Rotational Motion of a car on a curve

[ajmCane22]

Homework Statement

A 600 kg car is going around a banked curve with a radius of 110m at a speed of 24.5 m/s. What is the appropriate banking angle so that the car stays on its path without the assistance of friction?

Homework Equations

N cos{theta} = mg
N sin{theta} = mv^2/r

The Attempt at a Solution

I was told to divide the second equation by the first equation which gives tan{theta} = v^2/rg
I used this equation and got the right answer, but I'm just wondering if somebody could please explain WHY the second equation was divided by the first and not the other way around.

 

[cepheid]

I was told to divide the second equation by the first equation which gives tan{theta} = v^2/rg
I used this equation and got the right answer, but I'm just wondering if somebody could please explain WHY the second equation was divided by the first and not the other way around.

If you divided the first equation by the second, you would get:

cot(θ) = (gr/v²)

θ = arccot(gr/v²).

I imagine that ought to give you the right answer as well...

 

[ajmCane22]

This might be a really dumb question, but why do you divide instead of multiply?

 

[cepheid]

This might be a really dumb question, but why do you divide instead of multiply?

You do whatever algebraic manipulation makes it easiest to solve for theta. With N's on both lefthand sides, and m's on both righthand sides, it seems natural to get rid of both of them by dividing them out. Then you're left with something that is only in terms of theta on the lefthand side.

 

[rwisz]

As a scientist, it is important to understand the reasoning behind equations and how they are derived. In this case, the two equations provided are derived from the principles of rotational motion and the forces acting on the car on a banked curve.

The first equation, N cos{theta} = mg, represents the vertical forces acting on the car. N represents the normal force, which is the force exerted by the road on the car, and mg represents the weight of the car. Since the car is moving in a circular motion, the normal force must balance out the weight of the car in order for it to stay on the curved path.

The second equation, N sin{theta} = mv^2/r, represents the horizontal forces acting on the car. N sin{theta} represents the component of the normal force that is acting in the direction of the car's motion, and mv^2/r represents the centripetal force needed to keep the car on its curved path. This equation is derived from the fact that the car's acceleration is directed towards the center of the curve, and is equal to v^2/r.

Dividing the second equation by the first equation allows us to eliminate the normal force, which is a common factor in both equations. This leaves us with an equation that relates the tangential velocity of the car (v), the radius of the curve (r), and the angle of the banking (theta). By rearranging this equation, we can solve for theta and determine the appropriate banking angle for the car to stay on its path without the assistance of friction.

In summary, dividing the second equation by the first is a mathematical manipulation that allows us to eliminate the common factor (normal force) and solve for the desired variable. This approach is commonly used in physics and other scientific fields to simplify equations and make them more easily solvable.