Chapter 6: Uniform Circular Motion and Gravitation

Rotation Angle ( $\Delta \theta$ ): The angle of rotation defined by the ratio of arc length ( $\Delta s$ ) to the radius of curvature ( $r$ ): $\Delta \theta = \frac{\Delta s}{r}$

Angular Velocity: $\omega=\frac{\Delta\theta}{\Delta t}$

Tangential Velocity ( $v$ ): The linear speed tangent to the circular path relates to angular velocity ( $\omega$ ) and radius ( $r$ ) by: $v = r\omega$

$\omega=\frac{2\pi}{T}$ , ( $2\pi$ radians)

$v=\frac{2\pi r}{T}$

$T$ = period = time for one revolution to pass

Centripetal Acceleration ( $a_c$ ): Radial acceleration pointing directly toward the center of curvature caused by a change in velocity direction: $a_{c}=\frac{v^2}{r}=r\omega^2$
Uniform Circular Motion = an object moves in a circle at a constant speed

Centripetal Force ( $F_c$ ): Any net force perpendicular to velocity that directs an object toward the center of curvature: $F_{c}=ma_{c}=m\frac{v^2}{r}$

$T=F_{c}$

Unbanked turn with friction: $v=\sqrt{\mu_{s}gr}$

Steps to derive

1) Frictional force $F_{f}$ = $\mu_{s}\cdot N$ and Centripetal Force $F_{c}=m\frac{v^2}{r}$ are set equal in order to maintain circular motion

2) Use N = w = mg $\rightarrow$ $\mu_{s}\cdot mg=m\frac{v^2}{r}$

3) Cancel m from both sides, simplify $\rightarrow v^2=\mu_{s}\cdot gr$ , so $v=\sqrt{\mu_{s}gr}$

Banked turn with no friction $v=\sqrt{\tan\theta gr}$

Steps to derive

1) set Fnet(y) = 0, so $N_{y}$ - w = 0

2) Using trig angle (adjacent), $N_{y}$ = Ncos $\theta$ $\rightarrow$ Ncos $\theta$ - w = 0

3) Ncos $\theta$ = w $\rightarrow$ Ncos $\theta$ = mg $\rightarrow$ $N=\frac{mg}{cos\theta}$

4) set Fnet(x) = m $a_{c}$ , since Fnet only force nin x direction is normal force $\rightarrow$ $N_{x}$ = m $a_{c}$

5) Use $a_{c}$ = $\frac{v^2}{r}$ $\rightarrow$ $N_{x}=m\frac{v^2}{r}$

6) Using trig angle (opposite), $N_{x}$ = Nsin $\theta$ $\rightarrow$ Nsin $\theta$ = $m\frac{v^2}{r}$

7) plug in $N=\frac{mg}{cos\theta}$ and simplify to $g\tan\theta=\frac{v^2}{r}$ , so $v=\sqrt{\tan\theta gr}$

Unbanked (Level) Curves: Static friction ( $f$ ) between the tires and the roadway supplies the necessary centripetal force: $f = F_c \le \mu_s N = \mu_s m g$
Banked Curves: The normal force ( $N$ ) is tilted at an angle ( $\theta$ ) relative to the vertical to provide centripetal force without relying on friction:
X direction: $N\sin(\theta)=F_{c}=F_{\text{net}}\left(x\right)$
Y direction: $N \cos(\theta) = w = m g$

Universal Gravitation Law: Every mass attracts every other mass along a line intersecting their centers of mass ( $\text{CM}$ ):

$F=G\frac{m_1m_2}{r^2}$

G = gravitational constant, $6.674\cdot10^{-11}N\cdot\frac{m^2}{\operatorname{kg}^2}$
m1,m2 = two masses
r = radius
R is distance between their centers of mass

Surface Gravity: For objects near Earth's surface, the distance separating their centers of mass is approximated by Earth's radius ( $r_e$ ).
Earth-Moon System: Earth and the Moon revolve around their common center of mass once per month, producing detectable orbital "wiggles" in Earth's elliptical path around the Sun.
Tidal Dynamics: The gravitational differential exerted across Earth creates two high and two low ocean tides per day as Earth rotates under the tidal bulge

For the planet that feels the acceleration:

$g=\frac{G\left(M_{opposite}\right)}{R_{opposite}^{_{}2}}$

relative gravity = $\frac{4\pi G}{3}pR$