A highway curve has a radius of 123 m. At what angle should the road be banked so that a car traveling at 26.5 m/s has no tendency to skid sideways on the road? [Hint: No tendency to skid means the frictional force is zero.]
I can't get this one right either please someone explain this and help me out. Thank you.
Let the x-axis point toward the center of curvature and the y-axis point upward. Use Newton’s
second law.
ΣFy = N• cos θ − − f •sin θ = 0
ΣFx = N• sin θ + f •cos θ = m•v²/R,
Solving the 1st equation for N, we obtain
N =( f •sin θ + m•g)/cos θ,
Substitute N to the 2nd equation and obtain friction force
f =m[(v²•cosθ/R) - g•sinθ].
f=0 =>
(v²•cosθ/R) - g•sinθ = 0,
v²•cosθ/R = g•sinθ,
v²•/R•g = sinθ/cosθ = tanθ.
θ =arctan(v²•/R•g) =
= arctan(26.5²•/123•9.8) =
=arctan(0.158) = 8.98 º
Thanks Elena, but I tried the answer and its wrong I don't know why. Could you please help me. Thanks.
Adam?...Elena thanks for the answer, but it says its wrong.
The first line is
ΣFy = N• cos θ − mg − f •sin θ = 0
The general solution is correct.
May be your answer is 9º
Ohh!
I calculated
arctan(0.58) =30,2º
Thank you Elena! :)
To find the angle at which the road should be banked, we can consider the forces acting on the car as it moves around the curved road. In order to have no tendency to skid sideways, the frictional force between the car's tires and the road must be zero. This means that the horizontal component of the car's weight is balanced by the component of the normal force acting toward the center of the curve.
Here's how you can approach the problem:
1. Draw a free-body diagram of the forces acting on the car:
- The weight of the car acts vertically downward.
- The normal force (N) acts perpendicular to the road.
- The frictional force (f) acts opposite to the direction of motion.
2. Resolve the weight force into two components:
- The vertical component (Wv) points upward and is balanced by the normal force (N).
- The horizontal component (Wh) points inward and is balanced by the frictional force (f).
3. Use trigonometry to find the value of the components:
- Wv = mg, where m is the mass of the car and g is the acceleration due to gravity (approximately 9.8 m/s^2).
- Wh = mg sin(θ), where θ is the angle at which the road is banked.
4. Equate the horizontal components:
mg sin(θ) = f
Since we want the frictional force to be zero (no tendency to skid sideways), we set f = 0. This leads to:
mg sin(θ) = 0
We know that sin(θ) = 0 when θ = 0° and θ = 180°. However, a road cannot be vertical or upside down, so we consider the case when sin(θ) = 0. This occurs when θ = 0°, which means the road should not be banked (or perfectly horizontal) in order to prevent skidding.
Therefore, no banking angle is required for the given conditions.
Note: If the question were asking for the angle at which the car has maximum speed without skidding, the banking angle would be solved using the equation mg tan(θ) = μN, where μ is the coefficient of static friction between the tires and the road. But in this case, the question specifically asks for no tendency to skid, so the banking angle is 0°.