Question on Motion of a car round a banked track

In summary, the conversation discusses a question about the motion of a car around a banked track, with given parameters of its mass, speed, and track radius. The conversation includes equations and diagrams to help solve for the angle of inclination of the track and the reaction force at the wheels. After some discussion and calculations, it is determined that the angle is 42 degrees and the reaction force is 13450N. The conversation also explores the process of deriving the formula for this problem.
  • #1
Freyth
12
2
Question on Motion of a car round a banked track [SOLVED]

Homework Statement



A racing car of 1000kg moves round a banked track at a constant speed of 108km/h. Assuming the total reaction at the wheel is normal to the track, and the horizontal radius is 100m, calculate the angle of inclination of the track to the horizontal and the reaction at the wheels.

Homework Equations



tan(theta) = v^2 / rg where r is the radius and g is 10m/s

v = rw, w is the angular speed, r is the radius

F = mrw^2 = mv^2 / r
, F is the force towards the centre of the track, r is the radius, w is the angular speed

The Attempt at a Solution



Currently in a Junior College, not sure what that equates to in any part of the world but it's sort of a Pre-University education. So that should give you some idea on my knowledge.

I tried drawing a vector diagram as attached. But I can't figure out whether the car is at the extreme of the track so that its horizontal distance is 100m from the track.

However, I'm also unsure if the above equations with radius means horizontal radius. That said, I assumed it to be and I used the above equations to try solving but I couldn't get an answer. Spent over 2 hours and I'm stumped... Anyone care to help? Any help is appreciated!

P.S. the Answer is 42 degrees and 13450N.
 

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  • #2
The 100m means that the radius of the centrifugal force is 100m. The fact that the reaction at the wheel is normal to the surface means that the frictional force along the track and the centrifugal force along the track are equal. This should give you the angle.
 
  • #3
Draw the freebody diagram of the car... It is on an incline at an angle theta... what are the forces acting on it...

EDIT: nevermind... I'll have a look at the picture you've attached when it is approved...
 
  • #4
The tan(theta) equation you have posted gives you the angle... can you show your calculations?
 
  • #5
Okay guys, thanks for your input. I forgot that by converting 108km/h gives me 30m/s. I was under the impression that it was 108 x 1000 x 3600, when it should have been
(108 x 1000) / 3600 = 30m/s.

So by using tan (theta) = v^2 / rg,

(theta) = tan^-1 (900 / 100 x 10) = 41.9 ~ 42 degrees.

I've yet to find the reaction force but I'm sure I can handle that.

Thanks for the fast replies! Greatly appreciated.
 
  • #6
Freyth said:
Okay guys, thanks for your input. I forgot that by converting 108km/h gives me 30m/s. I was under the impression that it was 108 x 1000 x 3600, when it should have been
(108 x 1000) / 3600 = 30m/s.

So by using tan (theta) = v^2 / rg,

(theta) = tan^-1 (900 / 100 x 10) = 41.9 ~ 42 degrees.

I've yet to find the reaction force but I'm sure I can handle that.

Thanks for the fast replies! Greatly appreciated.

Although the formula works... you should be able to derive the formula yourself... using the freebody diagram... what are the vertical forces? what are the horizontal forces? what is the vertical acceleration? what is the horizontal acceleration?
 

1. What is the purpose of banking a track for a car?

Banking a track for a car allows for higher speeds to be achieved while maintaining control and reducing the risk of the car sliding off the track.

2. How does the angle of banking affect the car's motion?

The angle of banking determines the amount of centripetal force acting on the car, which allows the car to maintain a circular path around the track.

3. What is the relationship between the speed of the car and the angle of banking?

As the speed of the car increases, the angle of banking must also increase in order for the car to maintain a steady circular path on the track.

4. How does friction affect the motion of a car on a banked track?

Friction plays a crucial role in keeping the car on the track. The frictional force between the tires and the track provides the necessary centripetal force for the car to make the turn.

5. Can a car successfully navigate a banked track with no friction?

No, a car would not be able to navigate a banked track without friction. Without friction, there would be no centripetal force acting on the car, causing it to slide off the track.

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