Circular motion- finding angle of the banking

In summary, the problem involves a banked circular highway curve with a radius of 200m and a minimum coefficient of friction needed for cars to safely take the turn without sliding off the road. To find the angle of banking, the formula theta=tangent^-1(v^2/rg) is used, where r is the radius, v is the velocity, and g is gravity. However, in order to get the correct angle, standard units for speed and distance, such as m/s and m, should be used instead of km/h and km.
  • #1
wxwolf
1
0

Homework Statement


A banked circular highway curve is designed for traffic moving at 60km/h the radius of the curve is 200m traffic is moving along the highway at 40km/h on a rainy day. what is the minimum coefficient of friction between the tires and the road that will allow cars to take the turn without sliding off the road (assume the cars do not have negative lift)


Im trying to find the angle of the banking first. i used [theta=tangent^-1(v^2/rg)] where r is the radius, v is the velocity and g is gravity.

when i do the problem out though:

=tan^-1(60km/h)^2/(.2km)(9.8m/s^2)
=tan^-1(3600)/(1.96)
=tan^-1(1836.73)
=89.96

there is no way that banking is almost 90 degrees. Where am i going wrong?

once i do that, I can plug it in and solve the problem for the kinetic coefficient.
 
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  • #2
You're messing up with units. Use standard units for speed (m/s) and distance (m).
 
  • #3


First of all, it is important to note that the formula you are using to find the angle of banking assumes that the curve is frictionless. In reality, there will always be some friction between the tires and the road, so the actual angle of banking will be less than 90 degrees. This could be the reason why your calculation is giving you an unrealistic result.

To find the correct angle of banking, you need to consider the forces acting on the car. In this case, the centripetal force (F=mv^2/r) is balanced by the component of the normal force (N) acting towards the center of the curve. This can be represented by the equation N=mgcos(theta), where theta is the angle of banking.

Using this equation, we can rearrange it to solve for theta: theta=cos^-1(v^2/rg)

Plugging in the values given in the problem, we get: theta=cos^-1((40km/h)^2/(200m)(9.8m/s^2))=cos^-1(0.0816)=83.5 degrees.

This is a more realistic angle of banking for a highway curve and can be used to solve for the minimum coefficient of friction.
 

1. What is circular motion?

Circular motion is a type of motion in which an object moves in a circular path around a fixed point, also known as the center of rotation.

2. How do you find the angle of banking in circular motion?

To find the angle of banking in circular motion, you can use the formula θ = tan⁻¹(v²/rg), where v is the speed of the object, r is the radius of the circular path, and g is the acceleration due to gravity. This angle is also known as the angle of inclination or the angle of tilt.

3. Why is the angle of banking important in circular motion?

The angle of banking is important in circular motion because it helps to maintain the stability and safety of the object moving in the circular path. It also helps to reduce the friction between the object and the surface, allowing for a smoother motion.

4. What factors affect the angle of banking in circular motion?

The factors that affect the angle of banking in circular motion include the speed of the object, the radius of the circular path, and the coefficient of friction between the object and the surface. The angle of banking also varies depending on the type of vehicle or object in motion.

5. How does the angle of banking affect the speed of an object in circular motion?

The angle of banking affects the speed of an object in circular motion by providing the necessary centripetal force to keep the object moving in the circular path. If the angle of banking is too small, the object may slip or slide on the surface. If the angle is too large, it may lead to excessive centrifugal force and cause the object to lose control or stability.

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