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If a loop of chain is spun at high speed, it can roll along the ground like a circular hoop without collapsing. Consider a chain of uniform linear mass density μ whose center of mass travels to the right at a high speed υ0 as shown in Figure P16.67. (a) Determine the tension in the chain in terms of μ and υ0. Assume the weight of an individual link is negligible compared to the tension, (b) If the loop rolls over a small bump, the resulting deformation of the chain causes two transverse pulses to propagate along the chain, one moving clockwise and one moving counterclockwise. What is the speed of the pulses traveling along the chain? (c) Through what angle does each pulse travel during the time interval over which the loop makes one revolution?
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Chapter 16 Solutions
Physics for Scientists and Engineers, Technology Update, Hybrid Edition (with Enhanced WebAssign Multi-Term LOE Printed Access Card for Physics)
- Review. A string is wound around a uniform disk of radius R and mass M. The disk is released from rest with the string vertical and its top end tied to a fixed bar (Fig. P10.78). Show that (a) the tension in the string is one third of the weight of the disk, (b) the magnitude of the acceleration of the center of mass is 2g/3, and (c) the speed of the center of mass is (4gh/3)1/2 after the disk has descended through distance h. (d) Verify your answer to part (c) using the energy approach. Figure P10.78arrow_forwardA uniform solid sphere of mass m and radius r is releasedfrom rest and rolls without slipping on a semicircular ramp ofradius R r (Fig. P13.76). Ifthe initial position of the sphereis at an angle to the vertical,what is its speed at the bottomof the ramp? FIGURE P13.76arrow_forwardFigure P10.16 shows the drive train of a bicycle that has wheels 67.3 cm in diameter and pedal cranks 17.5 cm long. The cyclist pedals at a steady cadence of 76.0 rev/min. The chain engages with a front sprocket 15.2 cm in diameter and a rear sprocket 7.00 cm in diameter. Calculate (a) the speed of a link of the chain relative to the bicycle frame, (b) the angular speed of the bicycle wheels, and (c) the speed of the bicycle relative to the road. (d) What pieces of data, if any, are not necessary for the calculations? Figure P10.16arrow_forward
- A ball of mass M = 5.00 kg and radius r = 5.00 cm isattached to one end of a thin,cylindrical rod of length L = 15.0 cm and mass m = 0.600 kg.The ball and rod, initially at restin a vertical position and freeto rotate around the axis shownin Figure P13.70, are nudgedinto motion. a. What is therotational kinetic energy of thesystem when the ball and rodreach a horizontal position? b. What is the angular speed of the ball and rod when they reach a horizontal position? c. What is the linear speed of the centerof mass of the ball when the ball and rod reach a horizontalposition? d. What is the ratio of the speed found in part (c) tothe speed of a ball that falls freely through the same distance? FIGURE P13.70arrow_forwardA nylon siring has mass 5.50 g and length L = 86.0 cm. The lower end is tied to the floor, and the upper end is tied to a small set of wheels through a slot in a track on which the wheels move (Fig. P18.76). The wheels have a mass that is negligible compared with that of the siring, and they roll without friction on the track so that the upper end of the string is essentially free. Figure P18.76 At equilibrium, the string is vertical and motionless. When it is carrying a small-amplilude wave, you may assume the string is always under uniform tension 1.30 N. (a) Find the speed of transverse waves on the siring, (b) The string's vibration possibilities are a set of standing-wave states, each with a node at the fixed bottom end and an antinode at the free top end. Find the node-antinode distances for each of the three simplest states, (c) Find the frequency of each of these states.arrow_forwardA smaller disk of radius r and mass m is attached rigidly to the face of a second larger disk of radius R and mass M as shown in Figure P15.48. The center of the small disk is located at the edge of the large disk. The large disk is mounted at its center on a frictionless axle. The assembly is rotated through a small angle from its equilibrium position and released. (a) Show that the speed of the center of the small disk as it passes through the equilibrium position is v=2[Rg(1cos)(M/m)+(r/R)2+2]1/2 (b) Show that the period of the motion is v=2[(M/2m)+R2+mr22mgR]1/2 Figure P15.48arrow_forward
- A square plate with sides 2.0 m in length can rotatearound an axle passingthrough its center of mass(CM) and perpendicular toits surface (Fig. P12.53). There are four forces acting on the plate at differentpoints. The rotational inertia of the plate is 24 kg m2. Use the values given in the figure to answer the following questions. a. Whatis the net torque acting onthe plate? b. What is theangular acceleration of the plate? FIGURE P12.53 Problems 53 and 54.arrow_forwardA square plate with sides of length 4.0 m can rotate about an axle passing through its center of mass and perpendicular to the plate as shown in Figure P14.36. There are four forces acting on the plate at different points. The rotational inertia of the plate is 24 kgm2. Is the plate in equilibrium? FIGURE P14.36arrow_forwardConsider a “round” rigid body with moment of inertia I = BMR2, where M is the body’s mass, R is the body’s radius, and B is a constant depending on the type of the body. The center of the “round” rigid body is attached to a spring of force constant k, and then the body is made to roll without slipping on a rough horizontal surface. Due to the spring, it is expected the body will oscillate by rolling back and forth from its resting position. A. Determine the angular frequency and the period for small oscillations of the round rigid body. Express your answer in terms of B.arrow_forward
- Two metal disks, one with radius R1 = 2.59 cm and mass M1 = 0.850 kg and the other with radius R2 = 5.08 cm and mass M2 = 1.61 kg , are welded together and mounted on a frictionless axis through their common center. (Figure 1). (A) A light string is wrapped around the edge of the smaller disk, and a 1.50-kg block, suspended from the free end of the string. If the block is released from rest at a distance of 2.10 m above the floor, what is its speed just before it strikes the floor? (B) Repeat the calculation of part (A), this time with the string wrapped around the edge of the larger disk.arrow_forwardA uniform solid disk of mass m = 2.98 kg and radius r = 0.200 m rotates about a fixed axis perpendicular to its face with angular frequency 6.07 rad/s. (a)Calculate the magnitude of the angular momentum of the disk when the axis of rotation passes through its center of mass. (b) What is the magnitude of the angular momentum when the axis of rotation passes through a point midway between the center and the rim?arrow_forwardA spool of wire of mass M and radius R is unwound under a constant force F (Fig. P10.85). Assuming the spool is a uniform solid cylinder that doesn’t slip, show that (a) The acceleration of the center of mass is 4F/3M and (b) The force of friction is to the right and equal in magnitude to F/3. (c) If the cylinder starts from rest and rolls without slipping, what is the speed of its center of mass after it has rolled through a distance d?arrow_forward
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