Rotational Motion

Angular Measure (radian) Angular Speed and velocity Centripetal Acceleration Centripetal Force Angular Acceleration

Angular measurey

x

ry

x

x=r cos()

y=r sin()

Angular measurey

x

r

x

=

2 radians = 360°

s

s

rradians

1.0 radian = 57.3°

How far to the moon?

Early Greek scientists estimated that the earth’s diameter is 3.5 times larger than the moon’s.

If a penny covers up the moon when held 200 cm from your eye, How far is the moon from Earth? (the diameter of a penny is 1.9 cm)

Small angles

rh s

h≈ s & r ≈ x

Sin ≈ ≈ tan h/r ≈ s/r ≈ h/x

x

Angular speed and velocity

Average angular speed

€

ϖ =Δt

θ = ϖt (θ i = 0)

ω =dθdt

Linear speed and angular speed

rh s

x

v =st=

rt

=rϖ

v=rω

Linear speed and angular speed

rh s

Sin ≈ ≈ tan h/r ≈ s/r ≈ h/x

x

v =rω

Angular acceleration

€

α =ΔωΔt

ω = ωi +α t

Angular acceleration

α =Δv

rt

=Δvt ⋅r

=a t

ra t = rα

€

α =ΔωΔt

ω = ωi +α t

Circular Motion(with constant α

€

ϖ =(ω + ωi )

2

Circular Motion(constant α

€

ω =ω i +α t

θ =θ i +ω f +ω i

2t

θ =θ i +ωi t +12

α t2

ω2 = ωi2 + 2α (θ −θ i )

Kinetic Energy

€

Ki =12mivi

2 =12miri

2ω2

€

K =12miri

2ω2 =∑12

miri2

[ ]ω2 =∑

12Iω2

€

K =12Iω2

€

I = miri2

[ ] or I = r2dm∫∑

€

Iz = Icm +Md2

d

Torque

€

τ =rF sinφ = Fd

€

τ =τ1 + τ 2 = +F1d1 − F2d2

€

τ∑ A= IAα

a=?

T=?

a=?

T=?

F=mamg-T=ma

τ= IαTR = 1/2MR2 (a/R)

T= 1/2M a

mg-T=ma

mg- 1/2M a=ma

mg= 1/2M a+ma

mg= (1/2M +m)a

mg (1/2M +m)

T= 1/2M a

a =

Work and Power

€

work =r τ ⋅

r θ

Power =r τ ⋅

r ω