For each of the cases described below, sketch and label the total acceleration vector, the radial acceleration vector, and the tangential acceleration vector. A car slows down while rounding a curve of constant radius.

Find the tangential and centripetal components of the net force exerted on the car (by the ground) in Example 5-16 when its speed is $12 \mathrm{~m} / \mathrm{s}$. The car's mass is $950 \mathrm{~kg}$.

A curve of radius $75 \mathrm{~m}$ is banked for a design speed of $85 \mathrm{~km} / \mathrm{h}$. If the coefficient of static friction is $0.30$ (wet pavement), at what range of speeds can a car safely make the curve?

If a curve with a radius of $85 \mathrm{~m}$ is properly banked for a car traveling $65 \mathrm{~km} / \mathrm{h}$, what must be the coefficient of static friction for a car not to skid when traveling at a speed of $95 \mathrm{~km} / \mathrm{h}$?

The position of a particle moving in the $x y$ plane is given by $\overrightarrow{\mathbf{r}}=(2.0 \mathrm{~m}) \cos [(3.0 \mathrm{rad} / \mathrm{s}) t] \hat{\mathbf{i}}+(2.0 \mathrm{~m}) \sin [(3.0 \mathrm{rad} / \mathrm{s}) t] \hat{\mathbf{j}}$. where $r$ is in meters and $t$ is in seconds. Show that the acceleration vector always points toward the center of the circle.

The position of a particle moving in the $x y$ plane is given by $\overrightarrow{\mathbf{r}}=(2.0 \mathrm{~m}) \cos [(3.0 \mathrm{rad} / \mathrm{s}) t] \hat{\mathbf{i}}+(2.0 \mathrm{~m}) \sin [(3.0 \mathrm{rad} / \mathrm{s}) t] \hat{\mathbf{j}}$. where $r$ is in meters and $t$ is in seconds. Show that $a=v^2 / r$.