Topic 2.1 Extended L – Angular acceleration  When we discussed circular motion we looked at two variables:  We will also define a new rotational variable called angular acceleration α. x y  Consider the irregularly- shaped peanut-like solid shown here.  Two points are shown on an an arbitrary reference line.  We can superimpose the Cartesian coordinate axes for reference: θ r  Note that every point on the reference line has the same angular position θ.  And that every point on the reference line has a different radius r. r Angular position θ, and angular speed ω.

Topic 2.1 Extended L – Angular acceleration  Note that each point on our reference line traces out an arc of length s:  Recall the relationship between r, θ, and s: x y s r r s θ s = rθ θ in radians Arc length  Note that each point covers a different distance s in the same amount of time.  Thus each point moves at a different velocity v.

Topic 2.1 Extended L – Angular acceleration x y θ1θ1 Definition of average angular speed  Consider the position of the red dot at two times: = ΔθΔtΔθΔt θ2θ2 t1t1 t2t2 ΔθΔθ  We can define a change in angular position Δθ:  We define the average angular speed :  This is the same definition we had when we talked about circular motion.  Note that the units of are radians per second.  We define the instantaneous angular speed ω: ω = ΔθΔtΔθΔt lim Δt→0 = dθ dt Definition of angular speed

Topic 2.1 Extended L – Angular acceleration  The relation between linear velocity v and angular velocity ω comes from s = rθ:  Just as we define average acceleration a for the linear variables, v = ΔsΔtΔsΔt Relation between linear and angular speed = r ΔθΔtΔθΔt = rω v = rω a = ΔvΔtΔvΔt we define average angular acceleration α for the angular variables. α = ΔωΔtΔωΔt Definition of average angular acceleration FYI: Angular acceleration is measured in radians per second squared. FYI: In the limit as  t  0, average values become instantaneous values.

Topic 2.1 Extended L – Angular acceleration  The instantaneous angular acceleration is given by α = ΔωΔtΔωΔt lim Δt→0 = dω dt Definition of angular acceleration  The relation between linear acceleration a and angular acceleration α comes from v = rω: a = ΔvΔtΔvΔt Relation between linear and angular acceleration = r ΔωΔtΔωΔt = rα a t = rα FYI: We call this linear acceleration a t the tangential acceleration because it occurs along the arc length, and is thus tangent to the circle the point is following.  Don’t forget the centripetal acceleration for any object moving in a circle: a c = rω 2 = v2rv2r Centripetal acceleration FYI: a t and a c are mutually perpenducular. Question: a t = 0 in UCM. Why? FYI: In the limit as  t  0, average values become instantaneous values.

Topic 2.1 Extended L – Angular acceleration  Recall that velocity is a speed in a particular direction.  Thus, angular velocity is angular speed in a particular direction.  We define angular direction with another right- hand-rule:  Consider the record spinning on the axis (a):  Curl the fingers of the right hand as (c):  Your thumb points in the direction of the angular velocity (b): FYI: Angular direction always points along the rotational axis of the turning object. FYI: We can usually get by with clockwise (cw) and counterclockwise (ccw) when working with angular direction.

Topic 2.1 Extended L – Angular acceleration  Suppose the angular acceleration α is constant. Thus is also constant so that a t = rα a t = constant and all of the following equations are true: s = s o + v o t + a t t v = v o + a t t v 2 = v o 2 + 2a t  s s = rθ v = rω a t = rα The linear equations Angular to linear relations  Substitution of s, v, and a t yields the following: rθ = rθ o + rω o t + rαt rω = rω o + rαt r 2 ω 2 = r 2 ω o 2 + 2rα  (rθ) θ = θ o + ω o t + αt ω = ω o + αt ω 2 = ω o 2 + 2α  θ Constant angular acceleration s or x   v   a  

Topic 2.1 Extended L – Angular acceleration Suppose a grinding wheel of 10-cm radius is turned on, and reaches a speed of 6000 rpm in 20 seconds. (a) What is the final angular speed of the wheel? ω = 6000 rev min = rad/s · 1 min 60 s · 2  rad 1 rev (b) What is constant angular acceleration of the wheel during the first 20 seconds? ω = ω o + αt = 0 + α(20) α = rad/s 2 (c) Through what angle does the wheel rotate during the first 20 seconds? ω 2 = ω o 2 + 2α  θ = ( )  θ  θ = radians