Lesson 2 1

Angles can be measured in both degrees & radians : The angle  in radians is defined as the arc length / the radius For a whole circle, (360°) the arc length is the circumference, (2  r)  360° is 2  radians  Arc length r Common values : 45° =  /4 radians 90° =  /2 radians 180° =  radians Note. In S.I. Units we use “rad” How many degrees is 1 radian?

Angular velocity, for circular motion, has counterparts which can be compared with linear speed s=d/t. Time (t) remains unchanged, but linear distance (d) is replaced with angular displacement  measured in radians.  Angular displacement  r r Angular displacement is the number of radians moved

For a watch calculate the angular displacement in radians of the tip of the minute hand in 1.One second 2.One minute 3.One hour Each full rotation of the London eye makes takes 30 minutes. What is the angular displacement per second?

Consider an object moving along the arc of a circle from A to P at a constant speed for time t: Definition : The rate of change of angular displacement with time “The angle, (in radians) an object rotates through per second”  =  / t  Arc length r r P A This is all very comparable with normal linear speed, (or velocity) where we talk about distance/time Where  is the angle turned through in radians, (rad), yields units for  of rad/s

The period T of the rotational motion is the time taken for one complete revolution (2  radians). Substituting into :  =  /t  = 2  / T  T = 2  /  From our earlier work on waves we know that the period (T) & frequency (f) are related T = 1/f  f =  / 2 

Considering the diagram below, we can see that the linear distance travelled is the arc length  Linear speed (v) = arc length (AP) / t v = r  /t Substituting... (  =  / t) v = r   Arc length r r P A

A cyclist travels at a speed of 12m/s on a bike with wheels which have a radius of 40cm. Calculate: a.The frequency of rotation for the wheels b.The angular velocity for the wheels c.The angle the wheel turns through in 0.1s in i radians ii degrees

The frequency of rotation for the wheels Circumference of the wheel is 2  r = 2  x 0.4m = 2.5m Time for one rotation, (the period) is found using s =d/t rearranged for t t = d/s = T = circumference / linear speed T = 2.5/12 = 0.21s f = 1/T = 1/0.21 = 4.8Hz

The angular velocity for the wheels Using T = 2  / , rearranged for   = 2  /T  = 2  /0.21  = 30 rad/s

The angle the wheel turns through in 0.1s in i radians ii degrees Using  =  / t re-arranged for   =  t  = 30 x 0.1  = 3 rad = 3 x (360 ° /2  ) = 172 °

If an object is moving in a circle with a constant speed, it’s velocity is constantly changing.... Because the direction is constantly changing.... If the velocity is constantly changing then by definition the object is accelerating If the object is accelerating, then an unbalanced force must exist Velocity v acceleration

Velocity v B  Velocity v A Consider an object moving in circular motion with a speed v which moves from point A to point B in  t seconds (From speed=distance/time), the distance moved along the arc AB,  s is v  t Velocity v B Velocity v A vv CA B  The vector diagram shows the change in velocity  v : (v B – v A )

Velocity v B  Velocity v A The triangles ABC & the vector diagram are similar If  is small, then  v/v =  s/r Velocity v B Velocity v A vv CA B  Substituting for  s = v  t  v/v = v  t /r  v/v = v  t /r (a = change in velocity / time) a =  v/  t = v 2 /r

We can substitute for angular velocity.... a = v 2 /r From the last lesson we saw that: v = r  (substituting for v into above) a = (r  ) 2 /r a = r  2