Centripetal Acceleration and Finding an Angle

[David_W_2012]

Homework Statement

An old streetcar rounds a flat corner of radius 9.1 m, at 16 km/h. What angle with the vertical will be made by the loosely hanging hand straps?

Homework Equations

The equations I know:
acceleration = velocity squared / radius
(a = v^2 / R)

Force = mass * acceleration
(Fnet = m * a)

Friction = Coefficient * mass * g
(f = U * mass * g)

The Attempt at a Solution

I did some calculations and I am pretty sure I am right so far about these values:

a = 4.444 m / s^2
Not hard to find; just convert 16 km/h to m/s, use the given radius, then plug and chug to find the acceleration.

U = 0.453464
I found this by noting that the centripetal acceleration is due to a single force: the force of static friction pointing towards the center of the turn. Knowing that
f = U * mass * g
And that a = F / mass,
Then a = (U * mass * g) / mass = U * g,
Leaving us with U = a / g = 0.453464.

But now I'm stuck. I have no idea how to generate an angle in this problem. The only angle I see is the 90 degree angle between the force of friction and the normal force or gravity. I thought I might need to construct a triangle with legs l1 = |friction| and l2 = |normal force|, but I have no logic for thinking this and am unsure how to proceed.

 

[Dick]

There's no friction acting on the straps. They are hanging freely. That means the direction of the straps is parallel to the direction of total acceleration. If there were no centripetal acceleration they would be hanging vertically due to the gravitational acceleration. If there were no gravity they would be hanging horizontally due the centripetal acceleration. Suppose there is both? Then it should be someplace in between, right? What's the angle made by the sum of the two acceleration vectors?

 

[Delphi51]

Maybe check that acceleration calculation; I got 2.17 but I make lots of mistakes.

From the point of view of the car moving in circular motion, the hanging mass will have a centripetal force ma which causes it to angle outward. You will probably have to make a diagram showing the three forces acting on the strap and separate them into horizontal and vertical components. Still in the point of view of the car, the strap is not moving and has zero acceleration so the total force is zero in both directions.

 

[David_W_2012]

There's no friction acting on the straps. They are hanging freely. That means the direction of the straps is parallel to the direction of total acceleration. If there were no centripetal acceleration they would be hanging vertically due to the gravitational acceleration. If there were no gravity they would be hanging horizontally due the centripetal acceleration. Suppose there is both? Then it should be someplace in between, right? What's the angle made by the sum of the two acceleration vectors?

BRILLIANT! I was getting hung up because I didn't know what the masses were, and was trying to do a sum of the force vectors of friction and gravity. But like you said, there's no friction acting on the straps, so that doesn't make sense. But after realizing that I do not need the masses any longer because I have the accelerations, I was able to find the correct answer very promptly.

Also to the above poster, you were right, I found the wrong value for a and consequently the coefficient of friction. I was very frustrated at the time, haha