On a banked race track, the smallest circular path on which cars can move has a radius r1 = 106 m, while the largest has a radius r2 = 153 m, as the drawing illustrates. The height of the outer wall is 18 m.
Find the smallest speed at which cars can move on this track without relying on friction.
) Find the largest speed at which cars can move on this track without relying on friction.
To find the minimum speed at which cars can move on the track without relying on friction, we need to consider the forces acting on the car.
Let's start with the minimum speed. When a car is moving at the minimum possible speed, the frictional force between the car's tires and the track is zero. This means that the only force acting on the car is the normal force.
The normal force is the force exerted by a surface to support the weight of an object in contact with it. In this case, the normal force is equal to the weight of the car. So, we need to find the weight of the car.
The weight of an object is given by the formula: weight = mass * gravity
Since we don't have information about the mass of the car, we can use the assumption that all cars have the same mass. Let's assume the mass of the car is m.
Now, the weight of the car can be calculated as: weight = m * gravity
Next, we need to consider the forces acting on the car when it is moving on the banked track. There are two forces involved: the weight of the car acting vertically downward and the normal force acting perpendicular to the banked surface.
Since the minimum speed is when the car is just about to slide down the banked surface, the normal force needs to cancel out the component of the weight that acts down the banked surface.
To find the component of the weight that acts down the banked surface, we use trigonometry. The angle of the banked track can be found using the equation: tangent(angle) = height/radius
Given the height of the outer wall (18 m), and the radius of the smallest circular path (106 m), we can find the angle by substituting the values into the equation.
tangent(angle) = 18/106
angle = arctan(18/106)
angle ≈ 9.77 degrees
Now, we can calculate the component of the weight that acts down the banked surface using the formula: component = weight * cos(angle)
The normal force needs to be equal to the component of the weight, so the minimum speed can be calculated by setting the normal force equal to the component of the weight and solving for the speed.
Let's assume the minimum speed is v1. The equation becomes:
m * gravity * cos(angle) = m * v1^2 / radius1
Simplifying the equation, we can find the minimum speed:
v1 = sqrt(gravity * radius1 * cos(angle))
Now that we have calculated the minimum speed, we can move on to finding the maximum speed.
To find the maximum speed, we need to consider the forces acting on the car at this speed. At the maximum speed, the car is experiencing the maximum frictional force that can keep it in a circular path.
The maximum frictional force is acting towards the center of the circular path and can be calculated using the formula: frictional force = mass * centripetal acceleration
The centripetal acceleration is given by: centripetal acceleration = v2^2 / radius2
Let's assume the maximum speed is v2. The equation becomes:
frictional force = mass * v2^2 / radius2
From the equation, we can solve for the maximum speed:
v2 = sqrt(frictional force * radius2 / mass)
Since the frictional force is the maximum force that can be provided by friction, it is equal to the static friction coefficient multiplied by the normal force. So, the equation becomes:
v2 = sqrt(static friction coefficient * normal force * radius2 / mass)
However, we can substitute the normal force with the weight of the car because the normal force is equal to the weight of the car when the car is not sliding.
v2 = sqrt(static friction coefficient * weight * radius2 / mass)
So, to find the maximum speed, we need to know the static friction coefficient.
In summary, to find the smallest speed at which cars can move on the track without relying on friction, use the formula:
v1 = sqrt(gravity * radius1 * cos(angle))
To find the largest speed at which cars can move on the track without relying on friction, use the formula:
v2 = sqrt(static friction coefficient * weight * radius2 / mass)