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, a) what does the angle θ has to be if the road is frictionless?
b) If a car moving at constant speed, calculate friction force the is experiencing
a) If the road is frictionless, the only force acting on the car is the force of gravity and the normal force. The normal force is perpendicular to the surface of the road, and since the car is on a banked curve, it will have a component pointing towards the center of the curve, providing the centripetal force. Therefore:
N cosθ = mg
N sinθ = mv^2/r
where m is the mass of the car, g is the acceleration due to gravity, v is the speed of the car, and r is the radius of the curve.
Solving for θ, we get:
tanθ = v^2/(rg)
tanθ = (40.0 m/s)^2 / (900 kg x 9.81 m/s^2 x 100 m)
θ = 23.3 degrees
Therefore, the angle θ has to be 23.3 degrees if the road is frictionless.
b) If the car is moving at constant speed, the net force on it must be zero. Thus, the friction force must exactly balance the component of the normal force pointing down the slope:
f = N sinθ - mg cosθ
Using the same values as before, we get:
f = (900 kg x 9.81 m/s^2 x sin23.3°) - (900 kg x 9.81 m/s^2 x cos23.3°)
f = 198 N
Therefore, the friction force the car is experiencing is 198 N.
a) To find the angle θ, we can use the formula for the centripetal force:
F = m * v^2 / r
where F is the net force acting towards the center of the curve, m is the mass of the car, v is the velocity of the car, and r is the radius of the curve.
Since the road is frictionless, the only force acting towards the center of the curve is the gravitational force. Therefore, we have:
F_gravity = m * g
where g is the acceleration due to gravity.
At the angle θ, the gravitational force component acting towards the center of the curve can be written as:
F_gravity_component = F_gravity * sin(θ)
where sin(θ) is the component of the gravitational force perpendicular to the surface of the road.
For the car to move in the curved bank without slipping, the net force must be equal to the gravitational force component acting towards the center of the curve:
F = F_gravity_component
Substituting the expressions for F and F_gravity_component, we have:
m * v^2 / r = m * g * sin(θ)
Simplifying and solving for θ, we get:
sin(θ) = v^2 / (g * r)
θ = sin^(-1)(v^2 / (g * r))
Plugging in the given values, we have:
θ = sin^(-1)(40^2 / (9.8 * 100))
θ ≈ 81.1 degrees
Therefore, the angle θ has to be approximately 81.1 degrees if the road is frictionless.
b) Since the car is moving at a constant speed, the net force acting on the car must be zero. Therefore, the friction force must balance out the centrifugal force pushing the car outwards.
The centripetal force required to keep the car moving in a curve is given by:
F_centrifugal = m * v^2 / r
Since the car is moving at a constant speed, the friction force can be determined by the equation:
friction force = F_centrifugal
Plugging in the given values, we have:
friction force = 900 * 40^2 / 100
friction force ≈ 144,000 N
Therefore, the car is experiencing a friction force of approximately 144,000 N.