Solved: Objects of equal mass are oscillating up and down

When a tire pressure gauge is pressed against a tire valve, as in Figure 10.2, the air in the tire pushes against a plunger attached to a spring. Suppose the spring constant of the spring is k 320 N/m and the bar indicator of the gauge extends 2.0 cm when the gauge is pressed against the tire valve. What force does the air in the tire apply to the spring?

Figure 10.3a shows a 10-coil spring that has a spring constant k. When this spring is cut in half, so there are two 5-coil springs, is the spring constant of each of the shorter springs

A steel ball is dropped onto a very hard floor. Over and over again, the ball rebounds to its original height (assuming that no energy is lost during the collision with the floor). Is the motion of the ball simple harmonic motion?

The drawing shows identical springs that are attached to a box in two different ways. Initially, the springs are unstrained. The box is then pulled to the right and released. In each case the initial displacement of the box is the same. At the moment of release, which box, if either, experiences the greater net force due to the springs?

A 0.42-kg block is attached to the end of a horizontal ideal spring and rests on a frictionless surface. The block is pulled so that the spring stretches for 2.1 cm relative to its unstrained length. When the block is released, it moves with an acceleration of 9.0 m/s2 . What is the spring constant of the spring?

Two people pull on a horizontal spring that is attached to an immovable wall. Then, they detach it from the wall and pull on opposite ends of the horizontal spring. They pull just as hard in each case. In which situation, if either, does the spring stretch more?

The diaphragm of a loudspeaker moves back and forth in simple harmonic motion to create sound, as in Figure 10.11. The frequency of the motion is and the amplitude is (a) What is the maximum speed of the diaphragm? (b) Where in the motion does this maximum speed occur?

Over the entrance to a restaurant is mounted a strip of equally spaced light bulbs, as Figure 10.12a illustrates. Starting at the left end, each bulb turns on in sequence for one-half second. Thus, a lighted bulb appears to move from left to right. After the last bulb on the right turns on, the apparent motion reverses. The lighted bulb then appears to move to the left, as part b of the drawing indicates. As a result, the lighted bulb appears to oscillate back and forth. Is the apparent motion simple harmonic motion? (a) No (b) Yes

The loudspeaker diaphragm in Figure 10.11 is vibrating at a frequency of f 1.0 kHz, and the amplitude of the motion is A 0.20 mm. (a) What is the maximum acceleration of the diaphragm, and (b) where does this maximum acceleration occur? R

Astronauts who spend a long time in orbit measure their body masses as part of their health-maintenance programs. On earth, it is simple to measure body weight W with a scale and convert it to mass m using the magnitude g of the acceleration due to gravity, since W mg. However, this procedure does not work in orbit, because both the scale and the astronaut are in free fall and cannot press against each other (see Conceptual Example 12 in Chapter 5). Instead, astronauts use a body-mass measurement device, as Figure 10.14 illustrates. The device consists of a spring-mounted chair in which the astronaut sits. The chair is then started oscillating in simple harmonic motion. The period of the motion is measured electronically and is automatically converted into a value of the astronauts mass, after the mass of the chair is taken into account. The spring used in one such device has a spring constant of 606 N/m, and the mass of the chair is 12.0 kg. The measured oscillation period is 2.41 s. Find the mass of the astronaut.

The drawing shows plots of the displacement x versus the time t for three objects undergoing simple harmonic motion. Which objectI, II, or III has the greatest maximum velocity?

In Figure 10.13 the shadow moves in simple harmonic motion. Where on the path of the shadow is the acceleration equal to zero?

A particle is oscillating in simple harmonic motion. The time required for the particle to travel through one complete cycle is equal to the period of the motion, no matter what the amplitude is. But how can this be, since larger amplitudes mean that the particle travels farther?

Figure 10.17 shows an object of mass m 0.200 kg that is vibrating on a horizontal frictionless table. The spring has a spring constant of k 545 N/m. The spring is stretched initially to x0 4.50 cm and is then released from rest (see part A of the drawing). Determine the final translational speed vf of the object when the final displacement of the spring is (a) xf 2.25 cm and (b) xf 0 cm. Rea

Figure 10.18a shows a box of mass m attached to a spring that has a force constant k. The box rests on a horizontal, frictionless surface. The spring is initially stretched to x A and then released from rest. The box executes simple harmonic motion that is speed vx max, an amplitude A, and an angular frequency . When the box is passing through the point where the spring is unstrained (x 0 m), a second box of the same mass m and speed vx max is attached to it, as in part b of the drawing. Discuss what happens to (a) the maximum speed, (b) the amplitude, and (c) the angular frequency of the subsequent simple harmonic motion.characterized by a maximum C

A 0.20-kg ball is attached to a vertical spring, as in Figure 10.19. The spring constant of the spring is 28 N/m. The ball, supported initially so that the spring is neither stretched nor compressed, is released from rest. In the absence of air resistance, how far does the ball fall before being brought to a momentary stop by the spring?

Is more elastic potential energy stored in a spring when the spring is compressed by one centimeter than when it is stretched by the same amount?

A block is attached to the end of a horizontal ideal spring and rests on a frictionless surface. The block is pulled so that the spring stretches relative to its unstrained length. In each of the following three cases, the spring is stretched initially by the same amount. Rank the amplitudes of the resulting simple harmonic motion in decreasing order (largest first). (a) The block is released from rest. (b) The block is given an initial speed v0. (c) The block is given an initial speed

A block is attached to a horizontal spring and slides back and forth on a frictionless horizontal surface. A second identical block is suddenly attached to the first block. The attachment is accomplished by joining the blocks at one extreme end of the oscillation cycle. The velocities of the blocks are exactly matched at the instant of joining. How do the (a) amplitude, (b) frequency, and (c) maximum speed of the simple harmonic motion change?

Figure 10.21 shows a clock that uses a pendulum to keep time. Determine the length of a simple pendulum that will swing back and forth in simple harmonic motion with a period of 1.00 s.

When we walk, our legs alternately swing forward about the hip joint as a pivot. In this motion the leg is acting approximately as a physical pendulum. Treating the leg as a thin uniform rod of length 0.80 m, find the time it takes for the leg to swing forward.

Suppose that a grandfather clock (a simple pendulum) is running slowly; that is, the time it takes to complete each cycle is greater than it should be. Should you (a) shorten or (b) lengthen the pendulum to make the clock keep correct time?

Consult Concept Simulation 10.2 at www.wiley.com/college/cutnell to review the concept that is important here. In principle, the motions of a simple pendulum and an object on an ideal spring can both be used to provide the basic time interval or period used in a clock. Which of the two kinds of clocks becomes more inaccurate when carried to the top of a high mountain?

Concept Simulation 10.2 at www.wiley.com/college/cutnell deals with the concept on which this question is based. Suppose you were kidnapped and held prisoner by space invaders in a completely isolated room, with nothing but a watch and a pair of shoes (including shoe laces of known length). How could you determine whether you were on earth or on the moon?

Two people are sitting on identical playground swings. One is pulled back 4 from the vertical and the other is pulled back 8 . They are both released at the same instant. Will they both come back to their starting points at the same time? Assume simple-pendulum motion.

The shock absorbers on a car are badly in need of replacement and introduce very little damping. Does the number of occupants in the car affect the vibration frequency of the cars suspension system?

A car travels at a constant speed over a road that contains a series of equally spaced bumps. The spacing between bumps is d. The mass of the car is m, and the spring constant of the cars suspension springs is k. Because of resonance, a particularly jarring ride results. Ignoring the effect of the cars shock absorbers, derive an expression for the cars speed v in terms of d, m, and k, as well as some numerical constants.

A circus performer supports the combined weight (1080 N) of a number of colleagues (see Figure 10.28). Each thighbone (femur) of this performer has a length of 0.55 m and an effective cross-sectional area of 7.7  104 m2 . Determine the amount by which each thighbone compresses under the extra weight.

A helicopter is using a steel cable to lift a 2100-kg jeep. The unstretched length of the cable is 16 m, and its radius is 5.0  103 m. By what amount does the cable stretch when the jeep is hoisted straight upward with an acceleration of 1.5 m/s2 ?

A block of Jell-O is resting on a plate. Figure 10.31a gives the dimensions of the block. You are bored, impatiently waiting for dinner, and push tangentially across the top surface with a force of F 0.45 N, as in part b of the drawing. The top surface moves a distance X 6.0  103 m relative to the bottom surface. Use this idle gesture to measure the shear modulus of Jell-O. Re

Youngs modulus for steel is greater than that for a particular unknown material. What does this mean about how these materials compress when used in construction? (a) Steel compresses much more easily than the unknown material does. (b) The unknown material compresses more easily than steel does. (c) Youngs modulus has nothing to do with compression, so not enough information is given for an answer.

Two rods are made from the same material. One has a circular cross section, and the other has a square cross section. The circle just fits within the square. When the same force is applied to stretch these rods, they each stretch by the same amount. Which rod, if either, is longer?

A trash compactor crushes empty aluminum cans, thereby reducing the total volume, so that V/V0 0.75 in Equation 10.20. Can the value given in Table 10.3 for the bulk modulus of aluminum be used to calculate the change P in pressure generated in the trash compactor?

Both sides of the relation F S(X/L0)A (Equation 10.18) can be divided by the area A to give F/A on the left side. Can this F/A term be called a pressure, such as the pressure that appears in P B(V/V0) (Equation 10.20)? 2

The block in the drawing rests on the ground. Which face A, B, or Cexperiences the largest stress and which face experiences the smallest stress when the block is resting on it?

A 75-kg diver is standing at the end of a diving board while it is vibrating up and down in simple harmonic motion, as indicated in Figure 10.34. The diving board has an effective spring constant of k 4100 N/m, and the vertical distance between the highest and lowest points in the motion is 0.30 m. (a) What is the amplitude of the motion? (b) Starting when the diver is at the highest point, what is his speed one-quarter of a period later? (c) If the vertical distance between his highest and lowest points were doubled to 0.60 m, what would be the time required for the diver to make one complete motional cycle?

How is the amplitude A related to the vertical distance between the highest and lowest points of the divers motion?

Starting from the top, where is the diver located one-quarter of a period later, and what can be said about his speed at this point?

If the amplitude of the motion were to double, would the period also double?

A 68.0-kg bungee jumper is standing on a tall platform (h0 46.0 m), as indicated in Figure 10.35. The bungee cord has an unstrained length of L0 9.00 m and, when stretched, behaves like an ideal spring with a spring constant of k 66.0 N/m. The jumper falls from rest, and it is assumed that the only forces acting on him are his weight and, for the latter part of the descent, the elastic force of the bungee cord. What is his speed (it is not zero) when he is at the following heights above the water (see the drawing): (a) hA 37.0 m and (b) hB 15.0 m? Con

Can we use the conservation of mechanical energy to find his speed at any point during the descent?

What types of energy does he have when he is standing on the platform?

What types of energy does he have at point A?

What types of energy does he have at point B?

Which one of the following graphs correctly represents the restoring force F of an ideal spring as a function of the displacement x of the spring from its unstrained length?

You have two springs. One has a greater spring constant than the other. You also have two objects, one with a greater mass than the other. Which object should be attached to which spring, so that the resulting springobject system has the greatest possible period of oscillation? (a) The object with the greater mass should be attached to the spring with the greater spring constant. (b) The object with the greater mass should be attached to the spring with the smaller spring constant. (c) The object with the smaller mass should be attached to the spring with the smaller spring constant. (d) The object with the smaller mass should be attached to the spring with the greater spring constant.

An object is oscillating in simple harmonic motion with an amplitude A and an angular frequency . What should you do to increase the maximum speed of the motion? (a) Reduce both A and by 10%. + (b) Increase A by 10% and reduce by 10%. (c) Reduce A by 10% and increase by 10%. (d) Increase both A and by 10%. Sec

The kinetic energy of an object attached to a horizontal ideal spring is denoted by KE and the elastic potential energy by PE. For the simple harmonic motion of this object the maximum kinetic energy and the maximum elastic potential energy during an oscillation cycle are KEmax and PEmax, respectively. In the absence of friction, air resistance, and any other nonconservative forces, which of the following equations applies to the objectspring system? A. KE  PE constant B. KEmax PEmax (a) A, but not B (b) B, but not A (c) A and B (d) Neither A nor B 1

A block is attached to a horizontal spring. On top of this block rests another block. The two-block system slides back and forth in simple harmonic motion on a frictionless horizontal surface. At one extreme end of the oscillation cycle, where the blocks come to a momentary halt before reversing the direction of their motion, the top block is suddenly lifted vertically upward, without changing the zero velocity of the bottom block. The simple harmonic motion then continues. What happens to the amplitude and the angular frequency of the ensuing motion? (a) The amplitude remains the same, and the angular frequency increases. (b) The amplitude increases, and the angular frequency remains the same. (c) Both the amplitude and the angular frequency remain the same. (d) Both the amplitude and the angular frequency decrease. (e) Both the amplitude and the angular frequency increase.

Five simple pendulums are shown in the drawings. The lengths of the pendulums are drawn to scale, and the masses are either m or 2m, as shown. Which pendulum has the smallest angular frequency of oscillation? (a) A (b) B (c) C (d) D (e) E

An object on a spring is oscillating in simple harmonic motion. Suddenly, friction appears and causes the energy of the system to be dissipated. The system now exhibits ______. (a) driven harmonic motion (b) Hookes-law type of motion (c) damped harmonic motion

An external force (in addition to the spring force) is continually applied to an object of mass m attached to a spring that has a spring constant k. The frequency of this external force is such that resonance occurs. Then the frequency of this external force is doubled, and the force is applied to one of the spring systems shown in the drawing. With which system would resonance occur? (a) A (b) B (c) C (d) D (e) E

Drawings A and B show two cylinders that are identical in all respects, except that one is hollow. Identical forces are applied to each cylinder in order to stretch them. Which cylinder, if either, stretches more? (a) A and B both stretch by the same amount. (b) A stretches more than B. (c) B stretches more than A. (d) Insufficient information is given for an answer.

A material has a shear modulus of 5.0 109 N/m2 . A shear stress of 8.5 106 N/m2 is applied to a piece of the material. What is the resulting shear strain?

A hand exerciser utilizes a coiled spring. A force of 89.0 N is required to compress the spring by 0.0191 m. Determine the force needed to compress the spring by 0.0508 m.

The drawing shows three identical springs hanging from the ceiling. Nothing is attached to the first spring, whereas a 4.50-N block hangs from the second spring. A block of unknown weight hangs from the third spring. From the drawing, determine (a) the spring constant (in N/m) and (b) the weight of the block hanging from the third spring.

In a room that is 2.44 m high, a spring (unstrained length 0.30 m) hangs from the ceiling. A board whose length is 1.98 m is attached to the free end of the spring. The board hangs straight down, so that its 1.98-m length is perpendicular to the floor. The weight of the board (104 N) stretches the spring so that the lower end of the board just extends to, but does not touch, the floor. What is the spring constant of the spring?

A spring lies on a horizontal table, and the left end of the spring is attached to a wall. The other end is connected to a box. The box is pulled to the right, stretching the spring. Static friction exists between the box and the table, so when the spring is stretched only by a small amount and the box is released, the box does not move. The mass of the box is 0.80 kg, and the spring has a spring constant of 59 N/m. The coefficient of static friction between the box and the table on which it rests is s 0.74. How far can the spring be stretched from its unstrained position without the box moving when it is released? 4

A person who weighs 670 N steps onto a spring scale in the bathroom, and the spring compresses by 0.79 cm. (a) What is the spring constant? (b) What is the weight of another person who compresses the spring by 0.34 cm?

A spring (k 830 N/m) is hanging from the ceiling of an elevator, and a 5.0-kg object is attached to the lower end. By how much does the spring stretch (relative to its unstrained length) when the elevator is accelerating upward at a 0.60 m/s2 ?

A 0.70-kg block is hung from and stretches a spring that is attached to the ceiling. A second block is attached to the first one, and the amount that the spring stretches from its unstrained length triples. What is the mass of the second block?

A uniform 1.4-kg rod that is 0.75 m long is suspended at rest from the ceiling by two springs, one at each end of the rod. Both springs hang straight down from the ceiling. The springs have identical lengths when they are unstretched. Their spring constants are 59 N/m and 33 N/m. Find the angle that the rod makes with the horizontal.

In 0.750 s, a 7.00-kg block is pulled through a distance of 4.00 m on a frictionless horizontal surface, starting from rest. The block has a constant acceleration and is pulled by means of a horizontal spring that is attached to the block. The spring constant of the spring is 415 N/m. By how much does the spring stretch?

Review Conceptual Example 2 as an aid in solving this problem. An object is attached to the lower end of a 100-coil spring that is hanging from the ceiling. The spring stretches by 0.160 m. The spring is then cut into two identical springs of 50 coils each. As the drawing shows, each spring is attached between the ceiling and the object. By how much does each spring stretch?

A small ball is attached to one end of a spring that has an unstrained length of 0.200 m. The spring is held by the other end, and the ball is whirled around in a horizontal circle at a speed of 3.00 m/s. The spring remains nearly parallel to the ground during the motion and is observed to stretch by 0.010 m. By how much would the spring stretch if it were attached to the ceiling and the ball allowed to hang straight down, motionless?

To measure the static friction coefficient between a 1.6-kg block and a vertical wall, the setup shown in the drawing is used. A spring (spring constant 510 N/m) is attached to the block. Someone pushes on the end of the spring in a direction perpendicular to the wall until the block does not slip downward. The spring is compressed by 0.039 m. What is the coefficient of static friction?

A 30.0-kg block is resting on a flat horizontal table. On top of this block is resting a 15.0-kg block, to which a horizontal spring is attached, as the drawing illustrates. The spring constant of the spring is 325 N/m. The coefficient of kinetic friction between the lower block and the table is 0.600, and the coefficient of static friction between the two blocks is 0.900. A horizontal force is applied to the lower block as shown. This force is increasing in such a way as to keep the blocks moving at a constant speed. At the point where the upper block begins to slip on the lower block, determine (a) the amount by which the spring is compressed and (b) the magnitude of the force . F

A 15.0-kg block rests on a horizontal table and is attached to one end of a massless, horizontal spring. By pulling horizontally on the other end of the spring, someone causes the block to accelerate uniformly and reach a speed of 5.00 m/s in 0.500 s. In the process, the spring is stretched by 0.200 m. The block is then pulled at a constant speed of 5.00 m/s, during which time the spring is stretched by only 0.0500 m. Find (a) the spring constant of the spring and (b) the coefficient of kinetic friction between the block and the table.

When responding to sound, the human eardrum vibrates about its equilibrium position. Suppose an eardrum is vibrating with an amplitude of 6.3  107 m and a maximum speed of 2.9  103 m/s. (a) What is the frequency (in Hz) of the eardrums vibration? (b) What is the maximum acceleration of the eardrum?

The fan blades on a jet engine make one thousand revolutions in a time of 50.0 ms. Determine (a) the period (in seconds) and (b) the frequency (in Hz) of the rotational motion. (c) What is the angular frequency of the blades?

A block of mass m 0.750 kg is fastened to an unstrained horizontal spring whose spring constant is k 82.0 N/m. The block is given a displacement of 0.120 m, where the  sign indicates that the displacement is along the x axis, and then released from rest. (a) What is the force (magnitude and direction) that the spring exerts on the block just before the block is released? (b) Find the angular frequency of the resulting oscillatory motion. (c) What is the maximum speed of the block? (d) Determine the magnitude of the maximum acceleration of the block. 18

An 0.80-kg object is attached to one end of a spring, as in Figure 10.5, and the system is set into simple harmonic motion. The displacement x of the object as a function of time is shown in the drawing. With the aid of these data, determine (a) the amplitude A of the motion, (b) the angular frequency , (c) the spring constant k, (d) the speed of the object at t 1.0 s, and (e) the magnitude of the objects acceleration at t 1.0 s. 1

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?

Objects of equal mass are oscillating up and down in simple harmonic motion on two different vertical springs. The spring constant of spring 1 is 174 N/m. The motion of the object on spring 1 has twice the amplitude as the motion of the object on spring 2. The magnitude of the maximum velocity is the same in each case. Find the spring constant of spring 2.

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?

An object attached to a horizontal spring is oscillating back and forth along a frictionless surface. The maximum speed of the object is 1.25 m/s, and its maximum acceleration is 6.89 m/s2 . How much time elapses between an instant when the objects speed is at a maximum and the next instant when its acceleration is at a maximum?

A vertical spring (spring constant 112 N/m) is mounted on the floor. A 0.400-kg block is placed on top of the spring and pushed down to start it oscillating in simple harmonic motion. The block is not attached to the spring. (a) Obtain the frequency (in Hz) of the motion. (b) Determine the amplitude at which the block will lose contact with the spring.

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 5.00  102 m. S

A pen contains a spring with a spring constant of 250 N/m. When the tip of the pen is in its retracted position, the spring is compressed 5.0 mm from its unstrained length. In order to push the tip out and lock it into its writing position, the spring must be compressed an additional 6.0 mm. How much work is done by the spring force to ready the pen for writing? Be sure to include the proper algebraic sign with your answer.

The drawing shows three situations in which a block is attached to a spring. The position labeled 0 m represents the unstrained position of the spring. The block is moved from an initial position x0 to a final position xf, the magnitude of the displacement being denoted by the symbol s. Suppose the spring has a spring constant of k 46.0 N/m. Using the data provided in the drawing, determine the total work done by the restoring force of the spring for each situation.

A spring is hung from the ceiling. A 0.450-kg block is then attached to the free end of the spring. When released from rest, the block drops 0.150 m before momentarily coming to rest, after which it moves back upward. (a) What is the spring constant of the spring? (b) Find the angular frequency of the blocks vibrations.

A 3.2-kg block is hanging stationary from the end of a vertical spring that is attached to the ceiling. The elastic potential energy of this spring-block system is 1.8 J. What is the elastic potential energy of the system when the 3.2-kg block is replaced by a 5.0-kg block?

A vertical spring with a spring constant of 450 N/m is mounted on the floor. From directly above the spring, which is unstrained, a 0.30-kg block is dropped from rest. It collides with and sticks to the spring, which is compressed by 2.5 cm in bringing the block to a momentary halt. Assuming air resistance is negligible, from what height (in cm) above the compressed spring was the block dropped?

In preparation for shooting a ball in a pinball machine, a spring (k 675 N/m) is compressed by 0.0650 m relative to its unstrained length. The ball (m 0.0585 kg) is at rest against the spring at point A. When the spring is released, the ball slides (without rolling). It leaves the spring and arrives at point B, which is 0.300 m higher than point A. Ignore friction, and find the balls speed at point B. 3

A heavy-duty stapling gun uses a 0.140-kg metal rod that rams against the staple to eject it. The rod is attached to and pushed by a stiff spring called a ram spring (k 32 000 N/m). The mass of this spring may be ignored. The ram spring is compressed by 3.0  102 m from its unstrained length and then released from rest. Assuming that the ram spring is oriented vertically and is still compressed by 0.8  102 m when the downward-moving ram hits the staple, find the speed of the ram at the instant of contact. 3

A rifle fires a 2.10  102 -kg pellet straight upward, because the pellet rests on a compressed spring that is released when the trigger is pulled. The spring has a negligible mass and is compressed by 9.10  102 m from its unstrained length. The pellet rises to a maximum height of 6.10 m above its position on the compressed spring. Ignoring air resistance, determine the spring constant. 3

A 1.00  102 -kg block is resting on a horizontal frictionless surface and is attached to a horizontal spring whose spring constant is 124 N/m. The block is shoved parallel to the spring axis and is given an initial speed of 8.00 m/s, while the spring is initially unstrained. What is the amplitude of the resulting simple harmonic motion?

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 1.20  103 N/m. 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?

A horizontal spring is lying on a frictionless surface. One end of the spring is attached to a wall, and the other end is connected to a movable object. The spring and object are compressed by 0.065 m, released from rest, and subsequently oscillate back and forth with an angular frequency of 11.3 rad/s. What is the speed of the object at the instant when the spring is stretched by 0.048 m relative to its unstrained length?

A spring is resting vertically on a table. A small box is dropped onto the top of the spring and compresses it. Suppose the spring has a spring constant of 450 N/m and the box has a mass of 1.5 kg. The speed of the box just before it makes contact with the spring is 0.49 m/s. (a) Determine the magnitude of the springs displacement at an instant when the acceleration of the box is zero. (b) What is the magnitude of the springs displacement when the spring is fully compressed?

A spring is compressed by 0.0620 m and is used to launch an object horizontally with a speed of 1.50 m/s. If the object were attached to the spring, at what angular frequency (in rad/s) would it oscillate?

A 0.60-kg metal sphere oscillates at the end of a vertical spring. As the spring stretches from 0.12 to 0.23 m (relative to its unstrained length), the speed of the sphere decreases from 5.70 to 4.80 m/s. What is the spring constant of the spring?

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 5.08  102 m. 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 are 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.

A 1.1-kg object is suspended from a vertical spring whose spring constant is 120 N/m. (a) Find the amount by which the spring is stretched from its unstrained length. (b) The object is pulled straight down by an additional distance of 0.20 m and released from rest. Find the speed with which the object passes through its original position on the way up

A 1.00  102 -kg bullet is fired horizontally into a 2.50-kg wooden block attached to one end of a massless horizontal spring (k 845 N/m). The other end of the spring is fixed in place, and the spring is unstrained initially. The block rests on a horizontal, frictionless surface. The bullet strikes the block perpendicularly and quickly comes to a halt within it. As a result of this completely inelastic collision, the spring is compressed along its axis and causes the block/bullet to oscillate with an amplitude of 0.200 m. What is the speed of the bullet?

A simple pendulum is made from a 0.65-m-long string and a small ball attached to its free end. The ball is pulled to one side through a small angle and then released from rest. After the ball is released, how much time elapses before it attains its greatest speed?

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 magnitude of the acceleration due to gravity on this planet?

The length of a simple pendulum is 0.79 m and the mass of the particle (the bob) at the end of the cable is 0.24 kg. The pendulum is pulled away from its equilibrium position by an angle of 8.50 and released from rest. Assume that friction can be neglected and that the resulting oscillatory motion is simple harmonic motion. (a) What is the angular frequency of the motion? (b) Using the position of the bob at its lowest point as the reference level, determine the total mechanical energy of the pendulum as it swings back and forth. (c) What is the bobs speed as it passes through the lowest point of the swing?

A spiral staircase winds up to the top of a tower in an old castle. To measure the height of the tower, a rope is attached to the top of the tower and hung down the center of the staircase. However, nothing is available with which to measure the length of the rope. Therefore, at the bottom of the rope a small object is attached so as to form a simple pendulum that just clears the floor. The period of the pendulum is measured to be 9.2 s. What is the height of the tower?

Two physical pendulums (not simple pendulums) are made from meter sticks that are suspended from the ceiling at one end. The sticks are uniform and are identical in all respects, except that one is made of wood (mass 0.17 kg) and the other of metal (mass 0.85 kg). They are set into oscillation and execute simple harmonic motion. Determine the period of (a) the wood pendulum and (b) the metal pendulum. *

Multiple-Concept Example 11 explores the concepts that are important in this problem. Pendulum A is a physical pendulum made from a thin, rigid, and uniform rod whose length is d. One end of this rod is attached to the ceiling by a frictionless hinge, so the rod is free to swing back and forth. Pendulum B is a simple pendulum whose length is also d. Obtain the ratio TA/TB of their periods for small-angle oscillations.

Multiple-Concept Example 11 provides some pertinent background for this problem. A pendulum is constructed from a thin, rigid, and uniform rod with a small sphere attached to the end opposite the pivot. This arrangement is a good approximation to a simple pendulum (period 0.66 s), because the mass of the sphere (lead) is much greater than the mass of the rod (aluminum). When the sphere is removed, the pendulum is no longer a simple pendulum, but is then a physical pendulum. What is the period of the physical pendulum?

A small object oscillates back and forth at the bottom of a frictionless hemispherical bowl, as the drawing illustrates. The radius of the bowl is R, and the angle  is small enough that the object oscillates in simple harmonic motion. Derive an expression for the angular frequency of the motion. Express your answer in terms of R and g, the magnitude of the acceleration due to gravity.

A tow truck is pulling a car out of a ditch by means of a steel cable that is 9.1 m long and has a radius of 0.50 cm. When the car just begins to move, the tension in the cable is 890 N. How much has the cable stretched?

Two stretched cables both experience the same stress. The first cable has a radius of 3.5  103 m and is subject to a stretching force of 270 N. The radius of the second cable is 5.1  103 m. Determine the stretching force acting on the second cable.

The pressure increases by 1.0  104 N/m2 for every meter of depth beneath the surface of the ocean. At what depth does the volume of a Pyrex glass cube, 1.0  102 m on an edge at the oceans surface, decrease by 1.0  1010 m3 ?

When subjected to a force of compression, the length of a bone decreases by 2.7  105 m. When this same bone is subjected to a tensile force of the same magnitude, by how much does it stretch?

Multiple-Concept Example 13 presents a model for solving this type of problem. A 59-kg water skier is being pulled by a nylon tow rope that is attached to a boat. The unstretched length of the rope is 12 m, and its cross-sectional area is 2.0  105 m2 . As the skier moves, a resistive force (due to the water) of magnitude 130 N acts on her; this force is directed opposite to her motion. What is the change in the length of the rope when the skier has an acceleration whose magnitude is 0.85 m/s2 ?

A solid steel cylinder is standing (on one of its ends) vertically on the floor. The length of the cylinder is 3.6 m and its radius is 65 cm. When an object is placed on top of the cylinder, the cylinder compresses by an amount of 5.7  107 m. What is the weight of the object?

The drawing shows a 160-kg crate hanging from the end of a steel bar. The length of the bar is 0.10 m, and its cross-sectional area is 3.2  104 m2 . Neglect the weight of the bar itself and determine (a) the shear stress on the bar and (b) the vertical deflection Y of the right end of the bar.

A copper cube, 0.30 m on a side, is subjected to two shearing forces, each of which has a magnitude F 6.0  106 N (see the drawing). Find the angle  (in degrees), which is one measure of how the shape of the block has been altered by shear deformation.

Two metal beams are joined together by four rivets, as the drawing indicates. Each rivet has a radius of 5.0  103 m and is to be exposed to a shearing stress of no more than 5.0  108 Pa. What is the maximum tension that can be applied to each beam, assuming that each rivet carries one-fourth of the total load?

A copper cylinder and a brass cylinder are stacked end to end, as in the drawing. Each cylinder has a radius of 0.25 cm. A compressive force of F 6500 N is applied to the right end of the brass cylinder. Find the amount by which the length of the stack decreases.

A piece of aluminum is surrounded by air at a pressure of 1.01  105 Pa. The aluminum is placed in a vacuum chamber where the pressure is reduced to zero. Determine the fractional change V/V0 in the volume of the aluminum.

One end of a piano wire is wrapped around a cylindrical tuning peg and the other end is fixed in place. The tuning peg is turned so as to stretch the wire. The piano wire is made from steel (Y 2.0  1011 N/m2 ). It has a radius of 0.80 mm and an unstrained length of 0.76 m. The radius of the tuning peg is 1.8 mm. Initially, there is no tension in the wire, but when the tuning peg is turned, tension develops. Find the tension in the wire when the tuning peg is turned through two revolutions. Ignore the radius of the wire compared to the radius of the tuning peg.

A die is designed to punch holes with a radius of 1.00  102 m in a metal sheet that is 3.0  103 m thick, as the drawing illustrates. To punch through the sheet, the die must exert a shearing stress of 3.5  108 Pa. What force must be applied to the die?

A piece of mohair taken from an Angora goat has a radius of 31  106 m. What is the least number of identical pieces of mohair needed to suspend a 75-kg person, so the strain experienced by each piece is less than 0.010? Assume that the tension is the same in all the pieces.

Two rods are identical in all respects except one: one rod is made from aluminum and the other from tungsten. The rods are joined end to end, in order to make a single rod that is twice as long as either the aluminum or tungsten rod. What is the effective value of Youngs modulus for this composite rod? That is, what value YComposite of Youngs modulus should be used in Equation 10.17 when applied to the composite rod? Note that the change L Composite in the length of the composite rod is the sum of the changes in length of the aluminum and tungsten rods.

A square plate is 1.0  102 m thick, measures 3.0  102 m on a side, and has a mass of 7.2  102 kg. The shear modulus of the material is 2.0  1010 N/m2 . One of the square faces rests on a flat horizontal surface, and the coefficient of static friction between the plate and the surface is 0.91. A force is applied to the top of the plate, as in Figure 10.30a. Determine (a) the maximum possible amount of shear stress, (b) the maximum possible amount of shear strain, and (c) the maximum possible amount of shear deformation X (see Figure 10.30b) that can be created by the applied force just before the plate begins to move.

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 1.00  102 m and a hollow inner core with a radius of 4.0  103 m. 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?

Depending on how you fall, you can break a bone easily. The severity of the break depends on how much energy the bone absorbs in the accident, and to evaluate this let us treat the bone as an ideal spring. The maximum applied force of compression that one mans thighbone can endure without breaking is 7.0  104 N. The minimum effective cross-sectional area of the bone is 4.0  104 m2 , its length is 0.55 m, and Youngs modulus is Y 9.4  109 N/m2 . The mass of the man is 65 kg. He falls straight down without rotating, strikes the ground stiff-legged on one foot, and comes to a halt without rotating. To see that it is easy to break a thighbone when falling in this fashion, find the maximum distance through which his center of gravity can fall without his breaking a bone.

A 1.0  103 -kg spider is hanging vertically by a thread that has a Youngs modulus of 4.5  109 N/m2 and a radius of 13  106 m. Suppose that a 95-kg person is hanging vertically on an aluminum wire. What is the radius of the wire that would exhibit the same strain as the spiders thread, when the thread is stressed by the full weight of the spider?

The dimensions of a rectangular block of brass are 0.010 m, 0.020 m, and 0.040 m. The block is to be glued to a table and subjected to a horizontal force of 770 N, as in Figure 10.30. Note that there are three possibilities for the surface of the block that is in contact with the table. What is the maximum possible distance the top surface can move, relative to the bottom surface?

Consult Multiple-Concept Example 13 to review a model for solving this type of problem. A 61-kg snow skier is being pulled up a 12 slope by a steel cable. The cable has a cross-sectional area of 7.8  105 m2 . The cable applies a force to the skier, and, in doing so, the cable stretches by 2.0  104 m. A frictional force of magnitude 68 N acts on the skis and is directed opposite to the skiers motion. If the skiers acceleration up the slope has a magnitude of 1.1 m/s2 , what is the original (unstretched) length of the cable?

A 6.8-kg bowling ball is attached to the end of a nylon cord with a crosssectional area of 3.4  105 m2 . The other end of the cord is fixed to the ceiling. When the bowling ball is pulled to one side and released from rest,it swings downward in a circular arc. At the instant it reaches its lowest point, the bowling ball is 1.4 m lower than the point from which it was released, and the cord is stretched 2.7  103 m from its unstrained length. What is the unstrained length of the cord? Hint: When calculating any quantity other than the strain, ignore the increase in the length of the cord.

A solid brass sphere is subjected to a pressure of 1.0  105 Pa due to the earths atmosphere. On Venus the pressure due to the atmosphere is 9.0  106 Pa. By what fraction r/r0 (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.

A loudspeaker diaphragm is producing a sound for 2.5 s by moving back and forth in simple harmonic motion. The angular frequency of the motion is 7.54  104 rad/s. How many times does the diaphragm move back and forth?

A person bounces up and down on a trampoline, while always staying in contact with it. The motion is simple harmonic motion, and it takes 1.90 s to complete one cycle. The height of each bounce above the equilibrium position is 45.0 cm. Determine (a) the amplitude and (b) the angular frequency of the motion. (c) What is the maximum speed attained by the person?

A simple pendulum is swinging back and forth through a small angle, its motion repeating every 1.25 s. How much longer should the pendulum be made in order to increase its period by 0.20 s?

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 1.30  105 N/m. 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.

The femur is a bone in the leg whose minimum cross-sectional area is about 4.0  104 m2 . A compressional force in excess of 6.8  104 N will fracture this bone. (a) Find the maximum stress that this bone can withstand. (b) What is the strain that exists under a maximum-stress condition?

An archer, about to shoot an arrow, is applying a force of 240 N to a drawn bowstring. The bow behaves like an ideal spring whose spring constant is 480 N/m. What is the displacement of the bowstring?

Between each pair of vertebrae in the spinal column is a cylindrical disc of cartilage. Typically, this disc has a radius of about 3.0  102 m and a thickness of about 7.0  103 m. The shear modulus of cartilage is 1.2  107 N/m2 . Suppose that a shearing force of magnitude 11 N is applied parallel to the top surface of the disc while the bottom surface remains fixed in place. How far does the top surface move relative to the bottom surface?

A block rests on a frictionless horizontal surface and is attached to a spring. When set into simple harmonic motion, the block oscillates back and forth with an angular frequency of 7.0 rad/s. The drawing indicates the position of the block when the spring is unstrained. This position is labeled x 0 m. The drawing also shows a small bottle located 0.080 m to the right of this position. The block is pulled to the right, stretching the spring by 0.050 m, and is then thrown to the left. In order for the block to knock over the bottle, it must be thrown with a speed exceeding v0. Ignoring the width of the block, find v0.

A vertical ideal spring is mounted on the floor and has a spring constant of 170 N/m. A 0.64-kg block is placed on the spring in two different ways. (a) In one case, the block is placed on the spring and not released until it rests stationary on the spring in its equilibrium position. Determine the amount (magnitude only) by which the spring is compressed. (b) In a second situation, the block is released from rest immediately after being placed on the spring and falls downward until it comes to a momentary halt. Determine the amount (magnitude only) by which the spring is now compressed.

Multiple-Concept Example 6 reviews the principles that play roles in this problem. A bungee jumper, whose mass is 82 kg, jumps from a tall platform. After reaching his lowest point, he continues to oscillate up and down, reaching the low point two more times in 9.6 s. Ignoring air resistance and assuming that the bungee cord is an ideal spring, determine its spring constant.

Using the data given in Concepts & Calculations Example 16, determine how far the bungee jumper is from the water when he reaches the lowest point in his fall.

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 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?

When an object of mass m1 is hung on a vertical spring and set into vertical simple harmonic motion, it oscillates at a frequency of 12.0 Hz. When another object of mass m2 is hung on the spring along with the first object, the frequency of the motion is 4.00 Hz. Find the ratio m2/m1 of the masses.

An 8.0-kg stone at the end of a steel wire is being whirled in a circle at a constant tangential speed of 12 m/s. The stone is moving on the surface of a frictionless horizontal table. The wire is 4.0 m long and has a radius of 1.0  103 m. Find the strain in the wire.

A 0.200-m uniform bar has a mass of 0.750 kg and is released from rest in the vertical position, as the drawing indicates. The spring is initially unstrained and has a spring constant of k 25.0 N/m. Find the tangential speed with which end A strikes the horizontal surface.

The drawing shows two crates that are connected by a steel wire that passes over a pulley. The unstretched length of the wire is 1.5 m, and its cross-sectional area is 1.3  105 m2 . The pulley is frictionless and massless. When the crates are accelerating, determine the change in length of the wire. Ignore the mass of the wire.

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 3.0  102 N. The length and radius of the collagen are, respectively, 2.5 and 0.091 cm, and Youngs modulus is 3.1  106 N/m2 . (a) If the stretching obeys Hookes law, what is the spring constant k for collagen? (b) How much work is done by the variable force that stretches the collagen? (See Section 6.9 for a discussion of the work done by a variable force.)

The drawing shows a top view of a frictionless horizontal surface, where there are two springs with particles of mass m1 and m2 attached to them. Each spring has a spring constant of 120 N/m. The particles are pulled to the right and then released from the positions shown in the drawing. How much time passes before the particles are side by side for the first time at x 0 m if (a) m1 m2 3.0 kg and (b) m1 3.0 kg and m2 27 kg? ** 9

A copper rod (length 2.0 m, radius 3.0  103 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 rods mass, find the frequency f of the simple harmonic motion. 3

A 70.0-kg circus performer is fired from a cannon that is elevated at an angle of 40.0 above the horizontal. The cannon uses strong elastic bands to propel the performer, much in the same way that a slingshot fires a stone. Setting up for this stunt involves stretching the bands by 3.00 m from their unstrained length. At the point where the performer flies free of the bands, his height above the floor is the same as the height of the net into which he is shot. He takes 2.14 s to travel the horizontal distance of 26.8 m between this point and the net. Ignore friction and air resistance and determine the effective spring constant of the firing mechanism.