Maximum Velocity for a Banked Curve Without Sliding or Rolling Over

In summary: Hi sricho! :smile: Yes, you should be able to look that up :wink:In summary, the maximum velocity at which a car can travel if the banking angle is 15º is 23 meters per second.
  • #1
sricho
7
0
Determine the maximum velocity at which a car can travel if the banking angle is 15º for the following conditions
1) It is not to slide radially outwards.

2) It is not to roll over.

given:
coefficient of friction as 0.4
r = 60m
width of car is 2m and height from bank to centre of car is 0.5m

Homework Equations



Looking at some work i think the equation that i need to use is
V = square root (( tan feta + coefficent of friction/ 1-coefficent of friction tan feta) x r x g)

The Attempt at a Solution


my attempt at this solution gave me an answer of 22.9m/s so 23m/s
i wanted to check if this was correct and I am confused on why i have been given the length of the car and its distance from bank to centre of car.
 
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  • #2
Welcome to PF!

Hi sricho! Welcome to PF! :smile:
sricho said:
Determine the maximum velocity at which a car can travel if the banking angle is 15º for the following conditions

2) It is not to roll over.

given:
coefficient of friction as 0.4
r = 60m
length of car is 2m and height from bank to centre of car is 0.5m

im confused on why i have been given the length of the car and its distance from bank to centre of car.

because it will roll over if the net torque about the inside wheel is towards the centre …

for that, you need to know where the centre of mass is :wink:

hmm … are you sure 2m isn't the width of the car? :confused:
 
  • #3
yes sorry that is the width of the car. so i need to work out the centre of mass. rite that makes sense with the given lenghs.
was the equation for the velocity correct?
pretty new to this so any help would be ausome
thanks
 
  • #4
Hi sricho! :smile:

Sorry, I missed your reply. :redface:
sricho said:
… was the equation for the velocity correct?
sricho said:
Looking at some work i think the equation that i need to use is
V = square root (( tan feta + coefficent of friction/ 1-coefficent of friction tan feta) x r x g)

(have a theta: θ and a mu: µ :wink:)

Difficult to tell, since you haven't shown how you got it …

and do remember, the normal force won't be the usual mgcosθ, because of the centripetal acceleration :wink:
 
  • #5
well i got my answer by using the equation for velocity and subsituting the values i have.
so
sqr(tan15 degree + 0.4 / 1 - 0.4 tan 15 degrees) x 60 x 9.8

but this doesn't use the normall reaction force and i don't think this answers the question
 
  • #6
Ncos θ - µNsin θ = mg
N(cos θ - µsinθ) = mg
N = mg / cos θ - µsinθ

This is the equation i have for the normall reactiong force but you need the mass of the object where its not given.
so I am stuck?
 
  • #7
No, you don't need to know the mass: it always cancels.

And your equations haven't taken the centripetal acceleration into account.
 
  • #8
sorry had a busy weekend
so how do i use that acceleration in the equation i have.
and what is the equation for the acceleration?
thanks
sam
 
  • #9
is this the equation i need ?
Ac = - w^2 r
Ac = - v^2 / r

where w is omega

how do i add this into the velocity equation?
 
  • #10
sricho said:
… what is the equation for the acceleration?

you should be able to look that up :frown:

centripetal acceleration is v2/r (= ω2r)​
 
  • #11
well I am just all confused now, is there any chance you could show me how its done , i would be very happy if you could. it will help loads with the other questions i have to do.
 

What is a banked curve?

A banked curve is a type of curve in which the surface is tilted or angled towards the inside of the curve. This allows for vehicles or objects traveling along the curve to maintain a constant speed without having to use additional force or friction.

How does a banked curve work?

A banked curve utilizes the principles of centripetal force to keep an object moving along the curve without slowing down. The angled surface of the curve provides a normal force that acts perpendicular to the surface, allowing the object to move in a circular path without slipping or sliding.

What factors affect the banking angle of a curve?

The banking angle of a curve is affected by the speed of the object, the radius of the curve, and the coefficient of friction between the object and the surface of the curve. A higher speed or smaller radius will require a steeper banking angle, while a lower coefficient of friction will also require a steeper banking angle to prevent slipping.

What are some real-life examples of banked curves?

Banked curves are commonly seen in racetracks, where cars need to maintain high speeds while navigating sharp turns. Roller coasters also use banked curves to keep the cars on the track while maintaining a thrilling experience for riders.

What happens if the banking angle is too steep or too shallow?

If the banking angle is too steep, the object may experience excessive centripetal force and slide off the curve. If the banking angle is too shallow, the object may not have enough centripetal force and may slow down or even stop while traveling along the curve.

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