Circular Motion & Highway Curves
Sect. 5-3: Highway Curves: Banked & Unbanked
Case 1 - Unbanked Curve: When a car rounds a curve, there MUST
be a net force toward the circle center (a Centripetal Force) of
which the curve is an arc. If there weren’t such a force, the car couldn’t follow
the curve, but would (by Newton’s 2nd Law) go in a straight line. On a flat
road, this
Centripetal Force is the static friction force.
“Centripetal Force”
No static friction?
 No Centripetal Force
 The Car goes straight!
There is NEVER a
“Centrifugal Force”!!!
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Example 5-6: Skidding on a curve
A car, mass m = 1,000 kg car rounds a curve
on a flat road of radius r = 50 m at a constant
speed v = 14 m/s (50 km/h). Will the car follow
the curve, or will it skid? Assume:
a.
Dry pavement with the coefficient of
static friction μs = 0.6.
b.
Icy pavement with μs = 0.25.
Newton’s 2nd Law: ∑F = ma
x: ∑Fx = max  Ffr = maR = m(v2/r)
y: ∑Fy = may = 0  FN - mg = 0; FN = mg
The maximum static friction is Ffr = μsFN
Free Body
Diagram
If the friction force isn’t sufficient, the car will tend to move more
nearly in a straight line (Newton’s 1st Law) as the skid marks show.
As long as the tires don’t slip, the
friction is static. If the tires start
to slip, the friction is kinetic,
which is bad in 2 ways!!
1. The kinetic friction force is
smaller than the static friction
force.
2. The static friction force points
toward the circle center, but the
kinetic friction force opposes the
direction of motion, making it
difficult to regain control of the car
& continue around the curve.
Case 2- Banked Curve
Banking curves helps keep cars from skidding.
For every banked curve, there is one speed v at
which the entire Centripetal Force is
supplied by the horizontal component of the
normal force FN, so that no friction is required!!
Newton’s 2nd Law
Tells us what speed v this is:
x: ∑Fx = max  FNx = m(v2/r) or
Also y: ∑Fy = may = 0
 FN cosθ - mg = 0  FNcosθ = mg
So FN = (mg/cosθ)
Put this into the x equation:
 g(sinθ/cosθ) = (v2/r) or tanθ = (v2/gr)
Example 5-7: Banking angle
a. For a car traveling with speed v around a
curve of radius r, find a formula for the
angle θ at which a road should be banked
so that no friction is required.
b. Calculate this angle for an expressway
off-ramp curve of radius r = 50 m at a
design speed of v = 14 m/s (50 km/h).
Newton’s 2nd Law
x: ∑Fx = max  FNx = m(v2/r)
or FNsinθ = m(v2/r)
y: ∑Fy = may = 0  FNcosθ - mg = 0 or FNcosθ = mg
Dividing (2) by (1) gives:
tanθ = [(v2)/(rg)]
Putting in the given numbers
tanθ = 0.4 or θ = 22º
(1)
(2)