How Does Friction Affect a Car on a Rotating Platform?

[Swordplayk]

Homework Statement

I'm not given any signs of a correct answer in the book, so could I get a check here?

A car is driven on a large revolving platform which rotates with constant angular speed w. At t = 0 a driver leaves the origin and follows a line painted radially outward on the platform with constant speed v. The total weight of the car is W, and the coefficient of friction is
μ
between the car and platform.

a) Find the acceleration of the car as a function of time using polar coordinates.

b) Find the time at which the car starts to slide.

c) Find the direction of the friction force with respect to the instantaneous position vector r just before the car starts to slide.

2. Homework Equations

Ffriction<=μW

a=(r′′−rθ′2)rˆ+(rθ′′+2r′θ′)θˆ

The Attempt at a Solution

a) r' = v,
θ′
= w, and r'' =
θ′′
= 0, so
a=(−vw2)trˆ+2vwθˆ
.

b) In order for the car to follow the correct path (constant velocities in the radial and normal directions), the radial and normal accelerations must be zero. This is due to the frictional force counteracting these accelerations, and when the car starts to slide is the point at which
m∗|anet|=fmax
. This happens when:

m|a|=mv2w4t2+4v2w2−−−−−−−−−−−−√=μW=μmg⟹t=u2g2−4v2w2v2w4−−−−−−−−−−−√

However, for part C, I'm not sure what relationship I'm supposed to be looking for between r and the frictional Force. Thanks.

 

[anathema]

Thank you for your post. It seems like you are on the right track with your solution for parts a) and b). For part c), you are correct that you need to find a relationship between r and the frictional force in order to determine the direction of the friction force with respect to the instantaneous position vector r. One way to do this is to consider the forces acting on the car at the point just before it starts to slide. These forces include the normal force, the weight of the car, and the frictional force. You can then use the equation Ffriction<=μW to relate the magnitude of the frictional force to the weight of the car. From there, you can use trigonometry to determine the direction of the friction force with respect to the instantaneous position vector r. I hope this helps. Let me know if you have any further questions.