College Physics

# Section Summary

College PhysicsSection Summary

### 10.1Angular Acceleration

• Uniform circular motion is the motion with a constant angular velocity $ω=ΔθΔtω=ΔθΔt size 12{ω= { {Δθ} over {Δt} } } {}$.
• In non-uniform circular motion, the velocity changes with time and the rate of change of angular velocity (i.e. angular acceleration) is $α=ΔωΔtα=ΔωΔt size 12{α= { {Δω} over {Δt} } } {}$.
• Linear or tangential acceleration refers to changes in the magnitude of velocity but not its direction, given as $at=ΔvΔtat=ΔvΔt size 12{a rSub { size 8{t} } = { {Δv} over {Δt} } } {}$.
• For circular motion, note that $v=rωv=rω size 12{v=rω} {}$, so that
$at=ΔrωΔt.at=ΔrωΔt. size 12{a rSub { size 8{t} } = { {Δ left (rω right )} over {Δt} } } {}$
• The radius r is constant for circular motion, and so $Δrω=rΔωΔrω=rΔω size 12{Δ left (rω right )=rΔω} {}$. Thus,
$at=rΔωΔt.at=rΔωΔt. size 12{a rSub { size 8{t} } =r { {Δω} over {Δt} } } {}$
• By definition, $Δω/Δt=αΔω/Δt=α size 12{ {Δω} slash {Δt=α} } {}$. Thus,
$a t = rα a t = rα size 12{a rSub { size 8{t} } =rα} {}$

or

$α=atr.α=atr. size 12{α= { {a rSub { size 8{t} } } over {r} } } {}$

### 10.2Kinematics of Rotational Motion

• Kinematics is the description of motion.
• The kinematics of rotational motion describes the relationships among rotation angle, angular velocity, angular acceleration, and time.
• Starting with the four kinematic equations we developed in the One-Dimensional Kinematics, we can derive the four rotational kinematic equations (presented together with their translational counterparts) seen in Table 10.2.
• In these equations, the subscript 0 denotes initial values ($x0x0 size 12{x rSub { size 8{0} } } {}$ and $t0t0 size 12{t rSub { size 8{0} } } {}$ are initial values), and the average angular velocity $ω-ω- size 12{ { bar {ω}}} {}$ and average velocity $v-v- size 12{ { bar {v}}} {}$ are defined as follows:

### 10.3Dynamics of Rotational Motion: Rotational Inertia

• The farther the force is applied from the pivot, the greater is the angular acceleration; angular acceleration is inversely proportional to mass.
• If we exert a force $FF size 12{F} {}$ on a point mass $mm size 12{m} {}$ that is at a distance $rr size 12{r} {}$ from a pivot point and because the force is perpendicular to $rr size 12{r} {}$, an acceleration $a = F/ma = F/m size 12{F} {}$ is obtained in the direction of $FF size 12{F} {}$. We can rearrange this equation such that
$F = ma,F = ma, size 12{F} {","}$

and then look for ways to relate this expression to expressions for rotational quantities. We note that $a = rαa = rα size 12{F} {}$, and we substitute this expression into $F=maF=ma size 12{F} {}$, yielding

$F=mrαF=mrα size 12{F} {}$
• Torque is the turning effectiveness of a force. In this case, because $FF size 12{F} {}$ is perpendicular to $rr size 12{r} {}$, torque is simply $τ=rFτ=rF size 12{F} {}$. If we multiply both sides of the equation above by $rr size 12{r} {}$, we get torque on the left-hand side. That is,
$rF = mr 2 α rF = mr 2 α size 12{ ital "rF"= ital "mr" rSup { size 8{2} } α} {}$

or

$τ = mr 2 α . τ = mr 2 α . size 12{τ= ital "mr" rSup { size 8{2} } α "." } {}$
• The moment of inertia $II size 12{I} {}$ of an object is the sum of $MR2MR2 size 12{ ital "MR" rSup { size 8{2} } } {}$ for all the point masses of which it is composed. That is,
$I = ∑ mr 2 . I = ∑ mr 2 . size 12{I= sum ital "mr" rSup { size 8{2} } "." } {}$
• The general relationship among torque, moment of inertia, and angular acceleration is
$τ = Iα τ = Iα size 12{τ=Iα} {}$

or

$α = net τ I ⋅ α = net τ I ⋅ size 12{α= { { ital "net"`τ} over {I} } cdot } {}$

### 10.4Rotational Kinetic Energy: Work and Energy Revisited

• The rotational kinetic energy $KErotKErot size 12{ ital "KE" rSub { size 8{ ital "rot"} } } {}$ for an object with a moment of inertia $II$ and an angular velocity $ωω size 12{ω} {}$ is given by
$KErot=12Iω2.KErot=12Iω2. size 12{"KE" rSub { size 8{"rot"} } = { {1} over {2} } Iω rSup { size 8{2} } } {}$
• Helicopters store large amounts of rotational kinetic energy in their blades. This energy must be put into the blades before takeoff and maintained until the end of the flight. The engines do not have enough power to simultaneously provide lift and put significant rotational energy into the blades.
• Work and energy in rotational motion are completely analogous to work and energy in translational motion.
• The equation for the work-energy theorem for rotational motion is,
$net W=12Iω2−12I ω 0 2 .net W=12Iω2−12I ω 0 2 . size 12{"net "W= { {1} over {2} } Iω rSup { size 8{2} } - { {1} over {2} } Iω rSub { size 8{0} rSup { size 8{2} } } } {}$

### 10.5Angular Momentum and Its Conservation

• Every rotational phenomenon has a direct translational analog , likewise angular momentum $LL size 12{L} {}$ can be defined as $L=Iω.L=Iω. size 12{L=Iω} {}$
• This equation is an analog to the definition of linear momentum as $p=mvp=mv size 12{p= ital "mv"} {}$. The relationship between torque and angular momentum is $net τ= Δ L Δ t .net τ= Δ L Δ t . size 12{"net "τ= { {ΔL} over {Δt} } } {}$
• Angular momentum, like energy and linear momentum, is conserved. This universally applicable law is another sign of underlying unity in physical laws. Angular momentum is conserved when net external torque is zero, just as linear momentum is conserved when the net external force is zero.

### 10.6Collisions of Extended Bodies in Two Dimensions

• Angular momentum $LL$ is analogous to linear momentum and is given by $L=IωL=Iω size 12{L=Iω} {}$.
• Angular momentum is changed by torque, following the relationship $net τ = Δ L Δ t . net τ = Δ L Δ t .$
• Angular momentum is conserved if the net torque is zero $L = constant net τ = 0 L = constant net τ = 0$ or $L = L ′ net τ = 0 L = L ′ net τ = 0$ . This equation is known as the law of conservation of angular momentum, which may be conserved in collisions.

### 10.7Gyroscopic Effects: Vector Aspects of Angular Momentum

• Torque is perpendicular to the plane formed by $rr size 12{r} {}$ and $FF size 12{F} {}$ and is the direction your right thumb would point if you curled the fingers of your right hand in the direction of $F F size 12{F} {}$. The direction of the torque is thus the same as that of the angular momentum it produces.
• The gyroscope precesses around a vertical axis, since the torque is always horizontal and perpendicular to $L L size 12{L} {}$. If the gyroscope is not spinning, it acquires angular momentum in the direction of the torque ($L =ΔL L =ΔL size 12{L=ΔL} {}$), and it rotates about a horizontal axis, falling over just as we would expect.
• Earth itself acts like a gigantic gyroscope. Its angular momentum is along its axis and points at Polaris, the North Star.