Double
Objective:
Date: __________
Uniform Circular Motion II
Uniform Circular Motion III
Homework: Assignment (1-35)
Read 5.4, Do CONCEPT QUEST #(8),
Do PROBS (20, 21) Ch. 5
+ AP 1997 #2 (handout)
AP Physics “B”
Mr. Mirro
Date: ________
Uniform Circular Motion II
When a car travels without skidding around an unbanked curve, the static frictional force between the
tires and the road provides the centripetal force.
‰
The reliance on “friction” can be eliminated completely for a given speed
(Ie. Worst case = the speed limit)
‰
If the curve is “banked” at an angle relative to the horizontal, much in the same way a
plane is banked while making a turn.
Consider a car going around a friction-free banked curve
as shown in figure (a).
The radius of the curve, r, is measured parallel
to the horizontal and NOT to the slanted surface.
The normal force FN that the road applies to the car is perpendicular to the road. Because the roadbed
makes an angle θ with respect to the horizontal, the normal force has a component FN sin θ.
Where the centripetal force can be expressed as:
Fc ⇒ FN sin θ = mv2
r
(1)
Also, since the car does not accelerate in the y direction,
the vertical component of the normal force is equal to the
weight, that is:
FN cos θ = mg
(2)
Dividing equation (1) by equation (2) allows us to reduce the variables so that the equation becomes
independent of the mass of the car.
FN sin θ
FN cos θ
Where,
mv2/r
mg
Tan θ = v2
rg
If we put in the maximum allowable speed (the speed limit), we can then design safely banked curves !
AP Physics “B”
Mr. Mirro
Date: ________
Uniform Circular Motion II
Ex 1: When a car moves at a steady speed around an unbanked curve, the centripetal force keeping
the car on the curves comes from the static friction between the road and the tires. Compare the
maximum speeds at which a car can safely negotiate an unbanked turn of radius r = 50 m in dry
weather (μs = 0.90) and icy weather (μs = 0.10). [Cutnell5.7]
+y
+x
Ex 2: At what angle should a curve of radius 200 m be banked, so cars can travel safely at 30 m/s without
relying on friction ? [CutnellP5.20sim]
Ex 3: A car travels at a constant speed of 13.4 m/s (30 mph) on a level circular turn of radius 50 m.
[Serway7.6]
a. Determine the minimum coefficient of static friction (μs) between the tires an roadway in order
that the car makes the circular turn without skidding ?
b. What should the “bank-angle” be so that the car does not depend on friction to prevent skidding ?
Ex 4: A child swings a yo-yo of weight mg in a horizontal circle so that the cord makes an angle of 30°
with the vertical. [Serway7.7]
a. Using Newton’s Second Law, determine the centripetal
force of the toy in terms of m, g and θ.
30°
b. Determine is the centripetal acceleration of the toy (in m/s2).
AP Physics “B”
Mr. Mirro
Date: ________
Uniform Circular Motion III
Ex 1: Consider a model airplane traveling in a circular path under Uniform Circular Motion (UCM). The
plane has a mass of 0.90 kg and has a constant speed vtan. [Cutnell5.5]
a. Find the tension in a 17 m guideline when vtan = 20 m/s.
b. Find the tension in a 17 m guideline when vtan = 40 m/s.
c. By how much did the tension in the wire change when the speed doubled ?
______________
Ex 2: a. What is the minimum coefficient of static friction necessary to allow a penny to rotate along with
a 33 1/3 rpm (?? cps) vinyl record of diameter d = 0.30 m, when the penny is placed at the outer
edge of the record. [Cutnell4EP18]
b. If you were to take the record off the turn table and place the penny on the outer edge of the
record as it lay on a horizontal surface, by what angle could you tilt the record so that the
penny just starts to slip ?
Ex 3: The Daytona 500 is the major event of the NASCAR (National Association for Stock Car Auto
Racing) season. It is held at the Daytona International Speedway in Daytona, Florida. The turns
in this oval track have a maximum radius (at the top) of r = 316 m and are banked steeply, with
an angle of 31°. Suppose the maximum radius turns were frictionless, at what speed would the
cars have to travel around them without slipping ? [Cutnell5.4]
Ex 4: A car can negotiate an unbanked curve safely at a certain maximum speed when the coefficient of
static friction between the tires and the ground is μs = 0.88. [CutnellP5.22]
a. At what angle should the same curve be banked for the car to negotiate the curve safely at the
same maximum speed without relying on friction ?
b. What is the “maximum speed” a car could round this banked curve if the radius of the curve was
50 meters ?