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 angle θ refers to the banking angle of the road, which is the angle at which the road is tilted from the horizontal. If the road is frictionless, the car can only rely on the centripetal force provided by the banked curve to turn and avoid sliding off the road.
The centripetal force required to move the car in a circular path of radius 100m at a speed of 40.0 m/s can be calculated as:
Fc = mv²/r
where Fc is the centripetal force, m is the mass of the car (900kg), v is the speed of the car (40.0 m/s), and r is the radius of the curve (100m).
Plugging in the values, we get:
Fc = (900kg)(40.0 m/s)²/100m
Fc = 1,440,000 N
The centripetal force is provided by the horizontal component of the normal force (N) acting on the car, which is given by:
N*cos(θ) = Fc
where θ is the banking angle of the road.
Solving for θ, we get:
θ = arccos(Fc/N)
θ = arccos((900kg)(40.0 m/s)²/100m)/(9.81 m/s²)(cos(θ))
θ = 53.1°
Therefore, the road needs to be banked at an angle of 53.1° for the car to navigate the curve frictionlessly at a speed of 40.0 m/s.
To determine the angle θ, we can use the concept of centripetal force and the forces acting on the car. In this case, the centripetal force is provided entirely by the horizontal component of the normal force.
The given mass of the car is 900 kg, and the speed is 40.0 m/s. The radius of curvature is 100 m.
Step 1: Calculate the centripetal force
The centripetal force (F_c) is given by the formula:
F_c = (m * v^2) / r
Where:
- m is the mass of the car (900 kg)
- v is the velocity of the car (40.0 m/s)
- r is the radius of curvature (100 m)
Substituting the values:
F_c = (900 kg * (40.0 m/s)^2) / 100 m
Step 2: Calculate the normal force
Since the road is frictionless, the horizontal component of the normal force is equal to the centripetal force. Therefore, the normal force is given by:
N = F_c
Substituting the value of F_c:
N = (900 kg * (40.0 m/s)^2) / 100 m
Step 3: Calculate the vertical component of the normal force
To find the vertical component, we use the relationship between the normal force (N) and the weight (mg) of the car:
N = mg
Substituting the value of m:
mg = (900 kg * 9.8 m/s^2)
Step 4: Determine the angle θ
Since the normal force (N) has both vertical and horizontal components, we can use trigonometry to find the angle θ:
tan(θ) = (vertical component of N) / (horizontal component of N)
Substituting the values:
tan(θ) = ((900 kg * 9.8 m/s^2) / (900 kg * (40.0 m/s)^2) / 100 m)
Simplifying and solving for θ:
θ = atan(((900 kg * 9.8 m/s^2) / (900 kg * (40.0 m/s)^2) / 100 m))
Calculating this using a calculator, θ is approximately equal to 0.25 degrees.