Additional Problems

| 67 | ssm The shock absorbers in the suspension system of a car are in such bad shape that they have no effect on the behavior of the springs attached to the axles. Each of the identical springs attached to the front axle supports 320 kg. A person pushes down on the middle of the front end of the car and notices that it vibrates through five cycles in 3.0 s. Find the spring constant of either spring.

| 68 | Refer to Conceptual Example 2 as an aid in solving this problem. A 100-coil spring has a spring constant of 420 N/m. It is cut into four shorter springs, each of which has 25 coils. One end of a 25-coil spring is attached to a wall. An object of mass 46 kg is attached to the other end of the spring, and the system is set into horizontal oscillation. What is the angular frequency of the motion?

| 69 | ssm Concept Simulation 10.2 at allows you to explore the effect of the acceleration due to gravity on pendulum motion, which is the focus of this problem. Astronauts on a distant planet set up a simple pendulum of length 1.2 m. The pendulum executes simple harmonic motion and makes 100 complete vibrations in 280 s. What is the acceleration due to gravity?

| 70 | Multiple-Concept Example 6 presents a model for solving this problem. As far as vertical oscillations are concerned, a certain automobile can be considered to be mounted on four identical springs, each having a spring constant of . Four identical passengers sit down inside the car, and it is set into a vertical oscillation that has a period of 0.370 s. If the mass of the empty car is 1560 kg, determine the mass of each passenger. Assume that the mass of the car and its passengers is distributed evenly over the springs.

| 71 | ssm www A 3500-kg statue is placed on top of a cylindrical concrete () stand. The stand has a crosssectional area of and a height of 1.8 m. By how much does the statue compress the stand?

| 72 | The graph shows the force Fx that an archer applies to the string of a long bow versus the string’s displacement x. Drawing back this bow is analogous to stretching a spring. From the data in the graph determine the effective spring constant of the bow.

| 73 | The pressure increases by for every meter of depth beneath the surface of the ocean. At what depth does the volume of a Pyrex glass cube, on an edge at the ocean’s surface, decrease by ?

* | 74 | Review Conceptual Example 8 before starting this problem. A block is attached to a horizontal spring and oscillates back and forth on a frictionless horizontal surface at a frequency of 3.00 Hz. The amplitude of the motion is . At the point where the block has its maximum speed, it suddenly splits into two identical parts, only one part remaining attached to the spring. (a) What is the amplitude and the frequency of the simple harmonic motion that exists after the block splits? (b) Repeat part (a), assuming that the block splits when it is at one of its extreme positions.

* | 75 | ssm Interactive LearningWare 10.1 at reviews the concepts involved in this problem. A spring stretches by 0.018 m when a 2.8-kg object is suspended from its end. How much mass should be attached to this spring so that its frequency of vibration is f = 3.0 Hz?

* | 76 | An 86.0-kg climber is scaling the vertical wall of a mountain. His safety rope is made of nylon that, when stretched, behaves like a spring with a spring constant of . He accidentally slips and falls freely for 0.750 m before the rope runs out of slack. How much is the rope stretched when it breaks his fall and momentarily brings him to rest?

* | 77 | Refer to Interactive Solution 10.77 at to review a method by which this problem can be solved. An 11.2-kg block and a 21.7-kg block are resting on a horizontal frictionless surface. Between the two is squeezed a spring (spring constant = 1330 N/m). The spring is compressed by 0.141 m from its unstrained length and is not attached permanently to either block. With what speed does each block move away after the mechanism keeping the spring squeezed is released and the spring falls away?

* | 78 | A gymnast does a one-arm handstand. The humerus, which is the upper arm bone between the elbow and the shoulder joint, may be approximated as a 0.30-m-long cylinder with an outer radius of and a hollow inner core with a radius of . Excluding the arm, the mass of the gymnast is 63 kg. (a) What is the compressional strain of the humerus? (b) By how much is the humerus compressed?

* | 79 | ssm The front spring of a car’s suspension system has a spring constant of and supports a mass of 215 kg. The wheel has a radius of 0.400 m. The car is traveling on a bumpy road, on which the distance between the bumps is equal to the circumference of the wheel. Due to resonance, the wheel starts to vibrate strongly when the car is traveling at a certain minimum linear speed. What is this speed?

** | 80 | Interactive LearningWare 10.2 at explores the approach taken in problems such as this one. A spring is mounted vertically on the floor. The mass of the spring is negligible. A certain object is placed on the spring to compress it. When the object is pushed further down by just a bit and then released, one up/down oscillation cycle occurs in 0.250 s. However, when the object is pushed down by to point P and then released, the object flies entirely off the spring. To what height above point P does the object rise in the absence of air resistance?

** | 81 | ssm www A solid brass sphere is subjected to a pressure of due to the earth’s atmosphere. On Venus the pressure due to the atmosphere is . By what fraction (including the algebraic sign) does the radius of the sphere change when it is exposed to the Venusian atmosphere? Assume that the change in radius is very small relative to the initial radius.

** | 82 | A copper rod (length = 2.0 m, ) hangs down from the ceiling. A 9.0-kg object is attached to the lower end of the rod. The rod acts as a “spring,” and the object oscillates vertically with a small amplitude. Ignoring the rod’s mass, find the frequency f of the simple harmonic motion.

** | 83 | A tray is moved horizontally back and forth in simple harmonic motion at a frequency of f = 2.00 Hz. On this tray is an empty cup. Obtain the coefficient of static friction between the tray and the cup, given that the cup begins slipping when the amplitude of the motion is .

** | 84 | A cylindrically shaped piece of collagen (a substance found in the body in connective tissue) is being stretched by a force that increases from 0 to . The length and radius of the collagen are, respectively, 2.5 and 0.091 cm, and Young’s modulus is . (a) If the stretching obeys Hooke’s law, what is the spring constant k for collagen? (b) How much work is done by the variable force that stretches the collagen? (See 6.9 for a discussion of the work done by a variable force.)

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