How to find the direction of static frictional force - given angular acceleration? (coin on rotating platform)

Question

Hello,

I was wondering if someone could help me with this problem.

Given a coin resting on a rotating aluminum ruler (10cm from the center of the ruler), I am given an angular acceleration of the ruler and told to find the direction that the static friction acting on the coin is pointing.

I am confused because I'm not quite sure how static friction and angular acceleration would be related. I have reasoned that the direction the static friction is pointing is opposite the direction the coin would move given a frictionless surface, but I'm not sure how to proceed.

In case it is relevant, I also have two values for angular velocity (when the coin is not on the ruler), from which I have found a value for angular acceleration using the equation ω_f = ω_i + α * t. (was I meant to use this equation?). In addition, I have the initial and final angular positions (also when the coin was not on the ruler).

t = 0.83 sec

ω₀ = 1.58 rad/sec

ω₁ = 2.23 rad/sec

α = 0.783 rad/sec/sec

Θ₀ = pi radians

Θ₁ = 2pi radians

I am very confused, so any clarification would be much appreciated!

Answer 1

The direction of the force of friction involves the following:

r = 10 cm -- the radius of rotation

α = 0.783 rad/sec/sec -- the angular acceleration

ω -- angular velocity, which is changing.

As ω keeps changing, so does the force of friction:

When ω=0, then so is the force of friction;

as ω keeps increasing, so is the force of friction to the point

when the surface is not able to provide the friction and the coing flies away.

The coin is subject to two accelerations:

centripetal acceleration of magnitude ω²r, which is direction to the center, and

tangential acceleration αr, which is perpendicular to the ruler.

Their vector sum will have angle φ from the ruler.

If you draw a picture then you will see that

tan(φ) = αr/ω²r = α/ω²

Under the unstated assumtion that friction is the only force acting on the coin,

the force of friction must be responsible for that acceleration, and so

has the same direction.