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Race car drivers routinely cut corners as shown in [link] . Explain how this allows the curve to be taken at the greatest speed.
A number of amusement parks have rides that make vertical loops like the one shown in [link] . For safety, the cars are attached to the rails in such a way that they cannot fall off. If the car goes over the top at just the right speed, gravity alone will supply the centripetal force. What other force acts and what is its direction if:
(a) The car goes over the top at faster than this speed?
(b)The car goes over the top at slower than this speed?
What is the direction of the force exerted by the car on the passenger as the car goes over the top of the amusement ride pictured in [link] under the following circumstances:
(a) The car goes over the top at such a speed that the gravitational force is the only force acting?
(b) The car goes over the top faster than this speed?
(c) The car goes over the top slower than this speed?
As a skater forms a circle, what force is responsible for making her turn? Use a free body diagram in your answer.
Suppose a child is riding on a merry-go-round at a distance about halfway between its center and edge. She has a lunch box resting on wax paper, so that there is very little friction between it and the merry-go-round. Which path shown in [link] will the lunch box take when she lets go? The lunch box leaves a trail in the dust on the merry-go-round. Is that trail straight, curved to the left, or curved to the right? Explain your answer.
Do you feel yourself thrown to either side when you negotiate a curve that is ideally banked for your car’s speed? What is the direction of the force exerted on you by the car seat?
What is the ideal banking angle for a gentle turn of 1.20 km radius on a highway with a 105 km/h speed limit (about 65 mi/h), assuming everyone travels at the limit?
$4\text{.}\text{14\xba}$
(a) What is the radius of a bobsled turn banked at 75.0° and taken at 30.0 m/s, assuming it is ideally banked?
(b) Calculate the centripetal acceleration.
(c) Does this acceleration seem large to you?
a) 24.6 m
b) $\mathrm{36.6\; m}/{\text{s}}^{2}$
c) ${a}_{\text{c}}=3.73\phantom{\rule{0.25em}{0ex}}\mathrm{g.}$ This does not seem too large, but it is clear that bobsledders feel a lot of force on them going through sharply banked turns.
Part of riding a bicycle involves leaning at the correct angle when making a turn, as seen in [link] . To be stable, the force exerted by the ground must be on a line going through the center of gravity. The force on the bicycle wheel can be resolved into two perpendicular components—friction parallel to the road (this must supply the centripetal force), and the vertical normal force (which must equal the system’s weight).
(a) Show that $\theta $ (as defined in the figure) is related to the speed $v$ and radius of curvature $r$ of the turn in the same way as for an ideally banked roadway—that is, $\theta ={\text{tan}}^{\text{\u20131}}{v}^{2}/\mathrm{rg}$
(b) Calculate $\theta $ for a 12.0 m/s turn of radius 30.0 m (as in a race).
A large centrifuge, like the one shown in [link] (a), is used to expose aspiring astronauts to accelerations similar to those experienced in rocket launches and atmospheric reentries.
(a) At what angular velocity is the centripetal acceleration $\mathrm{10}\phantom{\rule{0.25em}{0ex}}g$ if the rider is 15.0 m from the center of rotation?
(b) The rider’s cage hangs on a pivot at the end of the arm, allowing it to swing outward during rotation as shown in [link] (b). At what angle $\theta $ below the horizontal will the cage hang when the centripetal acceleration is $\mathrm{10}\phantom{\rule{0.25em}{0ex}}g$ ? (Hint: The arm supplies centripetal force and supports the weight of the cage. Draw a free body diagram of the forces to see what the angle $\theta $ should be.)
a) 2.56 rad/s
b) $\mathrm{5.71\xba}$
Unreasonable Results
(a) Calculate the minimum coefficient of friction needed for a car to negotiate an unbanked 50.0 m radius curve at 30.0 m/s.
(b) What is unreasonable about the result?
(c) Which premises are unreasonable or inconsistent?
a) 1.84
b) A coefficient of friction this much greater than 1 is unreasonable .
c) The assumed speed is too great for the tight curve.
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