Last Time: Collisions in 1- and 2-Dimensions Today: Start Rotational Motion (now thru mid-Nov) Basics: Angular Speed, Angular Acceleration Recitation Quiz #6 tomorrow

Examples of Rotational Motion

Math Review: Radians, Circular Arc Length When discussing circular motion (such as rotations, revolutions, etc.), we will generally express angles in terms of radians : 360 degrees = 2π radians (rad) Quick Example: How many radians are there in 135 degrees ? [Radians are dimensionless] y The arc length s along a circle of radius r, for an angle θ measured in radians counterclockwise from the +x-axis, is : r θ x θ : angular displacement from +x-axis s : displacement along the circular arc

Angular Displacement Suppose an object undergoes a rotation, such as a CD or DVD. y y Δθ x x at some later time An object’s angular displacement, Δθ, is the difference between its initial and final angles: [ SI: Δθ, θf , and θi in radians ]

Example Suppose the CD rotates, so that the black dot rotates from an initial position of 0 degrees to a final position of 60 degrees. What is the angular displacement Δθ ? y y Δθ x x at some later time

Angular Speed Now suppose we want to quantify how “quickly” the CD rotates. y y Final Time tf Δθ x x Initial Time ti The average angular speed ωav of a rotating rigid object during the time interval Δt is the angular displacement divided by Δt : [ SI: radians/second (rad/s) ]

Instantaneous Angular Speed / Signs For very short time intervals, the average angular speed ωav “approaches” (becomes very close to) the instantaneous angular speed (just like instantaneous and average velocity) : [ SI: radians/second (rad/s) ] Sign Convention for Rotation Angles : + When θ is increasing  Counter-clockwise rotations – When θ is decreasing  Clockwise rotations

Example y x (a) Suppose a CD rotates in the counter-clockwise direction at 500 RPM (revolutions per minute). What is its angular speed ? (b) The radius of a standard CD is 6.0 cm. If the CD rotates at 500 RPM, what arc length does some point on the edge trace out in 1 minute ?

Example The Earth rotates about its axis once per day. What is its angular speed of rotation ? See also : http://en.wikipedia.org/wiki/Earth's_rotation As viewed from the North Pole, the Earth rotates counterclockwise about its axis !

Angular Acceleration When you ride a bike downhill, the wheels will start to rotate faster and faster (i.e., the angular speed increases !). Just like in linear motion, where we defined the acceleration to be the change in the velocity, we define … An object’s average angular acceleration αav during the time Δt is the change in its angular speed Δω divided by Δt : [ SI: rad/s2 ]

Instantaneous Angular Acceleration The instantaneous angular acceleration α is the limit of the average angular acceleration Δωav as the time interval Δt approaches zero : [ SI: rad/s2 ]

Angular Acceleration + – Sign Convention for Angular Acceleration : Means angular acceleration is counter-clockwise If object initially rotating counter-clockwise sense, it will start to rotate faster in counter-clockwise sense. If object initially rotating clockwise sense, its rate of rotation in clockwise sense will start to decrease. – Means angular acceleration is clockwise If object initially rotating clockwise sense, it will start to rotate faster in clockwise sense. If object initially rotating counter-clockwise sense, its rate of rotation in counter-clockwise sense will start to decrease.

Examples An object is initially rotating with an angular speed of 10.0 rad/s. It undergoes an angular acceleration of –1.0 rad/s2 for 5.0 s. What is its angular speed after these 5.0 s ? An object is initially rotating with an angular speed of –5.0 rad/s. It undergoes an angular acceleration of –1.0 rad/s2 for 3.0 s. What is its angular speed after these 3.0 s ?

Important Point About Angular Speed y x When a rigid object rotates about an axis (such as this CD), EVERY PORTION OF THE OBJECT has the SAME angular speed and the same angular acceleration. If the CD makes 500 revolutions in 1 minute, EVERY POINT on the CD will make 500 revolutions in the same 1 minute !!

Another Way To Think About It … It doesn’t matter if you are in Alaska or Kentucky. No matter where you are on Earth, you will make one rotation once every day. Thus, the Earth’s angular speed of rotation is the same everywhere !

Rotational Motion Under Constant Angular Acceleration Just like for linear motion under constant acceleration, we have kinematic equations for rotational motion. They are analogous … Rotational Motion with Constant α Linear Motion with Constant a

Relations Between Angular and Linear Quantities y v Angular rotation variables are closely related to linear quantities. Think about the red point on the CD. Assume it has rotated through Δθ in Δt. Δθ x During this time, it has moved through an arc length of Δs :

Relations Between Angular and Linear Quantities y v Angular rotation variables are closely related to linear quantities. Think about the red point on the CD. Assume it has rotated through Δθ in Δt. Δθ x The velocity vector is tangent to (or along) the circular arc. The magnitude of the velocity vector is the linear speed v = vt , called the tangential speed of an object moving in a circular path tangential speed = distance of point from axis multiplied by angular speed

Important Point y v Every point on the rotating object has the same angular speed. But the tangential speed depends on how far you are from the axis of rotation ! Δθ x Remember your merry-go-round days. The farther you are from the center, the faster you are moving (get sicker…).

Tangential Acceleration v If the object undergoes angular acceleration, its angular speed increases. At a fixed point, the tangential speed will then increase. Δθ x If you are on a merry-go-round, and it starts to spin faster, you will start to move faster. Your tangential speed increases. The tangential acceleration (change in vt / time) is given by tangential acceleration = distance of point from axis multiplied by angular acceleration

Example: 7.14 + Extra An electric motor rotating a wheel at a rate of 100 rev/min is turned off. Assume the wheel has a constant angular acceleration of –2.00 rad/s2. (a) How long does it take for the wheel to come to a stop ? (b) Through how many radians has the wheel turned during the time interval found in (a) ? Extra: Suppose the wheel has a radius of 0.20 m. At t = 2.0 s, what is the magnitude of the tangential speed and acceleration at a point on the edge of the wheel ?

Next Class 7.4 – 7.5 : Centripetal Acceleration, Newtonian Gravitation