Topic of Chapter: Objects rotating • First, rotating, without translating. • Then, rotating AND translating together. • Assumption:Rigid Body • Definite shape. Does not deform or change shape. • Rigid Body motion = Translational motion of center of mass (everything done up to now) + Rotational motion about an axis through center of mass. Can treat the two parts of motion separately.
COURSE THEME: NEWTON’S LAWS OF MOTION! • Chs. 4 - 7:Methods to analyze the dynamics of objects in TRANSLATIONAL MOTION. Newton’s Laws! • Chs. 4 & 5: Newton’s Laws using Forces • Ch. 6: Newton’s Laws using Energy & Work • Ch. 7: Newton’s Laws using Momentum. NOW • Ch. 8:Methods to analyze dynamics of objects inROTATIONAL LANGUAGE. Newton’s Laws in Rotational Language! • First, Rotational Language. Analogues of each translational concept we already know! • Then, Newton’s Laws in Rotational Language.
Rigid Body Rotation A rigid body is an extended object whose size, shape, & distribution of mass don’t change as the object moves and rotates. Example: a CD
Pure Rotational Motion All points in the object move in circles about the rotation axis (through the Center of Mass) r Reference Line The axis of rotation is through O & is to the picture. All points move in circles about O
In purely rotational motion, all points on the object move in circles around the axis of rotation (“O”). The radius of the circle is R. All points on a straight line drawn through the axis move through the same angle in the same time. r r
Sect. 8-1: Angular Quantities • Description of rotational motion: Need concepts: Angular Displacement Angular Velocity, Angular Acceleration • Defined in direct analogy to linear quantities. • Obey similar relationships! r Positive Rotation!
Rigid object rotation: • Each point (P) moves in a circle with the same center! • Look at OP: When P (at radius R) travels an arc length ℓ, OP sweeps out angle θ. θ Angular Displacementof the object r Reference Line
r • θ Angular Displacement • Commonly, measure θ in degrees. • Mathof rotation: Easier if θis measured in Radians • 1 Radian Angle swept out when the arc length = radius • When R, θ1 Radian • θin Radians is definedas: θ= ratio of 2 lengths (dimensionless) θMUST be in radians for this to be valid! Reference Line
θin Radians for a circle of radius r, arc length isdefinedas: θ (/r) • Conversion between radians & degrees: θfor a full circle = 360º = (/r) radians Arc length for a full circle = 2πr θfor a full circle = 360º = 2πradians Or 1 radian (rad) = (360/2π)º 57.3º Or 1º = (2π/360) rad 0.017 rad • In doing problems in this chapter,put your calculators in RADIAN MODE!!!!
Example 8-2: θ 310-4 rad = ? º r = 100 m, = ? a) θ= (310-4 rad) [(360/2π)º/rad] = 0.017º b) = rθ = (100) (310-4) = 0.03 m = 3 cm θMUST be in radians in part b!
Angular Velocity(Analogous to linear velocity!) Average Angular Velocity = angular displacement θ = θ2 – θ1 (rad) divided by time t: (Lower case Greek omega, NOT w!) Instantaneous Angular Velocity (Units = rad/s) The SAME for all points in the object! Valid ONLY if θis in rad!
Angular Acceleration(Analogous to linear acceleration!) • Average Angular Acceleration= change in angular velocity ω = ω2– ω1 divided by time t: (Lower case Greek alpha!) • Instantaneous Angular Acceleration = limit of α as t, ω0 (Units = rad/s2) TheSAMEfor all points in body! Valid ONLYfor θin rad & ω in rad/s!
Relations of Angular & Linear Quantities Ch. 5 (circular motion): A mass moving in a circle has a linear velocity v & a linear acceleration a. We’ve just seen that it also has an angular velocity & an angular acceleration. Δ Δθ r There MUST be relationships between the linear & the angular quantities!
Connection Between Angular & Linear Quantities Radians! v = (/t), = rθ v = r(θ/t) = rω v = rω Depends on r (ω is the same for all points!) vB = rBωB,vA = rAωA vB > vA since rB > rA
Summary: Every point on a rotating body has an angular velocity ωand a linear velocity v. They are related as:
Relation Between Angular & LinearAcceleration _____________ In direction of motion: (Tangential acceleration!) atan= (v/t), v = rω atan= r (ω/t) atan= rα atan : depends on r α: the same for all points
Angular & LinearAcceleration _____________ From Ch. 5: there is also an acceleration to the motion direction (radial or centripetal acceleration) aR = (v2/r) But v = rω aR= rω2 aR: depends on r ω: the same for all points
Total Acceleration _____________ Two vector components of acceleration • Tangential: atan= rα • Radial: aR= rω2 • Total acceleration = vector sum: a = aR+ atan a ---
Relation Between Angular Velocity & Rotation Frequency • Rotation frequency: f = # revolutions / second (rev/s) 1 rev = 2πrad f = (ω/2π) or ω = 2π f = angular frequency 1 rev/s 1 Hz (Hertz) • Period: Time for one revolution. T = (1/f) = (2π/ω)
Translational-Rotational Analogues & Connections ANALOGUES Translation Rotation Displacement x θ Velocity v ω Acceleration a α CONNECTIONS = rθ, v = rω atan= r α aR = (v2/r) = ω2 r
Conceptual Example 8-3: Is the lion faster than the horse? On a rotating merry-go-round, one child sits on a horse near the outer edge & another child sits on a lion halfway out from the center. a. Which child has the greater translational velocity v? b. Which child has the greater angular velocity ω?
Example 8-4: Angular & Linear Velocities & Accelerations A merry-go-round is initially at rest (ω0 = 0). At t = 0 it is given a constant angular acceleration α = 0.06 rad/s2. At t = 8 s, calculate the following: a. The angular velocity ω. b. The linear velocity v of a child located r = 2.5 m from the center. c. The tangential (linear) acceleration atan of that child. d. The centripetal acceleration aR of the child. e. The total linear acceleration a of the child.
Example 8-5: Hard Drive The platter of the hard drive of a computer rotates at frequency f = 7200 rpm(rpm = revolutions per minute = rev/min) a. Calculate the angular velocityω(rad/s) of the platter. b. The reading head of the drive r = 3 cm(= 0.03 m) from the rotation axis. Calculate the linear speed v of the point on the platter just below it. c. If a single bit requires 0.5 μm of length along the direction of motion, how many bits per second can the writing head write when it is r = 3 cm from the axis?