Hi everyone, finished all my utexas physics homework, save this last question... I don't quite why I'm not getting the right answer. Any help offered will be appreciated, it could lead me to the right answer :)
P.S.---- I've already imputed the following four answers which were incorrect. (3844N,8487N,11194N,12925N)
The next curve that the car approaches also
has a radius of curvature 67.4 m. It is banked
at an angle of 30◦. The ideal speed for this
curve is vc (banked so that the car experiences
no frictional force). The speed of the car vs as
it rounds this curve is vs = 0.673 vc.
If the mass of the car is 1500 kg, what is
the magnitude of the frictional force needed
to keep it from sliding sideways?
Answer in units of N.
To find the magnitude of the frictional force needed to keep the car from sliding sideways, we can start by calculating the net force acting on the car in the horizontal direction. Since the car is not sliding sideways, the net force must be equal to zero.
The net force acting on the car in the horizontal direction is given by the sum of the centripetal force and the frictional force. The centripetal force is responsible for keeping the car moving in a curved path, and it is given by the equation:
Fc = mv^2 / r
Where:
Fc = centripetal force
m = mass of the car
v = velocity of the car
r = radius of curvature
In this case, the centripetal force is provided by the vertical component of the normal force acting on the car due to its weight. The normal force can be given by:
N = mg
Where:
N = normal force
m = mass of the car
g = acceleration due to gravity
Since the car is banked, the vertical component of the normal force can be given by:
N * cos(theta) = mg
Where:
theta = angle of the banking
The centripetal force can be given by:
Fc = N * cos(theta)
Now, let's calculate the normal force:
N = mg / cos(theta)
Now, substitute the expression for N into the equation for centripetal force:
Fc = (mg / cos(theta)) * cos(theta)
Simplifying the expression:
Fc = mg
Now, let's calculate the acceleration of the car in terms of the velocity:
v = vs / cos(theta)
Since vs = 0.673 vc, we can write:
v = 0.673 vc / cos(theta)
We can equate this to vc:
0.673 vc / cos(theta) = vc
Simplifying the equation:
1 / cos(theta) = 1 / 0.673
Now, solve for cos(theta):
cos(theta) = 0.673
Next, substitute this value into the expression for the net force:
Fc = mg / cos(theta)
Substituting the given values:
Fc = (1500 kg * 9.8 m/s^2) / 0.673
Now, calculate the centripetal force:
Fc = 22035.5 N
Since there is no friction acting in the horizontal direction, the magnitude of the frictional force needed to keep the car from sliding sideways is equal to the centripetal force:
Frictional force = 22035.5 N
Therefore, the magnitude of the frictional force needed to keep the car from sliding sideways is 22035.5 N.
To find the magnitude of the frictional force needed to keep the car from sliding sideways, we need to analyze the forces acting on the car as it rounds the banked curve.
First, let's identify the forces acting on the car:
1. Gravity (mg): The force pulling the car downward due to its mass.
2. Normal force (N): The force exerted by the surface of the banked curve perpendicular to the car's motion. It can be decomposed into two components: one perpendicular to the surface and one parallel to the surface.
3. Frictional force (f): The force required to keep the car from sliding sideways. This is the force we need to determine.
Next, let's analyze the forces in the vertical direction since there is no vertical acceleration (the car is not going up or down):
The vertical component of the normal force (Nv) should balance the vertical component of the gravitational force (mgv). Since the car is not moving vertically, these two forces must cancel each other out, so Nv = mgv.
Next, let's analyze the forces in the horizontal direction since the car is moving in a circular path:
The horizontal component of the normal force (Nh) provides the centripetal force required to keep the car in a curved path. We can find Nh using the following equation:
Nh = mass x centripetal acceleration
In this case, the centripetal acceleration can be calculated using the speed of the car (vs) and the radius of the curved path (67.4 m):
Centripetal acceleration = (vs^2) / r
Since the car is not experiencing any frictional force, the horizontal component of the normal force should be equal to the centripetal force. So Nh = (mass x centripetal acceleration) = (mass x (vs^2) / r).
Now, we can find the frictional force (f) by considering the horizontal forces:
f = Nh
Substituting the equation for Nh, we have:
f = (mass x (vs^2) / r)
Plugging in the given values, we get:
f = (1500 kg x (0.673 vc)^2) / 67.4 m
Simplifying the equation, we can calculate the magnitude of the frictional force needed to keep the car from sliding sideways.