Unit 6 Notes: Rolling Motion and Orbital Motion (Energy + Angular Momentum Connections)

Rolling motion

Motion that combines translation of an object’s center of mass with rotation about its center of mass.

Translation (in rolling)

The forward motion of the center of mass (like a block sliding).

Rotation (in rolling)

Spinning motion about the object’s center of mass.

Rolling without slipping (pure rolling)

Rolling where the contact point with the surface is instantaneously at rest relative to the surface.

No-slip (rolling) constraint

For radius $R$: $v_{cm} = \theta R$ (and tangentially $a_{cm} = \theta R$); links linear and rotational motion only when there is no slipping.

Static friction in pure rolling

Friction that prevents relative motion at the contact point; it can provide torque even though the contact point does not slide.

Why static friction often does no work in pure rolling

In the ground frame, the point of contact has zero displacement, so W = F·d at that point is zero (even though friction can provide torque).

Kinetic friction (slipping)

Friction that occurs when surfaces slide; it typically dissipates mechanical energy as thermal energy.

Total kinetic energy of a rolling object

$K_{total} = \frac{1}{2}mv_{cm}^2 + \frac{1}{2}I_{cm}\theta^2$ (translational plus rotational kinetic energy).

Moment of inertia about the center of mass (I_cm)

A measure of how mass is distributed relative to the rotation axis; determines how hard an object is to spin up.

Hoop (thin ring) moment of inertia

For radius $R$ about its symmetry axis: $I_{cm} = mR^2$.

Solid disk/solid cylinder moment of inertia

For radius $R$ about its symmetry axis: $I_{cm} = \frac{1}{2}mR^2$.

Solid sphere moment of inertia

For radius $R$ about its symmetry axis: $I_{cm} = \frac{2}{5}mR^2$.

Thin spherical shell moment of inertia

For radius $R$ about its symmetry axis: $I_{cm} = \frac{2}{3}mR^2$.

Shape factor (β) for rolling

Defined by $I_{cm} = \beta mR^2$; larger $\beta$ means more rotational inertia relative to $mR^2$.

Center-of-mass acceleration for rolling down an incline

For pure rolling: $a_{cm} = \frac{g \text{ sin} \theta}{1 + \frac{I_{cm}}{mR^2}} = \frac{g \text{ sin} \theta}{1+\beta}$.

Instantaneous axis of rotation (rolling)

In pure rolling, the object can be treated as instantaneously rotating about the contact point (the axis changes continuously).

Speed of the top point of a rolling wheel

For pure rolling: vtop = vcm + ωR = 2v_cm (relative to the ground).

Orbit (physics definition)

Motion under gravity where an object continuously “falls around” a central body; gravity supplies the centripetal acceleration (it doesn’t disappear).

Circular orbital speed

For orbital radius $r$ around mass $M$: $v = \sqrt\frac{GM}{r}$; depends on $r$ and $M$, not the satellite’s mass.

Centripetal acceleration in circular orbit

$a_c = \frac{v^2}{r} = \frac{GM}{r^2}$; equals the local gravitational field magnitude for a circular orbit.

Orbital period for a circular orbit

$T = \frac{2\pi r}{v} = 2\pi\sqrt{\frac{r^3}{GM}}$; larger $r$ gives a much longer period ($T \propto r^{3/2}$).

Total mechanical energy in a circular orbit

E = K + U = −GMm/(2r); negative for a bound orbit and becomes less negative (higher) as r increases.

Angular momentum conservation (satellites)

For a central force (gravity toward the center), torque about the center is zero, so angular momentum L = mr v_⊥ is conserved; implies faster speed when closer in an elliptical orbit.

Escape speed

Minimum speed at distance $r$ to reach infinity with zero final speed: $v_e = \frac{\sqrt{2GM}}{r} = \sqrt2$ times the circular orbital speed at the same $r$.