Chapter 8: Rotational Motion

translated into rotational language. Topics of Chapter: Objects Rotating Newton’s Laws of Motion applied to rotating objects, translated into rotational language. First, rotating, without translating. Then, rotating AND translating together. Assumption: Rigid Bodies 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! Everything up to now: Methods to analyze the dynamics of objects in TRANSLATIONAL MOTION. Newton’s Laws! Chs. 3, 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 in ROTATIONAL LANGUAGE.

Newton’s Laws of Motion! Newton’s Laws in Rotational Language! Course Theme: Newton’s Laws of Motion! Ch. 8: Methods to analyze dynamics of objects in ROTATIONAL LANGUAGE. Newton’s Laws in Rotational Language! First, introduction to Rotational Language. Then, Rotational 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 & rotates. Example: a CD

There are 3 Basic Types of Rigid Body Motion

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

Pure Rotational Motion All points on object move in circles around the axis of rotation (“O”). The circle radius is r. All points on a straight line drawn through the axis move through the same angle in the same time. r Figure 10-1. Caption: Looking at a wheel that is rotating counterclockwise about an axis through the wheel’s center at O (axis perpendicular to the page). Each point, such as point P, moves in a circular path; l is the distance P travels as the wheel rotates through the angle θ. r

Angular Quantities To describe rotational motion, we need Positive Rotation! To describe rotational motion, we need Rotational Concepts: Angular Displacement, Angular Velocity, Angular Acceleration These are defined in direct analogy to linear quantities & they obey similar relationships!

θ  Angular Displacement Rigid Body Rotation Each point (P) moves in a circle with the same center! Look at OP: When P (at radius r) sweeps out angle θ. θ  Angular Displacement of the object r

θ  Angular Displacement Commonly, we measure θ in degrees. Math of rotation: Easier if θ is measured in Radians 1 Radian  Angle swept out when the arc length = radius When   r, θ  1 Radian θ in Radians is defined as: θ = ratio of 2 lengths (dimensionless) θ MUST be in radians for this to be valid! r  Reference Line   (/r)

θ for a full circle = 360º = (/r) radians θ in Radians for a circle of radius r, arc length  is defined as: θ  (/r) Conversion between radians & degrees: θ for a full circle = 360º = (/r) radians Arc length  for a full circle = 2πr  θ for a full circle = 360º = 2π radians 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!!!!

Angular Displacement Figure 10-4. Caption: A wheel rotates from (a) initial position θ1 to (b) final position θ2. The angular displacement is Δθ = θ2 – θ1.

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 θ is 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, ω 0 (Units = rad/s2) The SAME for all points in body! Valid ONLY for θ in rad & ω in rad/s!

Relations Between 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),  = rθ  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 ω and a linear velocity v. They are related as:

Relation Between Angular & Linear Acceleration In direction of motion: (Tangential acceleration!) atan= (v/t), v = rω  atan= r (ω/t) atan= rα atan : depends on r α : the same for all points _____________

Angular & Linear Acceleration 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 Tangential: atan= rα Radial: aR= rω2  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 f = # revolutions / second (rev/s) 1 rev = 2π rad  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 Translation Rotation Displacement x θ Velocity v ω Acceleration a α CONNECTIONS  = rθ, v = rω atan= r α aR = (v2/r) = ω2 r

Correspondence between Linear & Rotational quantities

Conceptual Example: 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? Answer: The horse has a greater linear velocity; the angular velocities are the same.

Example: 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. Figure 10-8. Caption: Example 10–3. The total acceleration vector a = atan + aR, at t = 8.0 s. Solution: a. The angular velocity increases linearly; at 8.0 s it is 0.48 rad/s. b. The linear velocity is 1.2 m/s. c. The tangential acceleration is 0.15 m/s2. d. The centripetal acceleration at 8.0 s is 0.58 m/s2. e. The total acceleration is 0.60 m/s2, at an angle of 15° to the radius. 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: 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 velocity ω (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? Solution: a. F = 120 Hz, so the angular velocity is 754 rad/s. b. V = 22.6 m/s. c. 45 megabits/s