rad/s$ and the axle is horizontal. The distance From the support to the wheel centre is $0.15 \mathrm{ m$ . Calculate the precession angular Velocity.

Step 1: Moment of inertia of the wheel (thin ring approximation)

$I = MR^2 = 2 \times 0.35^2 = 0.245 \mathrm{ kg\cdot\mathrm{m^2$

Step 2: Angular momentum of the wheel

$L = I\omega = 0.245 \times 50 = 12.25 \mathrm{ kg\cdot\mathrm{m^2/\mathrm{s$

Step 3: Torque due to gravity about the support

$\tau = Mg \times d = 2 \times 9.8 \times 0.15 = 2.94 \mathrm{ N\cdot\mathrm{m$

Step 4: Precession angular velocity

$\omega_p = \frac{\tau}{L} = \frac{2.94}{12.25} = 0.240 \mathrm{ rad/s$

The wheel precesses at $0.240 \mathrm{ rad/s$ Completing one revolution in $T = 2\pi/\omega_p = 26.2 \mathrm{ s$ .

Example 17: Moment of Inertia of a Composite Object

A uniform rod of mass $3 \mathrm{ kg$ and length $2 \mathrm{ m$ has a point mass of $2 \mathrm{ kg$ attached at one end. Calculate the moment of inertia about (a) the end with the Point mass, and (b) the centre of the rod.

Part (a): About the end with the point mass

Rod about its end: $I_{\mathrm{rod} = \frac{1}{3}ML^2 = \frac{1}{3} \times 3 \times 4 = 4 \mathrm{ kg\cdot\mathrm{m^2$

Point mass at the pivot: $I_{\mathrm{pm} = mr^2 = 2 \times 0^2 = 0 \mathrm{ kg\cdot\mathrm{m^2$

$I_a = 4 + 0 = 4 \mathrm{ kg\cdot\mathrm{m^2$

Part (b): About the centre of the rod

Rod about its centre: $I_{\mathrm{rod} = \frac{1}{12}ML^2 = \frac{1}{12} \times 3 \times 4 = 1 \mathrm{ kg\cdot\mathrm{m^2$

Point mass is $1 \mathrm{ m$ from the centre: $I_{\mathrm{pm} = 2 \times 1^2 = 2 \mathrm{ kg\cdot\mathrm{m^2$

$I_b = 1 + 2 = 3 \mathrm{ kg\cdot\mathrm{m^2$

Verify with parallel axis theorem (part a from part b):

$I_a = I_b + M_{\mathrm{total}d^2 = 3 + 5 \times 1^2 = 8 \mathrm{ kg\cdot\mathrm{m^2$

Wait — this gives $8$ Not $4$ . Let me recheck. The parallel axis theorem requires the total mass To be at the centre of mass of the entire system, not just the rod.

Centre of mass from the pivot (end with point mass):

$x_{\mathrm{cm} = \frac{3 \times 1 + 2 \times 0}{5} = 0.6 \mathrm{ m from pivot$

Now using the parallel axis theorem from the centre of mass:

$I_{\mathrm{cm} = I_b = 3 \mathrm{ kg\cdot\mathrm{m^2 \quad \mathrm{(already calculated)$

$I_a = I_{\mathrm{cm} + M_{\mathrm{total} \times x_{\mathrm{cm}^2 = 3 + 5 \times 0.36 = 3 + 1.8 = 4.8 \mathrm{ kg\cdot\mathrm{m^2$

This still does not match. Let me recalculate $I_b$ more carefully. The point mass is at the end Of the rod, which is $1 \mathrm{ m$ from the centre of the rod. So $I_b = 1 + 2 \times 1^2 = 3$ is Correct.

And $I_a$ directly: rod about its far end (away from point mass): use parallel axis from centre, $I_{\mathrm{rod, end} = \frac{1}{12}(3)(4) + 3(1)^2 = 1 + 3 = 4$ . Point mass at distance $2 \mathrm{ m$ : $I_{\mathrm{pm} = 2 \times 4 = 8$ . Wait — the point mass is at the pivot, so $r = 0$ Giving $I_a = 4 + 0 = 4$ .

The parallel axis check failed because I was not careful. The correct check: $I_{\mathrm{cm} = 3$ About the centre of mass at $0.6 \mathrm{ m$ from pivot, so $I_a = 3 + 5(0.6)^2 = 3 + 1.8 = 4.8$ . But direct calculation gives $4$ .

Let me recheck $I_b$ : the point mass is at one end of the rod, which is $1 \mathrm{ m$ from the Rod’s centre. $I_{\mathrm{pm} = 2 \times 1^2 = 2$ . $I_{\mathrm{rod, centre} = 1$ . $I_b = 3$ . Correct.

The discrepancy means $I_{\mathrm{cm} \ne I_b$ . The centre of mass of the system is at $0.6 \mathrm{ m$ from the pivot (not at the centre of the rod). So $I_{\mathrm{cm}$ should be Calculated about the system’s centre of mass, not the rod’s centre. This is a subtle but important Point.

Example 18: Yo-Yo Problem

A yo-yo consists of two disks of total mass $0.1 \mathrm{ kg$ and radius $3 \mathrm{ cm$ With an Axle of radius $0.5 \mathrm{ cm$ . The string unwinds from the axle. Find the acceleration of the Yo-yo as it falls and the tension in the string.

Step 1: Equations of motion

Translation: $mg - T = ma$

Rotation: $TR = I\alpha$ Where $R$ is the axle radius and $a = R\alpha$ .

$T \times 0.005 = I \times \frac{a}{0.005}$

For the yo-yo (solid disk): $I = \frac{1}{2}MR_{\mathrm{disk}^2 = \frac{1}{2} \times 0.1 \times 0.03^2 = 4.5 \times 10^{-5} \mathrm{ kg\cdot\mathrm{m^2$

$T = \frac{Ia}{R^2} = \frac{4.5 \times 10^{-5} \times a}{2.5 \times 10^{-5}} = 1.8a$

Step 2: Substitute into translational equation

$mg - 1.8a = ma$

$a = \frac{mg}{m + 1.8} = \frac{0.1 \times 9.8}{0.1 + 1.8} = \frac{0.98}{1.9} = 0.516 \mathrm{ m/s^2$

$T = 1.8 \times 0.516 = 0.929 \mathrm{ N$

Common Pitfalls Extended

Pitfall 6: Using the Wrong Radius in Torque Calculations

In problems involving wheels, pulleys, or drums, the radius used in $\tau = Fr$ must be the radius At which the force is applied, which may differ from the overall radius. For example, a force Applied at a string wound around an axle uses the axle radius, not the wheel radius.

Pitfall 7: Confusing Angular Velocity with Linear Velocity in Rolling

For rolling without slipping, $v = r\omega$ relates the centre-of-mass velocity to the angular Velocity about the centre of mass. A point on the rim has a velocity of $2v$ at the top and $0$ at The contact point (instantaneously at rest).

Pitfall 8: Forgetting Units in Moment of Inertia

Moment of inertia has units of $\mathrm{kg\cdot\mathrm{m^2$ . A common error is to use centimetres Instead of metres when calculating $I = mr^2$ Giving answers that are off by a factor of $10^4$ . Always convert to SI units before calculating.

Additional Practice Problems

A solid sphere of mass $4 \mathrm{ kg$ and radius $0.1 \mathrm{ m$ rolls without slipping up a $20^\circ$ incline at $5 \mathrm{ m/s$ . Calculate how far up the incline it travels before stopping and rolling back.
A figure skater with arms extended has moment of inertia $4.5 \mathrm{ kg\cdot\mathrm{m^2$ and rotates at $2 \mathrm{ rad/s$ . She pulls her arms in, reducing her moment of inertia to $1.5 \mathrm{ kg\cdot\mathrm{m^2$ . Calculate her new angular velocity and the ratio of final to initial rotational KE.
A uniform rod of mass $8 \mathrm{ kg$ and length $3 \mathrm{ m$ is hinged at one end and held horizontally. It is released from rest. Calculate the angular acceleration just after release, the angular velocity as it passes through the vertical, and the force at the hinge at that instant.
A $500 \mathrm{ g$ ball of radius $5 \mathrm{ cm$ rolls without slipping along a horizontal surface at $4 \mathrm{ m/s$ . It encounters a ramp of height $0.5 \mathrm{ m$ . Does it reach the top? If so, what is its speed at the top?
Two flywheels ( $I_1 = 2 \mathrm{ kg\cdot\mathrm{m^2$ spinning at $300 \mathrm{ rpm$ $I_2 = 5 \mathrm{ kg\cdot\mathrm{m^2$ at rest) are coupled together. Calculate the final angular velocity and the energy lost in the coupling process.

Practice Problems

Question 1: Rolling without slipping down an incline

A solid sphere of mass $2 \mathrm{ kg$ and radius $0.1 \mathrm{ m$ rolls without slipping down a $30^\circ$ incline from rest. Calculate (a) the acceleration of the centre of mass, (b) the angular Acceleration, and (c) the speed after rolling $3 \mathrm{ m$ along the incline.

Answer

For a solid sphere, $I = \frac{2}{5}mr^2$ .

Acceleration: $a = \frac{g\sin\theta}{1 + I/(mr^2)} = \frac{g\sin\theta}{1 + 2/5} = \frac{9.8 \times 0.5}{1.4} = 3.5 \mathrm{ m/s^2$ .

Angular acceleration: $\alpha = a/r = 3.5/0.1 = 35 \mathrm{ rad/s^2$ .

Speed: $v^2 = 2ad = 2 \times 3.5 \times 3 = 21$ , $v = 4.58 \mathrm{ m/s$ .

Question 2: Conservation of angular momentum

A figure skater with arms extended has a moment of inertia of $4.5 \mathrm{ kg\cdot m^2$ and Rotates at $2 \mathrm{ rad/s$ . When she pulls her arms in, her moment of inertia decreases to $1.5 \mathrm{ kg\cdot m^2$ . Calculate her new angular velocity and the ratio of her new kinetic Energy to the original.

Answer

Conservation of angular momentum: $L = I_1\omega_1 = I_2\omega_2$ .

$\omega_2 = \frac{I_1\omega_1}{I_2} = \frac{4.5 \times 2}{1.5} = 6 \mathrm{ rad/s$ .

KE ratio: $\frac{KE_2}{KE_1} = \frac{\frac{1}{2}I_2\omega_2^2}{\frac{1}{2}I_1\omega_1^2} = \frac{1.5 \times 36}{4.5 \times 4} = \frac{54}{18} = 3$ .

The kinetic energy triples. The extra energy comes from the work done by the skater in pulling her Arms inward against the centrifugal tendency.

Question 3: Torque and equilibrium of a beam

A uniform beam of length $4 \mathrm{ m$ and mass $20 \mathrm{ kg$ is supported at its left end by A hinge and at a point $1 \mathrm{ m$ from the right end by a cable making $30^\circ$ with the Horizontal. A $50 \mathrm{ kg$ mass hangs from the right end. Find the tension in the cable and the Force exerted by the hinge.

Answer

Taking torques about the hinge (left end):

Forces and distances from hinge:

Weight of beam: $20g$ at $2 \mathrm{ m$ .
Hanging mass: $50g$ at $4 \mathrm{ m$ .
Cable tension $T$ at $3 \mathrm{ m$ from hinge, at $30^\circ$ above horizontal. Vertical component = $T\sin(30^\circ) = T/2$ .

Clockwise torques: $20g \times 2 + 50g \times 4 = 40g + 200g = 240g = 2352 \mathrm{ N\cdot m$ .

Anticlockwise torque: $T\sin(30^\circ) \times 3 = 1.5T$ .

$1.5T = 2352$ So $T = 1568 \mathrm{ N$ .

Hinge force: Horizontal = $T\cos(30^\circ) = 1568 \times 0.866 = 1358 \mathrm{ N$ (outward). Vertical = $70g - T\sin(30^\circ) = 686 - 784 = -98 \mathrm{ N$ (downward).

Question 4: Physical pendulum

A uniform rod of length $1.0 \mathrm{ m$ and mass $2 \mathrm{ kg$ is pivoted at one end and swings As a physical pendulum. Calculate the period of small oscillations.

Answer

Moment of inertia about the end: $I = \frac{1}{3}mL^2 = \frac{1}{3}(2)(1)^2 = 0.667 \mathrm{ kg\cdot m^2$ .

Distance from pivot to centre of mass: $d = L/2 = 0.5 \mathrm{ m$ .

Period: $T = 2\pi\sqrt{\frac{I}{mgd}} = 2\pi\sqrt{\frac{0.667}{2 \times 9.8 \times 0.5}} = 2\pi\sqrt{\frac{0.667}{9.8}} = 2\pi\sqrt{0.0681} = 2\pi \times 0.261 = 1.64 \mathrm{ s$ .

Question 5: Angular momentum of a system

A disk of mass $3 \mathrm{ kg$ and radius $0.4 \mathrm{ m$ rotates at $10 \mathrm{ rad/s$ . A $1 \mathrm{ kg$ lump of clay is dropped onto the disk at a distance of $0.3 \mathrm{ m$ from the Centre and sticks. What is the new angular velocity?

Answer

Initial angular momentum: $L = I_{\mathrm{disk}\omega = \frac{1}{2}(3)(0.4)^2 \times 10 = 0.24 \times 10 = 2.4 \mathrm{ kg\cdot m^2/s$ .

Final moment of inertia: $I_f = I_{\mathrm{disk} + mr^2 = 0.24 + 1(0.3)^2 = 0.24 + 0.09 = 0.33 \mathrm{ kg\cdot m^2$ .

Conservation: $L_f = L_i$$I_f\omega_f = 2.4$$\omega_f = 2.4/0.33 = 7.27 \mathrm{ rad/s$ .

Cross-References

Newton’s Laws — Rotational dynamics extends Newton’s second law to torque and angular acceleration.
Work, Energy, and Power — Rotational kinetic energy and rotational work parallel the translational concepts.
Momentum and Impulse — Angular momentum conservation connects rotational motion to the conservation principles.
AP Calculus — Integrals: Moment of inertia and rotational work are computed via integrals over continuous mass distributions, applying integration to rotational systems.

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