Driving down Highway 401 in Ontario, it had rained and stopped raining by the time I came to the
freezing rain on the highway. I knew there was a problem when the cars in front of me started to fly
off the highway at the curve in the road. Assume the curve in the road is banked at an angle of 10.0
and has a radius of 300 m.
(a) At what speed is no friction required to make the turn?
(b) Suppose the coefficient of static friction between my tires and the freezing rain is 0.1, what are
the maximum and minimum speeds at which the curve can be traveled?
a)Fn sinθ = mv^2/r
Fn = mg cosθ
Work backward and solve for v (m's cancel)
b) For min Ff points up.
Fn sinθ - Ff cosθ = mv^2/r
Fn cosθ + Ff sinθ = mg
and to tie it all together
mu Fn = Ff
For max Ff points down.
Fn sinθ + Ff cosθ = mv^2/r
Fn cosθ = Ff sinθ + mg
and
mu Fn = Ff
Hope that helps. A FBD is a must for these problems.
To solve this problem, we need to consider the forces acting on the car as it goes around the curved road. There are two important forces at play here: the gravitational force and the friction force.
(a) To find the speed at which no friction is required to make the turn, we need to determine the minimum force required to keep the car moving in a circular path. At this speed, the friction force is zero.
The centripetal force required to keep the car moving in a curved path is provided solely by the gravitational force. The formula for centripetal force is:
F = m * a
where F is the force, m is the mass of the car, and a is the centripetal acceleration.
The centripetal acceleration can be calculated using the formula:
a = v^2 / r
where v is the velocity of the car, and r is the radius of the curve.
In this case, since the friction force is zero, the force required is the gravitational force acting inwards towards the center of the curve. The gravitational force is given by:
Fg = m * g
where g is the acceleration due to gravity.
Setting the gravitational force equal to the centripetal force:
Fg = m * g = m * v^2 / r
Simplifying the equation:
v^2 = g * r
Solving for v:
v = sqrt(g * r)
Now we can plug in the given values to calculate the speed:
g = 9.8 m/s^2 (acceleration due to gravity)
r = 300 m (radius of the curve)
v = sqrt(9.8 * 300) ≈ 55.9 m/s
So, to make the turn without requiring any friction, the car must have a speed of approximately 55.9 m/s.
(b) Now let's consider the maximum and minimum speeds at which the curve can be traveled, given a coefficient of static friction between the tires and the freezing rain of 0.1.
The maximum speed occurs when the friction force at the tires is at its maximum value, which is equal to the product of the coefficient of friction and the normal force:
F friction = μ * F normal
The normal force is the force exerted by the car perpendicular to the road surface. In this case, it is equal to the gravitational force:
F normal = m * g
Therefore,
F friction = μ * m * g
For the maximum speed, the friction force must supply the entire centripetal force required to keep the car on the curved path. So, we can equate the friction force with the centripetal force:
F friction = m * a = m * v^2 / r
Plugging in the known values:
μ * m * g = m * v^2 / r
Simplifying the equation:
v_max = sqrt(μ * g * r)
Now we can find the maximum speed:
v_max = sqrt(0.1 * 9.8 * 300) ≈ 17.2 m/s
Therefore, the maximum speed at which the car can travel safely on the curve is approximately 17.2 m/s.
For the minimum speed, the friction force must supply the difference between the centripetal force required and the gravitational force. The minimum speed occurs when the friction force reaches its minimum value and is equal to:
F friction = μ * F normal = μ * m * g
So, we can equate the friction force with the difference between the centripetal force and the gravitational force:
F friction = m * a - m * g = m * (a - g)
Plugging in the known values:
μ * m * g = m * (a - g)
Simplifying the equation:
v_min = sqrt(μ * g * r + g^2 * r)
Now we can find the minimum speed:
v_min = sqrt(0.1 * 9.8 * 300 + 9.8^2 * 300) ≈ 51.9 m/s
Therefore, the minimum speed at which the car can safely travel on the curve is approximately 51.9 m/s.