Section Summary

Rotation Angle and Angular Velocity

• Uniform circular motion is motion in a circle at constant speed. The rotation angle ΔθΔ𝜃 is defined as the ratio of the arc length to the radius of curvature:Δθ=Δsr,Δ𝜃=Δ𝑠𝑟,where arc length ΔsΔ𝑠 is distance traveled along a circular path and r𝑟 is the radius of curvature of the circular path. The quantity ΔθΔ𝜃 is measured in units of radians (rad), for which2πrad=360º= 1 revolution.2πrad=360º= 1 revolution.
• The conversion between radians and degrees is 1rad=57.3º1rad=57.3º.
• Angular velocity ω𝜔 is the rate of change of an angle,ω=ΔθΔt,𝜔=Δ𝜃Δ𝑡,where a rotation ΔθΔ𝜃 takes place in a time ΔtΔ𝑡. The units of angular velocity are radians per second (rad/s). Linear velocity v𝑣 and angular velocity ω𝜔 are related byv=rω or ω=vr.𝑣=rω or 𝜔=𝑣𝑟.

Centripetal Acceleration

• Centripetal acceleration ac𝑎c is the acceleration experienced while in uniform circular motion. It always points toward the center of rotation. It is perpendicular to the linear velocity v𝑣 and has the magnitudeac=v2r;ac=rω2.𝑎c=𝑣2𝑟;𝑎c=rω2.
• The unit of centripetal acceleration is m/s2m/s2.

Centripetal Force

• Centripetal force FcFc is any force causing uniform circular motion. It is a “center-seeking” force that always points toward the center of rotation. It is perpendicular to linear velocity v𝑣 and has magnitudeFc=mac,𝐹c=mac,which can also be expressed asFc=mv2rorFc=mrω2,⎫⎭⎬⎪⎪⎪⎪𝐹c=𝑚𝑣2𝑟or𝐹c=mr𝜔2,

Fictitious Forces and Non-inertial Frames: The Coriolis Force

• Rotating and accelerated frames of reference are non-inertial.
• Fictitious forces, such as the Coriolis force, are needed to explain motion in such frames.

Newton’s Universal Law of Gravitation

• Newton’s universal law of gravitation: Every particle in the universe attracts every other particle with a force along a line joining them. The force is directly proportional to the product of their masses and inversely proportional to the square of the distance between them. In equation form, this isF=GmMr2,𝐹=𝐺mM𝑟2,where F is the magnitude of the gravitational force. G𝐺 is the gravitational constant, given by G=6.674×10–11N⋅m2/kg2𝐺=6.674×10–11N⋅m2/kg2.
• Newton’s law of gravitation applies universally.

6.6 Satellites and Kepler’s Laws: An Argument for Simplicity

• Kepler’s laws are stated for a small mass m𝑚 orbiting a larger mass M𝑀 in near-isolation. Kepler’s laws of planetary motion are then as follows:Kepler’s first lawThe orbit of each planet about the Sun is an ellipse with the Sun at one focus.Kepler’s second lawEach planet moves so that an imaginary line drawn from the Sun to the planet sweeps out equal areas in equal times.Kepler’s third lawThe ratio of the squares of the periods of any two planets about the Sun is equal to the ratio of the cubes of their average distances from the Sun:T 21T 22=r 31r 32,𝑇1 2𝑇2 2=𝑟1 3𝑟2 3,where T𝑇 is the period (time for one orbit) and r𝑟 is the average radius of the orbit.
• The period and radius of a satellite’s orbit about a larger body M𝑀 are related byT2=4π2GMr3𝑇2=4π2GM𝑟3orr3T2=G4π2M.