A car is moving in curved bank ,the 900kg car is moving at speed of 40.0 m/s through
this curve of radius 100m .what does the angle θ has to be if the road is frictionless?
The necessary angle θ can be found using the centripetal force equation:
Fc = mv^2/r
where Fc is the centripetal force, m is the mass of the car, v is the velocity, and r is the radius of the curve.
Since there is no friction, the only force acting on the car is the force required to keep it moving in a circle, which is the centripetal force.
Rearranging the equation to solve for θ:
Fc = mv^2/r
Fc = tanθ * mg
tanθ = Fc/mg
where g is the acceleration due to gravity.
Substituting the values given:
Fc = (900 kg)(40.0 m/s)^2 / 100 m
Fc = 144000 N
tanθ = Fc/mg
tanθ = 144000 N / (900 kg)(9.81 m/s^2)
tanθ = 16.26
Taking the inverse tangent:
θ = 86.4 degrees
Therefore, the road needs to be banked at an angle of approximately 86.4 degrees if it is frictionless to allow the car to travel through the curve at 40.0 m/s. However, this is not practical or safe in real-world situations, as such steep angles could cause the car to lose contact with the road. In reality, friction is necessary to keep the car on the road, and the angle of the bank is typically lower than what is calculated here.
To find the angle θ required for the road to be frictionless, we can use the concept of centripetal force and gravitational force.
1. Find the centripetal force:
The centripetal force is provided by the net force acting towards the center of the circular path. In this case, it is the gravitational force acting downwards and the normal force acting perpendicular to the road.
The centripetal force (Fc) can be calculated using the equation:
Fc = m * v^2 / r
Where:
m = mass of the car = 900 kg
v = velocity of the car = 40.0 m/s
r = radius of the curve = 100 m
Plugging in the values, we get:
Fc = 900 kg * (40.0 m/s)^2 / 100 m
2. Calculate the gravitational force:
The gravitational force (Fg) acting downward is given by:
Fg = m * g
Where:
g = acceleration due to gravity = 9.8 m/s^2
Plugging in the values, we get:
Fg = 900 kg * 9.8 m/s^2
3. Set the net force to zero:
For the road to be frictionless, the net force must be zero. Therefore, the centripetal force and gravitational force must balance each other.
Fc = Fg
Substituting the values of Fc and Fg, we get:
900 kg * (40.0 m/s)^2 / 100 m = 900 kg * 9.8 m/s^2
4. Solve for θ:
Using the equation above, we can solve for θ.
θ = atan(v^2 / (g * r))
Substituting the values, we get:
θ = atan((40.0 m/s)^2 / (9.8 m/s^2 * 100 m))
Calculating this using a calculator, we find:
θ ≈ 20.2 degrees
Therefore, the angle θ has to be approximately 20.2 degrees for the road to be frictionless.