A car enters a banked curve with a radius of 100 meters. Calculate the minimum speed the car can navigate the curve without relying on friction, given that the banking angle is 30 degrees.
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To calculate the minimum speed, we can use the concept of centripetal force.
The centripetal force is the force that keeps an object moving in a curved path. In this case, the centripetal force is provided by the component of the car's weight that acts towards the center of the curve.
The force of gravity acting on the car can be split into two components: one perpendicular to the surface of the curve (normal force) and one parallel to the surface of the curve (tangential force).
Since the car is not relying on friction, the only force acting along the surface of the curve is the tangential force. To calculate this force, we need to find the component of the weight parallel to the surface.
The component of the weight parallel to the surface is given by the equation:
tangential force = weight * sin(banking angle)
where the banking angle is 30 degrees.
The tangential force provides the centripetal force to keep the car moving in a curved path. The equation for centripetal force is:
centripetal force = mass * velocity^2 / radius
where mass is the mass of the car, velocity is the speed of the car, and radius is the radius of the curve.
Since the minimum speed is required, we want to find the velocity when the centripetal force is equal to the tangential force. Therefore, we can set these two equations equal to each other and solve for velocity:
mass * velocity^2 / radius = weight * sin(banking angle)
Canceling the mass from both sides, we get:
velocity^2 = (weight * sin(banking angle) * radius) / mass
velocity = sqrt((weight * sin(banking angle) * radius) / mass)
Now, we need to substitute the values for weight, sin(banking angle), radius, and mass to find the minimum speed.
The weight of the car can be calculated using the equation:
weight = mass * gravity
where gravity is the acceleration due to gravity (approximately 9.8 m/s^2).
Substituting the known values:
weight = mass * gravity = mass * 9.8 m/s^2
sin(banking angle) = sin(30 degrees) = 0.5
radius = 100 meters
Substituting these values into the equation for velocity, we have:
velocity = sqrt((mass * gravity * sin(banking angle) * radius) / mass)
= sqrt((9.8 m/s^2 * 0.5 * 100 meters) / mass)
The mass of the car is not given in the problem statement. We cannot calculate an exact value for the minimum speed without knowing the mass of the car.
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