Part of riding a bicycle involves leaning at the correct angle when making a turn, as seen in Figure 6.36. 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 (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, =tan1v2/rg (b) Calculate for a 12.0 m/s turn of radius 30.0 m (as in a race). Figure 6.36 A bicyclist negotiating a turn on level ground must lean at the correct angle—the ability to do this becomes instinctive. The force of the ground on the wheel needs to be on a line through the center of gravity. The net external force on the system is the centripetal force. The vertical component of the force on the wheel cancels the weight of the system while its horizontal component must supply the centripetal force. This process produces a relationship among the angle , the speed v, and the radius of curvature r of the turn similar to that for the ideal banking of roadways.

We know from studying friction forces that static friction increases with increasing normal force between the surfaces, which becomes important for vehicles traveling on icy or snowy roads that have coefficients of static friction much smaller than those of dry pavement. In particular, the greater the normal force on the drive wheels (those coupled to the engine), the better the traction. The horizontal position of the center of mass of a typical compact automobile is located 1.1 m toward the rear as measured from the front wheel axle. The wheelbase (distance from the front wheel axle to the rear wheel axle) is 2.7 m. Assume the car is stationary on level ground and has a weight of 12,000 N. Determine the total normal force on the two front tires and on the two rear tires. Which do you suppose are the drive wheels in this case?

An office door is given a sharp push and swings open against a pneumatic device that slows the door down and then reverses its motion. At the moment the door is open the widest, (a) does the doorknob have a centripetal acceleration? (b) Does it have a tangential acceleration?

A car rounds an unbanked curve of radius 65 m. If the coefficient of static friction between the road and car is 0.70, what is the maximum speed at which the car can traverse the curve without slipping?

Jamal and Dayo are lifting a large chest, weighing 207 lb, by using the two rope handles attached to either side. As they lift and hold it up so that it is motionless, each handle makes a different angle with respect to the vertical side of the chest (Fig. P5.76). If the angle between Jamals handle and the vertical side is 25.0 and the angle between Dayos handle and the vertical side of the chest is 30.0, what are the tensions in each handle? FIGURE P5.76

By what factor does the drag force on a car increase as it goes from 65 to 110 km/h?

If a car takes a banked curve at less than a given speed, friction is needed to keep it from sliding toward the inside of the curve (a real problem on icy mountain roads). a) Calculate the minimum speed, in meters per second, required to take a 84 m radius curve banked at 14° so that you don't slide inwards, assuming there is no friction.  b)What is the minimum coefficient of friction needed for a frightened driver to take the same curve at 25 km/h?

If a car takes a banked curve at less than the ideal speed, friction is needed to keep it from sliding toward the inside of the curve (a real problem on icy mountain roads). (a) Calculate the ideal speed in (m/s) to take a 95 m radius curve banked at 15°. 15.8 m/s (b) What is the minimum coefficient of friction needed for a frightened driver to take the same curve at 10.0 km/h?

A circular curve of highway is designed for traffic moving at 60 km/h. Assume the traffic consists of cars without negative lift. (a) If the radius of the curve is 150 m, what is the correct angle of banking of the road? (b) If the curve were not banked, what would be the minimum coefficient of friction between tires and road that would keep traffic from skidding out of the turn when traveling at 60 km/h?