Problem $3.$ Resisting rollover.

Relevant learning goals:

$-$ Calculate the torque due to a force about a reference point or axis.

$-$ Solve statics problems using both the translational and rotational forms of Newton's $2$nd Law.

Subskills: $1,2,3$

If a vehicle tries to turn around a corner too quickly, it can rollover $($or tip over$)$ onto its side $($or worse$).$ Here we analyze the conditions of stability for a bicycle and a car.

$($a$)$ A bicycle and its rider have a combined center of mass a height $\backslash ($ h $\backslash )$ above the ground and a total mass $\backslash ($ m $\backslash ).$ The bike is moving at a constant speed $\backslash ($ v $\backslash )$ in a circle of radius $\backslash ($ r $\backslash ).$ You know from experience that you need to lean into a turn so that the bike makes an angle $\backslash (\backslash$theta $\backslash )$ with respect to the vertical.

What lean angle should the bike make so that the bike doesn't rollover or fall to the ground? In other words, what angle makes the bike stable?

Assume the coefficient of friction between the tires and the road is strong enough to prevent slipping. Assume that gravity acts solely on the center of mass, which is located a height $\backslash ($ h $\backslash )$ off the ground when in the upright position. Hint: using the apparent centrifugal force in an accelerating reference frame may be helpful to solve this problem, but it is not necessary. See Problem Set $4$ to reminder yourself of circular motion.

$($b$)$ A four$-$wheeled vehicle is more stable than a bike, but it can still topple over if it corners a turn too sharply. The car has a center of mass a height $\backslash ($ h $\backslash )$ off the ground and a width between the centerline of two wheels on the same axle $($called "track"$)$ of $\backslash ($ w $\backslash ).$ What is the maximum speed the car can go around a turn of radius $\backslash ($ r $\backslash )$ without rolling over? Again, assume the coefficient of static friction is great enough to prevent slipping.

$($c$)$ A friend tells you to add weight to the roof rack of an SUV to make the vehicle heavier to prevent rolling over. Would this work? Explain why or why not.