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.
We can use the following formula to solve this problem:
centripetal force = mv^2/r
where m is the mass of the car, v is its speed, and r is the radius of the curve.
In this case, the centripetal force is provided by the horizontal component of the normal force, since there is no friction. So we can write:
centripetal force = N sin(θ)
where N is the normal force and θ is the angle of banking.
Equating these two expressions for the centripetal force, we get:
N sin(θ) = mv^2/r
Solving for θ, we get:
θ = sin^-1(mv^2/(Nr))
Substituting the given values, we get:
θ = sin^-1((900 kg)(40.0 m/s)^2/((900 kg)(9.81 m/s^2)(100m)))
θ ≈ 32.83 degrees
Therefore, the angle θ has to be approximately 32.83 degrees if the road is frictionless.
To determine the angle θ, we can use the equation for centripetal force. The centripetal force is provided by the horizontal component of the normal force acting on the car. In the absence of any other horizontal forces (such as friction), the equation for centripetal force is as follows:
Fc = m * v^2 / r
Where:
Fc is the centripetal force,
m is the mass of the car (900 kg),
v is the velocity of the car (40.0 m/s),
and r is the radius of the curve (100 m).
In this case, the centripetal force is provided by the horizontal component of the normal force, which is given by:
Fn = m * g * cos(θ)
Where:
Fn is the normal force,
g is the acceleration due to gravity (approximately 9.8 m/s^2),
and θ is the angle between the normal force and the vertical direction (which we need to determine).
Since the road is frictionless, the vertical component of the normal force (Fn * sin(θ)) is equal to the weight of the car (m * g). Therefore, we can equate these two equations:
Fn * sin(θ) = m * g
Solving for Fn:
Fn = m * g / sin(θ)
Substituting this value for Fn in the equation for centripetal force, we get:
m * g / sin(θ) = m * v^2 / r
Simplifying the equation:
g / sin(θ) = v^2 / r
Now we can solve for θ:
sin(θ) = r * g / v^2
θ = arcsin(r * g / v^2)
θ = arcsin((100 m * 9.8 m/s^2) / (40.0 m/s)^2)
Let's calculate the value of θ using the given values.
θ ≈ arcsin(24.5 / 16) ≈ 61.06 degrees
Therefore, the angle θ has to be approximately 61.06 degrees if the road is frictionless.