College Physics

# Section Summary

College PhysicsSection Summary

### 6.1Rotation Angle and Angular Velocity

• Uniform circular motion is motion in a circle at constant speed. The rotation angle $ΔθΔθ size 12{Δθ} {}$ is defined as the ratio of the arc length to the radius of curvature:
$Δθ=Δsr,Δθ=Δsr, size 12{Δθ= { {Δs} over {r} } ","} {}$

where arc length $ΔsΔs size 12{Δs} {}$ is distance traveled along a circular path and $rr size 12{r} {}$ is the radius of curvature of the circular path. The quantity $ΔθΔθ size 12{Δθ} {}$ is measured in units of radians (rad), for which

• The conversion between radians and degrees is $1rad=57.3º1rad=57.3º size 12{1"rad"="57" "." 3°} {}$.
• Angular velocity $ωω size 12{ω} {}$ is the rate of change of an angle,
$ω=ΔθΔt,ω=ΔθΔt, size 12{ω= { {Δθ} over {Δt} } ","} {}$

where a rotation $ΔθΔθ size 12{Δθ} {}$ takes place in a time $ΔtΔt size 12{Δt} {}$. The units of angular velocity are radians per second (rad/s). Linear velocity $vv size 12{v} {}$ and angular velocity $ωω size 12{ω} {}$ are related by

### 6.2Centripetal Acceleration

• Centripetal acceleration $acac size 12{a rSub { size 8{c} } } {}$ is the acceleration experienced while in uniform circular motion. It always points toward the center of rotation. It is perpendicular to the linear velocity $vv size 12{v} {}$ and has the magnitude
$a c = v 2 r ; a c = rω 2 . a c = v 2 r ; a c = rω 2 . size 12{a rSub { size 8{c} } = { {v rSup { size 8{2} } } over {r} } ; a rSub { size 8{c} } =rω rSup { size 8{2} } } {}$
• The unit of centripetal acceleration is $m/s2m/s2 size 12{m/s rSup { size 8{2} } } {}$.

### 6.3Centripetal Force

• Centripetal force $FcFc size 12{F rSub { size 8{c} } } {}$ 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 $vv size 12{v} {}$ and has magnitude
$Fc=mac,Fc=mac,$

which can also be expressed as

$F c = m v 2 r or F c = mr ω 2 , F c = m v 2 r or F c = mr ω 2 ,$

### 6.4Fictitious 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.

### 6.5Newton’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 is
$F=GmMr2,F=GmMr2, size 12{F=G { { ital "mM"} over {r rSup { size 8{2} } } } } {}$

where F is the magnitude of the gravitational force. $GG size 12{G} {}$ is the gravitational constant, given by $G=6.674×10–11N⋅m2/kg2G=6.674×10–11N⋅m2/kg2 size 12{G=6 "." "673" times "10" rSup { size 8{"-11"} } N cdot m rSup { size 8{2} } "/kg" rSup { size 8{2} } } {}$.

• Newton’s law of gravitation applies universally.

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

• Kepler’s laws are stated for a small mass $mm size 12{m} {}$ orbiting a larger mass $MM size 12{M} {}$ in near-isolation. Kepler’s laws of planetary motion are then as follows:

Kepler’s first law

The orbit of each planet about the Sun is an ellipse with the Sun at one focus.

Kepler’s second law

Each planet moves so that an imaginary line drawn from the Sun to the planet sweeps out equal areas in equal times.

Kepler’s third law

The 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:

where $TT size 12{m} {}$ is the period (time for one orbit) and $rr size 12{m} {}$ is the average radius of the orbit.

• The period and radius of a satellite’s orbit about a larger body $MM size 12{m} {}$ are related by
$T 2 = 4π 2 GM r 3 T 2 = 4π 2 GM r 3 size 12{T rSup { size 8{2} } = { {4π rSup { size 8{2} } } over { ital "GM"} } r rSup { size 8{3} } } {}$

or

$r3T2=G4π2M.r3T2=G4π2M. size 12{ { {r rSup { size 8{3} } } over {T rSup { size 8{2} } } } = { {G} over {4π rSup { size 8{2} } } } M} {}$
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