π
<-

## notes105 physics peen

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### Description

Relating the Linear and Angular Variables (H&R Chapter 10-5)

????
????=????

Where θ is angle in radians, s is arc length, r is radius
????

Angular Displacement

???????? ????ℎ???????????????? ???????? ???????????? ????????????????????ℎ
???????? = angular displacement (in radians) =
???? ???????????????????????????????? ???????????????? ????????????????

Note: points on a rigid object all undergo the same angular displacement, but their arc length

is dependent upon the points distance from the center.

Angular Speed

Note: all points of a rigid object will have the same angular velocity

???????? ???????????????????????????? ????????????????????????????????????????????????
???????????????? = average angular speed = (usually expressed in radians / second)
???????? ???????????????? ????????????????????????????????

Angular acceleration

Note: all points of a rigid object will have the same angular acceleration

???????? ????ℎ???????????????? ???????? ???????????????????????????? ????????????????????
???????????????? = average angular acceleration = (usually expressed in radians /sec2)
???????? ???????????????? ????????????????????????????

Angular substitutes for linear quantities

Linear Angular
x θ
v ω
a α
10-2 Rotational and linear kinematic equations with constant acceleration

Rotational Motion with constant angular Linear Motion with constant acceleration
acceleration
???????? = ???????? + ???????????? ???????? = ???????? + ????????????
1 1
???????? = ???????? ???????? + ???????????? 2 ???????? = ???????? ???????? + ???????? 2
2 2
2 2 2 2
???????? = ???????? + 2????(????????) ???????? = ???????? + 2????(????????)
1 1
???????? = (???????? + ???????? )???????? ???????? = (???????? + ???????? )????????
2 2
10-5

Tangential speed

s
 = divide both sides by ∆t
r
 s 1
= .
t t r

1
avg = vT
r

vT =r
Tangential acceleration

vT =r divide both sides by ∆t

vT 
=r
t t

aT =r
Centripetal acceleration

2
v
ac = T substitute VT =r
r

r 2 2
ac =
r

ac =r 2

Force that maintains circular motion
2
v
ac = T
r
∑ ???????? = ???????????? (sum forces in the radial direction)

2
v
Fc =m T
r

(substituting v t = r )

Fc =mr 2

2r
Also vT = so
T

m4 2 r
Fc =
T2

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