5.95. Banked Curve I. A curve with a 120-m radius on a level (d) Before reaching the ground, the monkey grabs the rope to stop road is banked at the correct angle for a speed of 20 m/s. If an her fall. What do the bananas do? hatomobile rounds this curve at 30 m/s, what is the minimum 5.100. You throw a rock downward into water with a speed of coefficient of static friction needed between tires and road to pre- 3mg/k, where k is the coefficient in Eq. (5.7). Assume that the rela- vent skidding? tionship between fluid resistance and speed is as given in Eq. (5.7), 5.96. Banked Curve II. Consider a wet roadway banked as in and calculate the speed of the rock as a function of time. Example 5.23 (Section 5.4), where there is a coefficient of static 5.101. A rock with mass m = 3.00 kg falls from rest in a viscous friction of 0.30 and a coefficient of kinetic friction of 0.25 between medium. The rock is acted on by a net constant downward force of the tires and the roadway. The radius of the curve is R = 50 m. 18.0 N (a combination of gravity and the buoyant force exerted by (a) If the banking angle is 8 = 25', what is the maximum speed the medium) and by a fluid resistance force f = ku, where v is the the automobile can have before sliding up the banking? (b) What speed in m/s and & = 2.20 N . s/m (see Section 5.3). (a) Find the is the minimum speed the automobile can have before sliding down initial acceleration do (b) Find the acceleration when the speed is the banking? 3.00 m/s. (c) Find the speed when the acceleration equals 0.1a,. 5.97. Maximum Safe Speed. As you travel every day to campus, (d) Find the terminal speed up (e) Find the coordinate, speed, and the road makes a large turn that is approximately an arc of a circle. acceleration 2.00 s after the start of the motion. (f) Find the time You notice the warning sign at the start of the turn, asking for a max- required to reach a speed 0.9,. imum speed of 55 mi/h. You also notice that in the curved portion 5.102. A rock with mass m slides with initial velocity vo on a hori- the road is level-that is, not banked at all. On a dry day with very zontal surface. A retarding force F, that the surface exerts on the little traffic, you enter the turn at a constant speed of 80 mi/h and rock is proportional to the square root of the instantaneous veloc- feel that the car may skid if you do not slow down quickly. You con- ity of the rock (F. = -kulF ). (a) Find expressions for the veloc- clude that your speed is at the limit of safety for this curve and you ity and position of the rock as a function of time. (b) In terms of slow down. However, you remember reading that on dry pavement m, k, and vo. at what time will the rock come to rest? (c) In terms new tires have an average coefficient of static friction of about 0.76. of m. k, and vo. what is the distance of the rock from its starting while under the worst winter driving conditions, you may encounter point when it comes to rest? wet ice for which the coefficient of static friction can be as low as 5.103. A fluid exerts an upward buoyancy force on an object 0.20. Wet ice is not unheard of on this road, so you ask yourself immersed in it. In the derivation of Eq. (5.9) the buoyancy force whether the speed limit for the turn on the roadside warning sign is exerted on an object by the fluid was ignored. But in some situa- for the worst-case scenario. (a) Estimate the radius of the curve tions, where the density of the object is not much greater than the from your 80-mi/h experience in the dry turn. (b) Use this estimate density of the fluid, you cannot ignore the buoyancy force. For a to find the maximum speed limit in the turn under the worst wet-ice plastic sphere falling in water, you calculate the terminal speed to conditions. How does this compare with the speed limit on the sign? be 0.36 m/s when you ignore buoyancy, but you measure it to be Is the sign misleading drivers? (c) On a rainy day, the coefficient of 0.24 m/s. The buoyancy force is what fraction of the weight? static friction would be about 0.37. What is the maximum safe speed 5.104. The 4.00-kg block in Figure 5.78 Problem 5.104. for the turn when the road is wet? Does your answer help you under- Fig. 5.78 is attached to a verti- stand the maximum speed sign? cal rod by means of two 5.98. You are riding in a school bus. As the bus rounds a flat curve strings. When the system at constant speed, a lunch box with mass 0.500 kg. suspended from rotates about the axis of the the ceiling of the bus by a string 1.80 m long, is found to hang at rod. the strings are extended as 1.25 m rest relative to the bus when the string makes an angle of 30.0" shown in the diagram and the with the vertical. In this position the lunch box is 50.0 m from the tension in the upper string is 2:00 m 4.00 kg center of curvature of the curve. Figure 5.77 Problem 5,99 80.0 N. (a) What is the tension What is the speed u of the bus? in the lower cord? (b) How 1.25 m 5.99. The Monkey and many revolutions per minute Bananas Problem. A 20-kg does the system make? (c) Find monkey has a firm hold on a the number of revolutions per light rope that passes over a minute at which the lower cord frictionless pulley and is just goes slack. (d) Explain attached to a 20-kg bunch of what happens if the number of revolutions per minute is less than bananas (Fig. 5.77). The monkey in part (c). looks up, sees the bananas, and 5.105. Equation (5.10) applies to the case where the initial velocity starts to climb the rope to get is zero, (a) Derive the corresponding equation for u, () when the them. (a) As the monkey climbs, do the bananas move up. down. kg falling object has an initial downward velocity with magnitude vo- (b) For the case where Do up. (d) Discuss what your result says about v. (r) when lance between the monkey and the bananas decrease. increase. 5.106. A small rock moves in water, and the force exerted on it by or remain constant? (c) The 20 kg the water is given by Eq. (5.7). The terminal speed of the rock is monkey releases her hold on the measured and found to be 2.0 m/s. The rock is projected upward at rope. What happens to the dis- an initial speed of 6.0 m/s. You can ignore the buoyancy force on tance between the monkey and the rock. (a) In the absence of fluid resistance, how high will the