Physics 106 Lecture 1 Introduction to Rotation SJ 7th Ed.: Chap. 10.1 to 3
• Course Introduction • Course Rules & Assignment • TiTopics Overvi ew • Rotation (rigid body) versus translation (point particle) • Rotation concepts and variables • Rotational kinematic quantities Angular position and displacement Angular velocity & acceleration • Rotation kinematics formulas for constant angular acceleration • Analogy with linear kinematics
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Physics B Topics Overview
PHYSICS A motion of point bodies COVERED: kinematics - translation dynamics ∑Fext = ma conservation laws: energy & momentum
motion of “Rigid Bodies” (extended, finite size) PHYSICS B rotation + translation, more complex motions possible COVERS: rigid bodies: fixed size & shape, orientation matters kinematics of rotation dynamics ∑Fext = macm and ∑ Τext = Iα
rotational modifications to energy conservation conservation laws: energy & angular momentum
TOPICS: 3 weeks: rotation: ▪ angular versions of kinematics & second law ▪ angular momentum ▪ equilibrium 2 weeks: gravitation, oscillations, fluids
1 Angular variables: language for describing rotation Measure angles in radians simple rotation formulas Definition: 360o 180o • 2π radians = full circle 1 radian = = = 57.3o • 1 radian = angle that cuts off arc length s = radius r 2π π
s arc length ≡ s = r θ (in radians) θ ≡ rad r
r s θ’ θ
Example: r = 10 cm, θ = 100 radians Æ s = 1000 cm = 10 m.
Rigid body rotation: angular displacement and arc length Angular displacement is the angle an object (rigid body) rotates through during some time interval...... also the angle that a reference line fixed in a body sweeps out A rigid body rotates about some rotation axis – a line located somewhere in or near it, pointing in some y direction in space • One polar coordinate θ specifies position of the whole body about this rotation axis. θ • By convention, θ is measured CCW from the x-axis Reference x line rotates • It keeps increasing past 2π, can be negative, etc. with body • Each point of the body moves around the axis in a circle with some specific radius rotation axis “o” rigid body fixed to body parallel to z-axis Angular displacement: • Net change in the angular coordinate Reference line rotating with body y Δθ ≡ θfinal − θinital (an angle in rad.) Δs = r Δθ
Arc length: Δs θf • Measures distance covered by a point as it moves r r through Δθ (constant r) θo Δs ≡ rΔθ (a distance along a circular arc) x
2 Rigid body rotation in 3D
Choose Z - axis as rotation axis (marks a constant direction in space) Use reference line perpendicular to rotation axis to measure rotation angles for the body
EACH POINT ON A BODY SWEEPS OUT A CIRCLE PARALLEL TO X-Y PLANE
3D VIEW – Z axis up TOP VIEW – Z axis out of paper Reference line rotates around it
REFERENCE LINE ROTATES WITH BODY
Measure θ CCW from x axis
THE ROTATION AXIS DIRECTION TAKES 2 ANGLES TO SPECIFY, e.g. (LATITUDE, LONGITUDE)
For any point in the body, r is the perpendicular distance to the rotation axis
Rigid body rotation: angular & tangential velocity
Angular velocity ω: For any point, r is the perpendicular • Rate of change of the angular displacement distance to the rotation axis Δθ Δθ dθ ωave ≡ ωinst ≡ Lim ≡ Δt Δt → 0 Δt dt vT • Units: radians/sec. Positive in CCW sense • If is CONSTANT (uniform circular motion) ω r
θf = θ0 + ωΔt θ = ωΔτ x • Frequency f = # of complete revolutions/unit time • f = 1/T T = period (time for 1 complete revolution
ω = 2πf = 2π/T f = ω/2π
Δs = rΔθ
ds rd (r constant) Tangential velocity v = θ T: ds dθ • Units: distance / time v ≡ = r = rω T dt dt • Rate at which a point sweeps out arc length along circular path • Proportional to r, same ω vT = ωr
3 Angular velocity, period, and linear velocity
1.1. The period of a rotating wheel is 12.57 seconds. The radius of the wheel is 3 meters. It’s angular speed is closest to:
A. 79 rpm B. 0.5 rad/s C. 2.0 rad/s D. .08 rev/s E. 6.28 rev/s
1.2. A point on the rim of the same wheel has a tangential speed closest to:
A. 12.57 rev/s B. 0.8 rev/s C. 0.24 m/s Δs ≡ rΔθ D. 1.5 m/s E. 6.28 m/s vT = ωr ω = 2πf = 2π/T
Rigid body rotation: angular and tangential acceleration
Centripetal (radial) acceleration ac or ar: vT • Body rotates at rate ω. r • Radial acceleration component, points toward rotation axis ac • Constant magnitude if ω is constant (UCM). Units: length/time2 ωτ • Changing if ω not constant (angular acceleration not zero) x 2 vT 2 F = ma a = = ω r (use v = ωr) centripetal c c r T
So far: nothing about angular velocity changing Angular acceleration α: Δω Δω αave ≡ αinst ≡ Lim • Rate of change of the angular velocity Δt Δt→0 Δt aT • Units: radians/sec/sec. 2 dω d θ vT • CCW considered positive α ≡ = 2 r dt dt ac
• for CONSTANT α: ωf = ω0 + αΔt ω,α x Tangential acceleration a : T aT = αr • Tangential acceleration component • Proportional to angular acceleration a and also to radius r dvT dω aT ≡ = r = rα • Units: length / time 2 dt dt
4 Note: for displacement, speed, and acceleration
• The tangential quantities depend on r • r varies for different points on the object • All points on a rotating rigid body have the same angular displacement, but do not move through the same path length. • All points on a rigid body have the same angular speed, but not the same tangential speed. • All points on a rigid object have the same angular acceleration, but not the same tangential acceleration.
Rotational Motion Example • For a compact disc player, the tangential speed must be constant to readdd dat a properly l • The angular speed must vary:
(vt = ωr) • For inner tracks, the angular speed must be larger than at the outer edge • The player has to vary the angular speed (rotation rate) accordingly.
Rotation variables: angular and linear Angular motion variables Δθ, ω, α: • apply to the whole rotating body 2 dθ dω d θ v Δθ ω ≡ α ≡ = T 2 dt dt dt aT atotal Linear motion variables r, s, vT, aT, ac: r • applifiiily to a specific point on a rotating a body & are signed magnitudes c • values all proportional to r – distance from the rotation axis ω,α
s ==rΔθ vT ωr 2 x ac = ω r aT = αr Total linear acceleration of a point (rotation): • vector sum of radial and tangential components (normal to each other)
2 2 atotal = ac + aT Some basic implications:
• IF ω is constant, then α=0, aT=0. But ac = ar is not 0
• IF ω is not constant, then α and aT are not 0, ac and VT are varying with time
• IF α is constant, then aT is constant for a particular point, different for different r Δω is proportional to Δt (angular kinematics)
vT is not constant (so this is not UCM) }
5 Rotational variables are vectors, having direction
The angular displacement, speed, and acceleration ( θ, ω, α ) are the magnitudes of rotational displacement, velocity, and acceleration vectors
The directions are actually given by the right-hand rule
Point thumb along + axis fingers curl CCW (positive) sense
Example: Find the angular speed ω for the hour, minute, and second hands of an analog clock [rad/sec]
Second Period T = 60 sec Hand f = Frequency = 1/T = 1/60 revolution/sec
ω = #rev/sec x # rad/rev = 2πf = 2π/60 sec
ω = 0.1 rad/sec
Minute Period T = 1 hour = 3600 sec Hand ω = 2πf = 2π/T = π/1800
ω = 1.75 x 10-3 rad/sec Physicists’s Clock
Hour Period T = 12 hours = 12 x 3600 sec = 43.200 sec Hand ω = 2πf = 2π/T = 2π/(3600 x 12)
ω = 1.45 x 10-4 rad/sec
6 Example: Assume the displacement θ is given as the following function of time 2 where v θ(t) = θ0 + bt + st b = 10, s = 5, r = 2 t
Actually, b = ω0 and s = α/2 r 1. Find ω(t) = angular speed (differentiate)
d this is not UCM, since ω(t) = θ(t) = 0 + b + 2st [rad/s] dt ω is not constant v (r) = ω(t) r = br + 2srt = 20 + 20t [m/s] t vt is not constant either 2. Find α(t) = angular acceleration (differentiate again)
d d2 α is constant because θ(t) α ≡ ω(t) = θ(t) = 2s = 10 [rad/s2 ] dt dt2 does not have t dependence higher than t2 3. Find tangential and radial accelerations
2 tangential acceleration is constant, at ≡ αr = 2sr = 20 [m/s ] but centripetal acceleration is not 2 2 2 2 ac ≡ ω r = (b + 2st) r [m/s ] unless s = 0 (no t term in θ(t))
2 2 1/2 atot ≡ ( at + ac ) Angular kinematics: constant α
Linear and Angular Kinematics Equations (Same mathematical forms)
Linear motion Angular motion • constant acceleration a • constant angular acceleration α x(t), v(t), a(t) variables θ(t), ω(t), α(t)
dv dx dθ dω a = v = Definitions ω = α = dt dt dt dt
vf (t) = v0 + at Kinematic ωf (t) = ω0 + αt Equations x (t) = x + v t + 1 at 2 θ (t) = θ + ω t + 1 αt2 f 0 0 2 f 0 0 2 v2 (t) = v2 + 2a[x − x ] 2 2 f 0 f 0 ωf (t) = ω0 + 2α[θf − θ0 ] 1 1 xf (t) = x0 + (vo + vf )t θ (t) = θ + (ω + ω )t 2 f 0 2 o f Check by differentiating: dx dθ v = = v + 1 (2)at = v + at ω = =+ωαωα1 (2) tt =+ dt 0 2 0 dt 002 Both sets of kinematic equations follow from the definitions of velocity and acceleration by integrating
7 Hints for solving rotational kinematics problems
Similar to the techniques used in linear motion problems With constant angular acceleration, the techniques are much like those with constant linear acceleration. Velocities and accelerations (linear and angular) are respectively first and second derivatives of displacements, so.... The equations for angular kinematics look like those for linear kinematics, but with different symbols There are some differences to keep in mind For rotational motion, a rotation axis is involved Once the axis is set, it must be maintained until you finish the problem In some problems, the physical situation may suggest a natural axis The object keeps returning to its original orientation (if there is no translational motion), so you can find the number of revolutions made by the body.
For constant angular acceleration: The rotational kinematics equations have the same mathematical form as the linear kinematics equations
Angular kinematics example: A turntable initially at rest reaches a speed of 33 1/3 rpm in 15 seconds. Determine the average angular acceleration.
Δω ω − ω Use the definition α ≡ = f 0 ave Δt Δt or
a kinematic formula ωf (t) = ω0 + αΔt α is constant and = αave
Either way: ω − ω α = f 0 ave Δt
Units: 33 1/3 rev/min = 100/3 x 2π/60 rad/s = 3.49 rad/sec
3.49 − 0 2 Substitute: α = = 0.233 rad/sec ave 15
8 Angular kinematics example: A grindstone is rotating with constant angular acceleration about a fixed axis in space. Initial conditions 2 at t = 0: α = 0.35 rad/s ω0 = - 4.6 rad/s initially rotating in negative sense with positive angular acceleration Positive directions: θ (t) = θ + ω t + 1 αt2 right hand rule Use kinematics formula: f 0 0 2 a) When is the displacement Δθ = 5 rev. θ (t) − θ = 5.0 rev = 10π rad = - 4.6 t + 1 0.35 t2 f 0 2 2 Solve quadratic equation: 0.175 t − 4.6 t − 10 π = 0
t = 32 sec (second root is negative)
b) When is Δθ = 0 again? 1 2 Set: 0 = +ω t + αt 0 2 Reverses direction, t = 0 (just the initial condition) rotates back and Two roots are: { α ω through original position ω = t = 0 ⇒ t = - 2 0 = 26.3 s 0 2 α
Wheel rotating and accelerating
1.3. At t = 0, a wheel rotating about a fixed axis at a constant angular acceleration has an angular velocity of 2.0 rad/s. Two seconds later it has turned through 5.0 complete revolutions. Find the angular acceleration of this wheel?
A. 17 rad/s2 B. 14 rad/s2 C. 20 rad/s2 D. 23 rad/s2 E. 13 rad/s2
ωf (t) = ω0 + αt θ (t) = θ + ω t + 1 αt2 f 0 0 2 2 2 ωf (t) = ω0 + 2α[θf − θ0 ]
9 Example: Uniform circular motion
Assume Circular Orbit
Angular kinematics example: Discus Thrower
Arm rotates as rigid body: r = 0.8 m Constant angular acceleration α = 50 rad/s2 Find the tangential and radial (centripetal) r components of the acceleration 2 atang = α r = 50 x 0.8 = 40 m/ s • constant magnitude • direction: tangent to rotary motion 2 2 arad = ω r = (ω0 + α t) r • ω not constant
Start from rest Æ ω0 = 0 2 ∴ arad = (α t) r • centrilipetal accellieration growing rapidly as square of time
time t arad # of “g”s 000 0.1 s 20 m/s2 ~2 0.2 s 80 m/s2 ~8 What would happen to 1.0 s 2000 m/s2 ~200 the throwers arm at 200 g’s
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