Physics 11
Uniform Circular
Motion
Uniform Circular Motion
Uniform circular motion is the motion of an object traveling
at a constant speed on a circular path.
Uniform Circular Motion
Centripetal Acceleration
Centripetal Force
Unbanked Curves
Banked Curves
Satellites
Vertical Circular Motion
Uniform Circular Motion
The period T is the time it takes for the object to
travel once around the circle.
The speed v is given by:
v=
2π r
T
r
Centripetal Acceleration
⇒ An object can accelerate even if it is traveling
at a constant speed if its direction is changing.
⇒ An object traveling in a
circular path of radius r at
speed v undergoes centripetal
acceleration.
v2
ac =
r
1
Unbanked Curves
Centripetal Force
⇒ Centripetal force is the name given to the net
force (ΣF) that causes an object to change
direction.
⇒ Centripetal force is not a new kind of force.
⇒ For a car rounding an
unbanked (flat) curve,
static friction supplies
the centripetal force.
⇒ Suppose we want to find the maximum speed
that a car can take an unbanked curve without
slipping.
⇒ The maximum speed
implies that the car is on
the verge of slipping.
⇒ If the speed were any
greater, the car would
slip.
⇒ This implies that static friction is at its
maximum value: fs = fs,max = μsN
Unbanked Curves
Unbanked Curves
N
N
fs
mg
∑F
ax =
y
2
v
r
to the left
(define left as +)
mg
= ma y = 0
η − mg = 0
η = mg
x
= max = m
fs = m
fs
ay = 0
∑F
μs g =
v2
r
μ sη = m
(
v2
r
v2
r
(define left as +)
f s = f s ,max = μ sη )
→ μ s (mg ) = m
v2
r
v2
→ v 2 = μ s gr → v = μ s gr
r
2
Banked Curves
Banked Curves
⇒ Curves are often banked to make turning easier.
⇒ A car can round a banked curve even if the
road is frictionless.
⇒ One of the components of the normal force
(N sin θ) supplies the centripetal force.
⇒ If there is no friction, what supplies the
centripetal force???
Banked Curves
N cosθ
Banked Curves
∑F
N cosθ
y
N sinθ
= ma y = 0
η cos θ = mg
N sinθ
η = mg / cos θ
mg
mg
∑F
ay = 0
ax =
y
2
v
r
to the left
(define left as +)
= ma y = 0
η cos θ = mg
η = mg / cos θ
∑ Fx = max = m
η sin θ = m
v2
r
(define left as +)
v2
⎛ mg
→ ⎜
r
⎝ cosθ
mg tan θ = m
v2
r
→
v
⎞
⎟ sin θ = m
r
⎠
2
tan θ =
v2
rg
3
Satellites
Satellites
There is only one speed that a satellite can have if the
satellite is to remain in an orbit with a fixed radius.
Fc = G
v=
mM E
v2
=
m
r2
r
GM E
r
Vertical Circular Motion
FN 1 − mg = m
FN 2 = m
v22
r
FN 4 = m
v42
r
v12
r
FN 3 + mg = m
v32
r
4