Chapter 5: Circular Motion and Gravitation

Centripetal on the FBD - force is directly to the center of the circle

  • Mass is never a factor of any uniform circular motion


Uniform Circular Motion

Motion of an object traveling at a constant speed on a circular path Centripetal force is the F\sum F that keeps an object in the circular path

Frad=maac=mv2/rF_{rad} = ma_{ac} = mv^{2}/r

v=2πr/Tv=2\pi r/T  where T represent the period of the motion

+ac=v2/rac\underline{+ a_{c} = v^{2}/r \Rightarrow a_{c}} represents the centripetal acceleration

a[c]=4π2R/T2Ra_[c] = 4\pi^{2}R/T^{2} \Rightarrow R represents the radius


  • An object will travel tangential to its original circular motion once it loses the centripetal force. (**Inertia**)
  • Cannot be taken to image on the FBD


Circular motion in reality

Friction - responsible for the centripetal force of a car making a curve with a specific μ\mu  .

Fc=μsFN=μsmg=mv2/rv=γμsgrF_{c}= \mu _{s}F_{N}=\mu _{s}mg=mv^{\wedge }2/r\Rightarrow v= \gamma \mu _{s}gr


Banking

On a tilt, the motion of an object in a circular direction

Fc=FNSinΘ=mv2/r\underline{F_c = F_N Sin\Theta = mv^2 / r} FN CosΘ=mg\underline{F_{N}\ Cos\Theta = mg } 

TanΘ=v2/rg\underline{Tan\Theta = v^2 /rg}


On a Ferris wheel

NBmg=mv2/RNB=mg+mv2/R=m(g+v2/R)N_B - mg = mv^2 / R \Rightarrow N_B = mg + mv^2 /R = m(g+v^2/R)

MgNT=mv2/RNT=mgmv2/R=m(gv2/R)Mg - N_T = mv^2 / R \Rightarrow N_T = mg - mv^2 /R = m(g-v^2/R)

  • The Normal force is greater @ bottom giving a greater pressurized force on a person.

@v2/R=g@ v^2 / R= g normal force 0=00=\rightarrow 0 weight / weightless (at top)

  • Walking is considered uniform circular motion Fcent=mg=mv2/gγgr=vF_{cent} = mg = mv^2 /g\Rightarrow \gamma gr = v


Newton’s Law of Gravitation

G=6.6741011Nm2kg2G = 6.674 \cdot 10^{-11} \frac{N \cdot m^2}{kg^2}  - Gravitational Constant

FG=Gm1m2/r2F_G = Gm_1m_2 / r^2 - calculates attraction force between 2 masses

M12=M_{\frac{1}{2}}= the 2 masses, given r = distance between the 2 masses where: Fg12=Fg21Fg_{12}=-Fg_{21}

Weight = [m]g = G[mass] (Mearth)/rearth2(M_{earth}) / r_{earth}^2


Circular motion around the earth

Fc=GmME/r2=mv2/rv=γ(GME/r)F_c = GmM_E / r^2 = mv^2/r\Rightarrow v = \gamma(GM_E/r) - Radius from center of earth to object

Keplar’s law

V=γ(GM/r)=2πr/TT=2πr3/2V = \gamma (GM/r) = 2\pi r/T \Rightarrow T =2\pi r^3 /2  GM\sqrt{GM} and M=4π2r3/GT2M = 4\pi ^2r^3 / GT^2

Escape Velocity - the speed required to leave earth’s gravitational force


Apparent Weight

  • The weight that we feel with other applied force on us.
  • It is different from FgaF_{ga}


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