Suppose the coefficient of static friction between a quarter and the back wall of a rocket car is 0.330. At what minimum rate would the car have to accelerate so that a quarter placed on the back wall would remain in place?

To determine the minimum acceleration required for the quarter to not slide down the back wall of the accelerating rocket car, we will use the formula that relates the force of static friction to the frictional coefficient and the normal force.

The force of static friction (f_s) that acts to keep the quarter from sliding is given by: f_s = μ_s * N

where μ_s is the coefficient of static friction (0.330 in this case) and N is the normal force, which, under normal conditions (when not in an accelerating frame), is equivalent to the force due to gravity on the object.

However, since the rocket car is accelerating, the normal force in this context is provided by the force due to acceleration, not gravity, because the wall the quarter rests upon is perpendicular to the direction of acceleration. For the quarter to stay put, the force of static friction must equal the force due to gravity that's trying to pull the quarter down.

Let's represent the mass of the quarter as m, the acceleration of gravity as g (which is approximately 9.8 m/s^2), and the acceleration of the car as a. The force of static friction in this scenario equals m * a and the force of gravity equals m * g.

Setting the force of static friction equal to the force of gravity, we get: μ_s * m * a = m * g

mass m cancels out from both sides, leading to: μ_s * a = g

Now, we can solve for a: a = g / μ_s

With μ_s = 0.330 and g = 9.8 m/s^2, the calculation becomes: a = 9.8 m/s^2 / 0.330 a ≈ 29.7 m/s^2

So, the minimum acceleration needed for the rocket car to keep the quarter in place is approximately 29.7 m/s^2.

Extra: It's interesting to consider this scenario as an example of non-inertial reference frames, where the usual forces that we think of, like gravity, behave differently due to the acceleration of the frame of reference. When objects are in a non-inertial frame that is accelerating, they experience what are called fictitious forces, like the centrifugal force, that seem to act on them. In our problem, the acceleration of the rocket car creates a situation where the "normal force" the quarter experiences is due to this acceleration.

Understanding the coefficient of friction is also crucial. It is a dimensionless scalar value which describes the ratio of the force of friction between two bodies and the force pressing them together. There are two coefficients of friction: static (μ_s) when objects are not moving relative to each other, and kinetic (μ_k) when objects are sliding past each other. The coefficient of static friction is usually higher than that of kinetic friction. This means it takes more force to start moving an object than it does to keep it moving, once it's already in motion.