Question

If a plastic cup on a plastic dash has a static coefficient of friction of around 0.15, what is the shortest amount of time (in s) you can decelerate from 70mph to rest in so that the cup does not slide?

a. Draw a free-body diagram for the plastic cup.

b. Write down the force-acceleration law of motion for the horizontal and the vertical components of all the forces.

c. Use the inequality relating the normal and friction forces to solve the equation for horizontal acceleration (assuming no vertical acceleration).

d. Use the constant acceleration equation relating change in velocity with acceleration and time to solve symbolically for the time above which you can safely decelerate with no sliding of the cup.

e. Put in the numbers for coefficient of friction and initial and final velocity to find the time in s.

Answer 1

a. We assume that the plastic cup is accelerating forward in the positive x direction with frictional force acting in the opposite direction. Hence these are the only two forces acting on the cup in the horizontal direction. The weight of the cup acts vertically downwards.

b. Horizontal component: F = ma

Vertical W = mg

c. We know that μmg >= ma

=> μmg = ma

=> μg = a

d. 70mph = 31.29 m/s

From 1st equation of motion

v = u + at

=> v = 0 + μgt

=> t = v/μg

=> t = 31.29/0.15*9.8

=> t = 21.28 s