If a car takes a banked curve at less than the ideal speed, friction is needed to keep it from sliding toward the inside of the curve (a real problem on icy mountain roads).
(a) Calculate the ideal speed to take a 95 m radius curve banked at 15°.
m/s
(b) What is the minimum coefficient of friction needed for a frightened driver to take the same curve at 30.0 km/h?
fsdkflg
asjkdbaksdb
dsfhgsdf
To find the ideal speed for a car to take a banked curve, we can use the concept of centripetal force. Here's how you can calculate the ideal speed for a given radius and banking angle:
(a) Calculating the ideal speed:
Step 1: Identify the forces acting on the car:
- The force of gravity acting vertically downward.
- The normal force acting perpendicular to the banked surface.
- The frictional force providing the necessary centripetal force to keep the car on the curve.
Step 2: Resolve the forces:
- The vertical component of the normal force counteracts the force of gravity.
- The horizontal component of the normal force provides the centripetal force.
- The frictional force acts in the opposite direction of the horizontal component of the normal force to provide the centripetal force.
Step 3: Write the equations:
- The vertical component of the normal force (Nv) is equal to the force of gravity (mg).
- The horizontal component of the normal force (Nh) is equal to the centripetal force (mv^2 / r), where v is the velocity and r is the radius.
- The frictional force (f) is equal to the horizontal component of the normal force times the coefficient of friction (μ).
Step 4: Substitute values and solve for the ideal speed:
Given: radius (r) = 95 m, banking angle (θ) = 15°, coefficient of friction (μ) = ?
From the equation Nh = mv^2 / r, we can solve for v:
Nh = mv^2 / r
(v^2 / g) Nh = m v^2 / r
(v^2 / g) mg cos(θ) = m v^2 / r
v^2 = r g cos(θ)
v = sqrt(r g cos(θ))
Substituting the values:
v = sqrt(95 m * 9.8 m/s^2 * cos(15°))
v ≈ 31.95 m/s (rounded to two decimal places)
Therefore, the ideal speed to take the 95 m radius curve banked at 15° is approximately 31.95 m/s.
(b) Calculating the minimum coefficient of friction:
To find the minimum coefficient of friction, we need to consider the fact that the car is traveling at a speed lower than the ideal speed. This means that an additional frictional force is needed to prevent the car from sliding toward the inside of the curve.
Step 1: Identify the forces:
- The force of gravity acting vertically downward.
- The normal force acting perpendicular to the banked surface.
- The frictional force, which consists of the static friction stopping the car from sliding and the kinetic friction acting if the car starts sliding.
Step 2: Write the equations:
- The vertical component of the normal force is equal to the force of gravity.
- The horizontal component of the normal force requires two parts: the static friction and the kinetic friction. The sum of these two forces provides the centripetal force.
Step 3: Find the minimum coefficient of friction:
Given: radius (r) = 95 m, banking angle (θ) = 15°, velocity (v) = 30.0 km/h, coefficient of friction (μ) = ?
Firstly, convert the velocity to m/s:
30.0 km/h * (1000 m / 3600 s) ≈ 8.33 m/s (rounded to two decimal places)
From the equation Nh = mv^2 / r:
Nh = m(v^2 / r)
Nh = mv^2 / r
Similarly to the previous equation, we can substitute the values and solve for the minimum coefficient of friction (μ):
mv^2 / r = mg sin(θ) + μ mg cos(θ)
μ mg cos(θ) = mv^2 / r - mg sin(θ)
μ = (v^2 - g r sin(θ)) / (g cos(θ))
Substituting the values:
μ = (8.33 m/s)^2 - (9.8 m/s^2)(95 m) sin(15°) / [(9.8 m/s^2)(95 m) cos(15°)]
μ ≈ 0.35 (rounded to two decimal places)
Therefore, the minimum coefficient of friction needed for a frightened driver to take the same curve at 30.0 km/h is approximately 0.35.