Solution: Two pendulums have the same dimensions (length L)

An object is moving with SHM of amplitude A on the end of a spring. If the amplitude is doubled, what happens to the total distance the object travels in one period? What happens to the period? What happens to the maximum speed of the object? Discuss how these answers are related.

Think of several examples in everyday life of motions that are, at least approximately, simple harmonic. In what respects does each differ from SHM?

Does a tuning fork or similar tuning instrument undergo SHM? Why is this a crucial question for musicians?

A box containing a pebble is attached to an ideal horizontal spring and is oscillating on a friction-free air table. When the box has reached its maximum distance from the equilibrium point, the pebble is suddenly lifted out vertically without disturbing the box. Will the following characteristics of the motion increase, decrease, or remain the same in the subsequent motion of the box? Justify each answer. (a) Frequency; (b) period; (c) amplitude; (d) the maximum kinetic energy of the box; (e) the maximum speed of the box

If a uniform spring is cut in half, what is the force constant of each half? Justify your answer. How would the frequency of SHM using a half-spring differ from the frequency using the same mass and the entire spring?

A glider is attached to a fixed ideal spring and oscillates on a horizontal, friction-free air track. A coin rests atop the glider and oscillates with it. At what points in the motion is the friction force on the coin greatest? The least? Justify your answers.

Two identical gliders on an air track are connected by an ideal spring. Could such a system undergo SHM? Explain. How would the period compare with that of a single glider attached to a spring whose other end is rigidly attached to a stationary object? Explain.

You are captured by Martians, taken into their ship, and put to sleep. You awake some time later and find yourself locked in a small room with no windows. All the Martians have left you with is your digital watch, your school ring, and your long silverchain necklace. Explain how you can determine whether you are still on earth or have been transported to Mars.

The system shown in Fig. 14.17 is mounted in an elevator. What happens to the period of the motion (does it increase, decrease, or remain the same) if the elevator (a) accelerates upward at \(5.0 \ \mathrm {m/s}^2\) ; (b) moves upward at a steady 5.0 m/s; (c) accelerates downward at \(5.0 \ \mathrm {m/s}^2\)? Justify your answers.

If a pendulum has a period of 2.5 s on earth, what would be its period in a space station orbiting the earth? If a mass hung from a vertical spring has a period of 5.0 s on earth, what would its period be in the space station? Justify your answers.

A simple pendulum is mounted in an elevator. What happens to the period of the pendulum (does it increase, decrease, or remain the same) if the elevator (a) accelerates upward at 5.0 m>s 2 ; (b) moves upward at a steady 5.0 m>s; (c) accelerates downward at 5.0 m>s 2 ; (d) accelerates downward at 9.8 m>s 2? Justify your answers.

What should you do to the length of the string of a simple pendulum to (a) double its frequency; (b) double its period; (c) double its angular frequency?

If a pendulum clock is taken to a mountaintop, does it gain or lose time, assuming it is correct at a lower elevation? Explain.

When the amplitude of a simple pendulum increases, should its period increase or decrease? Give a qualitative argument; do not rely on Eq. (14.35). Is your argument also valid for a physical pendulum?

Why do short dogs (like Chihuahuas) walk with quicker strides than do tall dogs (like Great Danes)?

At what point in the motion of a simple pendulum is the string tension greatest? Least? In each case give the reasoning behind your answer

Could a standard of time be based on the period of a certain standard pendulum? What advantages and disadvantages would such a standard have compared to the actual present-day standard discussed in Section 1.3?

For a simple pendulum, clearly distinguish between v (the angular speed) and v (the angular frequency). Which is constant and which is variable?

In designing structures in an earthquake-prone region, how should the natural frequencies of oscillation of a structure relate to typical earthquake frequencies? Why? Should the structure have a large or small amount of damping?

(a) Music. When a person sings, his or her vocal cords vibrate in a repetitive pattern that has the same frequency as the note that is sung. If someone sings the note B flat, which has a frequency of 466 Hz, how much time does it take the persons vocal cords to vibrate through one complete cycle, and what is the angular frequency of the cords? (b) Hearing. When sound waves strike the eardrum, this membrane vibrates with the same frequency as the sound. The highest pitch that young humans can hear has a period of 50.0 ms. What are the frequency and angular frequency of the vibrating eardrum for this sound? (c) Vision. When light having vibrations with angular frequency ranging from 2.7 * 1015 rad>s to 4.7 * 1015 rad>s strikes the retina of the eye, it stimulates the receptor cells there and is perceived as visible light. What are the limits of the period and frequency of this light? (d) Ultrasound. High-frequency sound waves (ultrasound) are used to probe the interior of the body, much as x rays do. To detect small objects such as tumors, a frequency of around 5.0 MHz is used. What are the period and angular frequency of the molecular vibrations caused by this pulse of sound?

If an object on a horizontal, frictionless surface is attached to a spring, displaced, and then released, it will oscillate. If it is displaced 0.120 m from its equilibrium position and released with zero initial speed, then after 0.800 s its displacement is found to be 0.120 m on the opposite side, and it has passed the equilibrium position once during this interval. Find (a) the amplitude; (b) the period; (c) the frequency

The tip of a tuning fork goes through 440 complete vibrations in 0.500 s. Find the angular frequency and the period of the motion.

The displacement of an oscillating object as a function of time is shown in Fig. E14.4. What are (a) the frequency; (b) the amplitude; (c) the period; (d) the angular frequency of this motion?

A machine part is undergoing SHM with a frequency of 4.00 Hz and amplitude 1.80 cm. How long does it take the part to go from x = 0 to x = -1.80 cm?

The wings of the blue-throated hummingbird (Lampornis clemenciae), which inhabits Mexico and the southwestern United States, beat at a rate of up to 900 times per minute. Calculate (a) the period of vibration of this birds wings, (b) the frequency of the wings vibration, and (c) the angular frequency of the birds wing beats.

A 2.40-kg ball is attached to an unknown spring and allowed to oscillate. Figure E14.7 shows a graph of the balls position x as a function of time t. What are the oscillations (a) period, (b) frequency, (c) angular frequency, and (d) amplitude? (e) What is the force constant of the spring?

In a physics lab, you attach a 0.200-kg air-track glider to the end of an ideal spring of negligible mass and start it oscillating. The elapsed time from when the glider first moves through the equilibrium point to the second time it moves through that point is 2.60 s. Find the springs force constant

When a body of unknown mass is attached to an ideal spring with force constant 120 N/m, it is found to vibrate with a frequency of 6.00 Hz. Find (a) the period of the motion; (b) the angular frequency; (c) the mass of the body.

When a 0.750-kg mass oscillates on an ideal spring, the frequency is 1.75 Hz. What will the frequency be if 0.220 kg are (a) added to the original mass and (b) subtracted from the original mass? Try to solve this problem without finding the force constant of the spring.

An object is undergoing SHM with period 0.900 s and amplitude 0.320 m. At t = 0 the object is at x = 0.320 m and is instantaneously at rest. Calculate the time it takes the object to go (a) from x = 0.320 m to x = 0.160 m and (b) from x = 0.160 m to x = 0.

A small block is attached to an ideal spring and is moving in SHM on a horizontal, frictionless surface. When the block is at x = 0.280 m, the acceleration of the block is -5.30 m>s 2 . What is the frequency of the motion?

A 2.00-kg, frictionless block is attached to an ideal spring with force constant 300 N/m. At t = 0 the spring is neither stretched nor compressed and the block is moving in the negative direction at 12.0 m/s. Find (a) the amplitude and (b) the phase angle. (c) Write an equation for the position as a function of time.

Repeat Exercise 14.13, but assume that at t = 0 the block has velocity -4.00 m>s and displacement +0.200 m

The point of the needle of a sewing machine moves in SHM along the x-axis with a frequency of 2.5 Hz. At t = 0 its position and velocity components are +1.1 cm and -15 cm>s, respectively. (a) Find the acceleration component of the needle at t = 0. (b) Write equations giving the position, velocity, and acceleration components of the point as a function of time.

A small block is attached to an ideal spring and is moving in SHM on a horizontal, frictionless surface. When the amplitude of the motion is 0.090 m, it takes the block 2.70 s to travel from x = 0.090 m to x = -0.090 m. If the amplitude is doubled, to 0.180 m, how long does it take the block to travel (a) from x = 0.180 m to x = -0.180 m and (b) from x = 0.090 m to x = -0.090 m?

Weighing Astronauts. This procedure has been used to weigh astronauts in space: A 42.5-kg chair is attached to a spring and allowed to oscillate. When it is empty, the chair takes 1.30 s to make one complete vibration. But with an astronaut sitting in it, with her feet off the floor, the chair takes 2.54 s for one cycle. What is the mass of the astronaut?

A 0.400-kg object undergoing SHM has ax = -1.80 m>s 2 when x = 0.300 m. What is the time for one oscillation?

On a frictionless, horizontal air track, a glider oscillates at the end of an ideal spring of force constant 2.50 N>cm. The graph in Fig. E14.19 shows the acceleration of the glider as a function of time. Find (a) the mass of the glider; (b) the maximum displacement of the glider from the equilibrium point; (c) the maximum force the spring exerts on the glider.

A 0.500-kg mass on a spring has velocity as a function of time given by vx1t2 = -13.60 cm>s2 sin314.71 rad>s2t - 1p>224. What are (a) the period; (b) the amplitude; (c) the maximum acceleration of the mass; (d) the force constant of the spring?

A 1.50-kg mass on a spring has displacement as a function of time given by x1t2 = 17.40 cm2 cos314.16 rad>s2t - 2.424 Find (a) the time for one complete vibration; (b) the force constant of the spring; (c) the maximum speed of the mass; (d) the maximum force on the mass; (e) the position, speed, and acceleration of the mass at t = 1.00 s; (f) the force on the mass at that time.

Weighing a Virus. In February 2004, scientists at Purdue University used a highly sensitive technique to measure the mass of a vaccinia virus (the kind used in smallpox vaccine). The procedure involved measuring the frequency of oscillation of a tiny sliver of silicon (just 30 nm long) with a laser, first without the virus and then after the virus had attached itself to the silicon. The difference in mass caused a change in the frequency. We can model such a process as a mass on a spring. (a) Show that the ratio of the frequency with the virus attached 1 fS+V2 to the frequency without the virus 1S2 is given by fS+V>fS = 1>11 + 1mV>mS2, where mV is the mass of the virus and mS is the mass of the silicon sliver. Notice that it is not necessary to know or measure the force constant of the spring. (b) In some data, the silicon sliver has a mass of 2.10 * 10-16 g and a frequency of 2.00 * 1015 Hz without the virus and 2.87 * 1014 Hz with the virus. What is the mass of the virus, in grams and in femtograms?

The difference in mass caused a change in the frequency. We can model such a process as a mass on a spring. (a) Show that the ratio of the frequency with the virus attached 1 fS+V2 to the frequency without the virus 1S2 is given by fS+V>fS = 1>11 + 1mV>mS2, where mV is the mass of the virus and mS is the mass of the silicon sliver. Notice that it is not necessary to know or measure the force constant of the spring. (b) In some data, the silicon sliver has a mass of 2.10 * 10-16 g and a frequency of 2.00 * 1015 Hz without the virus and 2.87 * 1014 Hz with the virus. What is the mass of the virus, in grams and in femtograms?

For the oscillating object in Fig. E14.4, what are (a) its maximum speed and (b) its maximum acceleration?

A small block is attached to an ideal spring and is moving in SHM on a horizontal frictionless surface. The amplitude of the motion is 0.165 m. The maximum speed of the block is 3.90 m/s. What is the maximum magnitude of the acceleration of the block?

A small block is attached to an ideal spring and is moving in SHM on a horizontal, frictionless surface. The amplitude of the motion is 0.250 m and the period is 3.20 s. What are the speed and acceleration of the block when x = 0.160 m?

A 0.150-kg toy is undergoing SHM on the end of a horizontal spring with force constant k = 300 N>m. When the toy is 0.0120 m from its equilibrium position, it is observed to have a speed of 0.400 m>s. What are the toys (a) total energy at any point of its motion; (b) amplitude of motion; (c) maximum speed during its motion?

A harmonic oscillator has angular frequency v and amplitude A. (a) What are the magnitudes of the displacement and velocity when the elastic potential energy is equal to the kinetic energy? (Assume that U = 0 at equilibrium.) (b) How often does this occur in each cycle? What is the time between occurrences? (c) At an instant when the displacement is equal to A>2, what fraction of the total energy of the system is kinetic and what fraction is potential?

A 0.500-kg glider, attached to the end of an ideal spring with force constant k = 450 N>m, undergoes SHM with an amplitude of 0.040 m. Compute (a) the maximum speed of the glider; (b) the speed of the glider when it is at x = -0.015 m; (c) the magnitude of the maximum acceleration of the glider; (d) the acceleration of the glider at x = -0.015 m; (e) the total mechanical energy of the glider at any point in its motion.

The tip of a tuning fork goes through 440 complete vibrations in 0.500 s. Find the angular frequency and the period of the motion.

For the situation described in part (a) of Example 14.5, what should be the value of the putty mass m so that the amplitude after the collision is one-half the original amplitude? For this value of m, what fraction of the original mechanical energy is converted into thermal energy?

A block with mass m = 0.300 kg is attached to one end of an ideal spring and moves on a horizontal frictionless surface. The other end of the spring is attached to a wall. When the block is at \(x=+0.240\mathrm{\ m}\), its acceleration is \(a_x=-12.0\mathrm{\ m}/\mathrm{s}^2\) and its velocity is \(v_x=+4.00\mathrm{\ m}/\mathrm{s}\). What are (a) the spring’s force constant k; (b) the amplitude of the motion; (c) the maximum speed of the block during its motion; and (d) the maximum magnitude of the block’s acceleration during its motion?

You are watching an object that is moving in SHM. When the object is displaced 0.600 m to the right of its equilibrium position, it has a velocity of 2.20 m>s to the right and an acceleration of 8.40 m>s 2 to the left. How much farther from this point will the object move before it stops momentarily and then starts to move back to the left?

A 2.00-kg frictionless block is attached to an ideal spring with force constant 315 N>m. Initially the spring is neither stretched nor compressed, but the block is moving in the negative direction at 12.0 m>s. Find (a) the amplitude of the motion, (b) the blocks maximum acceleration, and (c) the maximum force the spring exerts on the block.

A 2.00-kg frictionless block attached to an ideal spring with force constant 315 N>m is undergoing simple harmonic motion. When the block has displacement +0.200 m, it is moving in the negative x-direction with a speed of 4.00 m>s. Find (a) the amplitude of the motion; (b) the blocks maximum acceleration; and (c) the maximum force the spring exerts on the block.

A mass is oscillating with amplitude A at the end of a spring. How far (in terms of A) is this mass from the equilibrium position of the spring when the elastic potential energy equals the kinetic energy?

A 175-g glider on a horizontal, frictionless air track is attached to a fixed ideal spring with force constant 155 N/m. At the instant you make measurements on the glider, it is moving at 0.815 m/s and is 3.00 cm from its equilibrium point. Use energy conservation to find (a) the amplitude of the motion and (b) the maximum speed of the glider. (c) What is the angular frequency of the oscillations?

A proud deep-sea fisherman hangs a 65.0-kg fish from an ideal spring having negligible mass. The fish stretches the spring 0.180 m. (a) Find the force constant of the spring. The fish is now pulled down 5.00 cm and released. (b) What is the period of oscillation of the fish? (c) What is the maximum speed it will reach?

A thrill-seeking cat with mass 4.00 kg is attached by a harness to an ideal spring of negligible mass and oscillates vertically in SHM. The amplitude is 0.050 m, and at the highest point of the motion the spring has its natural unstretched length. Calculate the elastic potential energy of the spring (take it to be zero for the unstretched spring), the kinetic energy of the cat, the gravitational potential energy of the system relative to the lowest point of the motion, and the sum of these three energies when the cat is (a) at its highest point; (b) at its lowest point; (c) at its equilibrium position.

A uniform, solid metal disk of mass 6.50 kg and diameter 24.0 cm hangs in a horizontal plane, supported at its center by a vertical metal wire. You find that it requires a horizontal force of 4.23 N tangent to the rim of the disk to turn it by 3.34, thus twisting the wire. You now remove this force and release the disk from rest. (a) What is the torsion constant for the metal wire? (b) What are the frequency and period of the torsional oscillations of the disk? (c) Write the equation of motion for u1t2 for the disk.

A certain alarm clock ticks four times each second, with each tick representing half a period. The balance wheel consists of a thin rim with radius 0.55 cm, connected to the balance shaft by thin spokes of negligible mass. The total mass of the balance wheel is 0.90 g. (a) What is the moment of inertia of the balance wheel about its shaft? (b) What is the torsion constant of the coil spring (Fig. 14.19)?

A thin metal disk with mass \(2.00 \times 10^{-3} \ \mathrm {kg}\) and radius 2.20 cm is attached at its center to a long fiber (Fig. E14.42). The disk, when twisted and released, oscillates with a period of 1.00 s. Find the torsion constant of the fiber.

You want to find the moment of inertia of a complicated machine part about an axis through its center of mass. You suspend it from a wire along this axis. The wire has a torsion constant of 0.450 N # m>rad. You twist the part a small amount about this axis and let it go, timing 165 oscillations in 265 s. What is its moment of inertia?

The balance wheel of a watch vibrates with an angular amplitude , angular frequency v, and phase angle f = 0. (a) Find expressions for the angular velocity du>dt and angular acceleration d2 u>dt2 as functions of time. (b) Find the balance wheels angular velocity and angular acceleration when its angular displacement is , and when its angular displacement is >2 and u is decreasing. (Hint: Sketch a graph of u versus t.)

You pull a simple pendulum 0.240 m long to the side through an angle of 3.50 and release it. (a) How much time does it take the pendulum bob to reach its highest speed? (b) How much time does it take if the pendulum is released at an angle of 1.75 instead of 3.50?

An 85.0-kg mountain climber plans to swing down, starting from rest, from a ledge using a light rope 6.50 m long. He holds one end of the rope, and the other end is tied higher up on a rock face. Since the ledge is not very far from the rock face, the rope makes a small angle with the vertical. At the lowest point of his swing, he plans to let go and drop a short distance to the ground. (a) How long after he begins his swing will the climber first reach his lowest point? (b) If he missed the first chance to drop off, how long after first beginning his swing will the climber reach his lowest point for the second time?

A building in San Francisco has light fixtures consisting of small 2.35-kg bulbs with shades hanging from the ceiling at the end of light, thin cords 1.50 m long. If a minor earthquake occurs, how many swings per second will these fixtures make?

A Pendulum on Mars. A certain simple pendulum has a period on the earth of 1.60 s. What is its period on the surface of Mars, where \(g=3.71 \mathrm{\ m} / \mathrm{s}^{2} ?\)?

After landing on an unfamiliar planet, a space explorer constructs a simple pendulum of length 50.0 cm. She finds that the pendulum makes 100 complete swings in 136 s. What is the value of g on this planet?

In the laboratory, a student studies a pendulum by graphing the angle u that the string makes with the vertical as a function of time t, obtaining the graph shown in Fig. E14.50. (a) What are the period, frequency, angular frequency, and amplitude of the pendulums motion? (b) How long is the pendulum? (c) Is it possible to determine the mass of the bob?

A simple pendulum 2.00 m long swings through a maximum angle of 30.0 with the vertical. Calculate its period (a) assuming a small amplitude, and (b) using the first three terms of Eq. (14.35). (c) Which of the answers in parts (a) and (b) is more accurate? What is the percentage error of the less accurate answer compared with the more accurate one?

A small sphere with mass m is attached to a massless rod of length L that is pivoted at the top, forming a simple pendulum. The pendulum is pulled to one side so that the rod is at an angle \(\theta\) from the vertical, and released from rest. (a) In a diagram, show the pendulum just after it is released. Draw vectors representing the forces acting on the small sphere and the acceleration of the sphere. Accuracy counts! At this point, what is the linear acceleration of the sphere? (b) Repeat part (a) for the instant when the pendulum rod is at an angle \(\theta/2\) from the vertical. (c) Repeat part (a) for the instant when the pendulum rod is vertical. At this point, what is the linear speed of the sphere?

Two pendulums have the same dimensions (length L) and total mass 1m2. Pendulum A is a very small ball swinging at the end of a uniform massless bar. In pendulum B, half the mass is in the ball and half is in the uniform bar. Find the period of each pendulum for small oscillations. Which one takes longer for a swing?

We want to hang a thin hoop on a horizontal nail and have the hoop make one complete small-angle oscillation each 2.0 s. What must the hoops radius be?

A 1.80-kg connecting rod from a car engine is pivoted about a horizontal knife edge as shown in Fig. E14.55. The center of gravity of the rod was located by balancing and is 0.200 m from the pivot. When the rod is set into small-amplitude oscillation, it makes 100 complete swings in 120 s. Calculate the moment of inertia of the rod about the rotation axis through the pivot.

A 1.80-kg monkey wrench is pivoted 0.250 m from its center of mass and allowed to swing as a physical pendulum. The period for small-angle oscillations is 0.940 s. (a) What is the moment of inertia of the wrench about an axis through the pivot? (b) If the wrench is initially displaced 0.400 rad from its equilibrium position, what is the angular speed of the wrench as it passes through the equilibrium position?

The two pendulums shown in Fig. E14.57 each consist of a uniform solid ball of mass M supported by a rigid massless rod, but the ball for pendulum A is very tiny while the ball for pendulum B is much larger. Find the period of each pendulum for small displacements. Which ball takes longer to complete a swing?

A holiday ornament in the shape of a hollow sphere with mass M = 0.015 kg and radius R = 0.050 m is hung from a tree limb by a small loop of wire attached to the surface of the sphere. If the ornament is displaced a small distance and released, it swings back and forth as a physical pendulum with negligible friction. Calculate its period. (Hint: Use the parallel-axis theorem to find the moment of inertia of the sphere about the pivot at the tree limb.)

A 1.35-kg object is attached to a horizontal spring of force constant 2.5 N>cm. The object is started oscillating by pulling it 6.0 cm from its equilibrium position and releasing it so that it is free to oscillate on a frictionless horizontal air track. You observe that after eight cycles its maximum displacement from equilibrium is only 3.5 cm. (a) How much energy has this system lost to damping during these eight cycles? (b) Where did the lost energy go? Explain physically how the system could have lost energy.

A 50.0-g hard-boiled egg moves on the end of a spring with force constant k = 25.0 N/m. Its initial displacement is 0.300 m. A damping force \(F_x = -bv_x\) acts on the egg, and the amplitude of the motion decreases to 0.100 m in 5.00 s. Calculate the magnitude of the damping constant b.

An unhappy 0.300-kg rodent, moving on the end of a spring with force constant k = 2.50 N>m, is acted on by a damping force Fx = -bvx. (a) If the constant b has the value 0.900 kg>s, what is the frequency of oscillation of the rodent? (b) For what value of the constant b will the motion be critically damped?

A mass is vibrating at the end of a spring of force constant 225 N>m. Figure E14.62 shows a graph of its position x as a function of time t. (a) At what times is the mass not moving? (b) How much energy did this system originally contain? (c) How much energy did the system lose between t = 1.0 s and t = 4.0 s? Where did this energy go?

A sinusoidally varying driving force is applied to a damped harmonic oscillator of force constant k and mass m. If the damping constant has a value b1, the amplitude is A1 when the driving angular frequency equals 2k>m. In terms of A1, what is the amplitude for the same driving frequency and the same driving force amplitude Fmax, if the damping constant is (a) 3b1 and (b) b1>2?

An object is undergoing SHM with period 0.300 s and amplitude 6.00 cm. At t = 0 the object is instantaneously at rest at x = 6.00 cm. Calculate the time it takes the object to go from x = 6.00 cm to x = -1.50 cm.

An object is undergoing SHM with period 1.200 s and amplitude 0.600 m. At t = 0 the object is at x = 0 and is moving in the negative x-direction. How far is the object from the equilibrium position when t = 0.480 s?

Four passengers with combined mass 250 kg compress the springs of a car with worn-out shock absorbers by 4.00 cm when they get in. Model the car and passengers as a single body on a single ideal spring. If the loaded car has a period of vibration of 1.92 s, what is the period of vibration of the empty car?

At the end of a ride at a winter-theme amusement park, a sleigh with mass 250 kg (including two passengers) slides without friction along a horizontal, snow-covered surface. The sleigh hits one end of a light horizontal spring that obeys Hookes law and has its other end attached to a wall. The sleigh latches onto the end of the spring and subsequently moves back and forth in SHM on the end of the spring until a braking mechanism is engaged, which brings the sleigh to rest. The frequency of the SHM is 0.225 Hz, and the amplitude is 0.950 m. (a) What was the speed of the sleigh just before it hit the end of the spring? (b) What is the maximum magnitude of the sleighs acceleration during its SHM?

A block with mass M rests on a frictionless surface and is connected to a horizontal spring of force constant k. The other end of the spring is attached to a wall (Fig. P14.68). A second block with mass m rests on top of the first block. The coefficient of static friction between the blocks is ms. Find the maximum amplitude of oscillation such that the top block will not slip on the bottom block.

A 1.50-kg, horizontal, uniform tray is attached to a vertical ideal spring of force constant 185 N>m and a 275-g metal ball is in the tray. The spring is below the tray, so it can oscillate up and down. The tray is then pushed down to point A, which is 15.0 cm below the equilibrium point, and released from rest. (a) How high above point A will the tray be when the metal ball leaves the tray? (Hint: This does not occur when the ball and tray reach their maximum speeds.) (b) How much time elapses between releasing the system at point A and the ball leaving the tray? (c) How fast is the ball moving just as it leaves the tray?

A 10.0-kg mass is traveling to the right with a speed of 2.00 m/s on a smooth horizontal surface when it collides with and sticks to a second 10.0-kg mass that is initially at rest but is attached to a light spring with force constant 170.0 N/m. (a) Find the frequency, amplitude, and period of the subsequent oscillations. (b) How long does it take the system to return the first time to the position it had immediately after the collision?

An apple weighs 1.00 N. When you hang it from the end of a long spring of force constant 1.50 N>m and negligible mass, it bounces up and down in SHM. If you stop the bouncing and let the apple swing from side to side through a small angle, the frequency of this simple pendulum is half the bounce frequency. (Because the angle is small, the back-and-forth swings do not cause any appreciable change in the length of the spring.) What is the unstretched length of the spring (with the apple removed)?

SHM of a Floating Object. An object with height h, mass M, and a uniform cross-sectional area A floats upright in a liquid with density r. (a) Calculate the vertical distance from the surface of the liquid to the bottom of the floating object at equilibrium. (b) A downward force with magnitude F is applied to the top of the object. At the new equilibrium position, how much farther below the surface of the liquid is the bottom of the object than it was in part (a)? (Assume that some of the object remains above the surface of the liquid.) (c) Your result in part (b) shows that if the force is suddenly removed, the object will oscillate up and down in SHM. Calculate the period of this motion in terms of the density r of the liquid, the mass M, and the cross-sectional area A of the object. You can ignore the damping due to fluid friction (see Section 14.7).

A square object of mass m is constructed of four identical uniform thin sticks, each of length L, attached together. This object is hung on a hook at its upper corner (Fig. P14.73). If it is rotated slightly to the left and then released, at what frequency will it swing back and forth?

An object with mass 0.200 kg is acted on by an elastic restoring force with force constant 10.0 N>m. (a) Graph elastic potential energy U as a function of displacement x over a range of x from -0.300 m to +0.300 m. On your graph, let 1 cm = 0.05 J vertically and 1 cm = 0.05 m horizontally. The object is set into oscillation with an initial potential energy of 0.140 J and an initial kinetic energy of 0.060 J. Answer the following questions by referring to the graph. (b) What is the amplitude of oscillation? (c) What is the potential energy when the displacement is one-half the amplitude? (d) At what displacement are the kinetic and potential energies equal? (e) What is the value of the phase angle f if the initial velocity is positive and the initial displacement is negative?

A 2.00-kg bucket containing 10.0 kg of water is hanging from a vertical ideal spring of force constant 450 N>m and oscillating up and down with an amplitude of 3.00 cm. Suddenly the bucket springs a leak in the bottom such that water drops out at a steady rate of 2.00 g>s. When the bucket is half full, find (a) the period of oscillation and (b) the rate at which the period is changing with respect to time. Is the period getting longer or shorter? (c) What is the shortest period this system can have?

A uniform beam is suspended horizontally by two identical vertical springs that are attached between the ceiling and each end of the beam. The beam has mass 225 kg, and a 175-kg sack of gravel sits on the middle of it. The beam is oscillating in SHM with an amplitude of 40.0 cm and a frequency of 0.600 cycle/s. (a) The sack falls off the beam when the beam has its maximum upward displacement. What are the frequency and amplitude of the subsequent SHM of the beam? (b) If the sack instead falls off when the beam has its maximum speed, repeat part (a).

A 5.00-kg partridge is suspended from a pear tree by an ideal spring of negligible mass. When the partridge is pulled down 0.100 m below its equilibrium position and released, it vibrates with a period of 4.20 s. (a) What is its speed as it passes through the equilibrium position? (b) What is its acceleration when it is 0.050 m above the equilibrium position? (c) When it is moving upward, how much time is required for it to move from a point 0.050 m below its equilibrium position to a point 0.050 m above it? (d) The motion of the partridge is stopped, and then it is removed from the spring. How much does the spring shorten?

A 0.0200-kg bolt moves with SHM that has an amplitude of 0.240 m and a period of 1.500 s. The displacement of the bolt is +0.240 m when t = 0. Compute (a) the displacement of the bolt when t = 0.500 s; (b) the magnitude and direction of the force acting on the bolt when t = 0.500 s; (c) the minimum time required for the bolt to move from its initial position to the point where x = -0.180 m; (d) the speed of the bolt when x = -0.180 m.

SHM of a Butchers Scale. A spring of negligible mass and force constant k = 400 N>m is hung vertically, and a 0.200-kg pan is suspended from its lower end. A butcher drops a 2.2-kg steak onto the pan from a height of 0.40 m. The steak makes a totally inelastic collision with the pan and sets the system into vertical SHM. What are (a) the speed of the pan and steak immediately after the collision; (b) the amplitude of the subsequent motion; (c) the period of that motion?

A 40.0-N force stretches a vertical spring 0.250 m. (a) What mass must be suspended from the spring so that the system will oscillate with a period of 1.00 s? (b) If the amplitude of the motion is 0.050 m and the period is that specified in part (a), where is the object and in what direction is it moving 0.35 s after it has passed the equilibrium position, moving downward? (c) What force (magnitude and direction) does the spring exert on the object when it is 0.030 m below the equilibrium position, moving upward?

Dont Miss the Boat. While on a visit to Minnesota (Land of 10,000 Lakes), you sign up to take an excursion around one of the larger lakes. When you go to the dock where the 1500-kg boat is tied, you find that the boat is bobbing up and down in the waves, executing simple harmonic motion with amplitude 20 cm. The boat takes 3.5 s to make one complete up-and-down cycle. When the boat is at its highest point, its deck is at the same height as the stationary dock. As you watch the boat bob up and down, you (mass 60 kg) begin to feel a bit woozy, due in part to the previous nights dinner of lutefisk. As a result, you refuse to board the boat unless the level of the boats deck is within 10 cm of the dock level. How much time do you have to board the boat comfortably during each cycle of up-and-down motion?

An interesting, though highly impractical example of oscillation is the motion of an object dropped down a hole that extends from one side of the earth, through its center, to the other side. With the assumption (not realistic) that the earth is a sphere of uniform density, prove that the motion is simple harmonic and find the period. [Note: The gravitational force on the object as a function of the objects distance r from the center of the earth was derived in Example 13.10 (Section 13.6). The motion is simple harmonic if the acceleration ax and the displacement from equilibrium x are related by Eq. (14.8), and the period is then T = 2p>v.]

A rifle bullet with mass 8.00 g and initial horizontal velocity 280 m>s strikes and embeds itself in a block with mass 0.992 kg that rests on a frictionless surface and is attached to one end of an ideal spring. The other end of the spring is attached to the wall. The impact compresses the spring a maximum distance of 15.0 cm. After the impact, the block moves in SHM. Calculate the period of this motion.

Two uniform solid spheres, each with mass M = 0.800 kg and radius R = 0.0800 m, are connected by a short, light rod that is along a diameter of each sphere and are at rest on a horizontal tabletop. A spring with force constant k = 160 N/m has one end attached to the wall and the other end attached to a frictionless ring that passes over the rod at the center of mass of the spheres, which is midway between the centers of the two spheres. The spheres are each pulled the same distance from the wall, stretching the spring, and released. There is sufficient friction between the tabletop and the spheres for the spheres to roll without slipping as they move back and forth on the end of the spring. Show that the motion of the center of mass of the spheres is simple harmonic and calculate the period.

In Fig. P14.85 the upper ball is released from rest, collides with the stationary lower ball, and sticks to it. The strings are both 50.0 cm long. The upper ball has mass 2.00 kg, and it is initially 10.0 cm higher than the lower ball, which has mass 3.00 kg. Find the frequency and maximum angular displacement of the motion after the collision.

The Silently Ringing Bell. A large, 34.0-kg bell is hung from a wooden beam so it can swing back and forth with negligible friction. The bell’s center of mass is 0.60 m below the pivot. The bell’s moment of inertia about an axis at the pivot is \(18.0 \ \mathrm {kg} \cdot \mathrm{m}^2\) . The clapper is a small, 1.8-kg mass attached to one end of a slender rod of length L and negligible mass. The other end of the rod is attached to the inside of the bell; the rod can swing freely about the same axis as the bell. What should be the length L of the clapper rod for the bell to ring silently—that is, for the period of oscillation for the bell to equal that of the clapper?

A slender, uniform, metal rod with mass M is pivoted without friction about an axis through its midpoint and perpendicular to the rod. A horizontal spring with force constant k is attached to the lower end of the rod, with the other end f the spring attached to a rigid support. If the rod is displaced by a small angle from the vertical (Fig. P14.87) and released, show that it moves in angular SHM and calculate the period. (Hint: Assume that the angle is small enough for the approximations sin and cos 1 to be valid. The motion is simple harmonic if d2 u>dt2 = -v2 u, and the period is then T = 2p>v.)

f the spring attached to a rigid support. If the rod is displaced by a small angle from the vertical (Fig. P14.87) and released, show that it moves in angular SHM and calculate the period. (Hint: Assume that the angle is small enough for the approximations sin and cos 1 to be valid. The motion is simple harmonic if d2 u>dt2 = -v2 u, and the period is then T = 2p>v.)

A mass m is attached to a spring of force constant 75 N/m and allowed to oscillate. Figure P14.89 shows a graph of its velocity component vx as a function of time t. Find (a) the period, (b) the frequency, and (c) the angular frequency of this motion. (d) What is the amplitude (in cm), and at what times does the mass reach this position? (e) Find the maximum acceleration magnitude of the mass and the times at which it occurs. (f) What is the value of m?

You hang various masses m from the end of a vertical, 0.250-kg spring that obeys Hookes law and is tapered, which means the diameter changes along the length of the spring. Since the mass of the spring is not negligible, you must replace m in the equation T = 2p1m>k with m + meff, where meff is the effective mass of the oscillating spring. (See Challenge Problem 14.93.) You vary the mass m and measure the time for 10 complete oscillations, obtaining these data: m 1kg2 0.100 0.200 0.300 0.400 0.500 Time 1s2 8.7 10.5 12.2 13.9 15.1 (a) Graph the square of the period T versus the mass suspended from the spring, and find the straight line of best fit. (b) From the slope of that line, determine the force constant of the spring. (c) From the vertical intercept of the line, determine the springs effective mass. (d) What fraction is meff of the springs mass? (e) If a 0.450-kg mass oscillates on the end of the spring, find its period, frequency, and angular frequency.

Experimenting with pendulums, you attach a light string to the ceiling and attach a small metal sphere to the lower end of the string. When you displace the sphere 2.00 m to the left, it nearly touches a vertical wall; with the string taut, you release the sphere from rest. The sphere swings back and forth as a simple pendulum, and you measure its period T. You repeat this act for strings of various lengths L, each time starting the motion with the sphere displaced 2.00 m to the left of the vertical position of the string. In each case the spheres radius is very small compared with L. Your results are given in the table: L 1m2 12.00 10.00 8.00 6.00 5.00 4.00 3.00 2.50 2.30 T 1s2 6.96 6.36 5.70 4.95 4.54 4.08 3.60 3.35 3.27 (a) For the five largest values of L, graph T2 versus L. Explain why the data points fall close to a straight line. Does the slope of this line have the value you expected? (b) Add the remaining data to your graph. Explain why the data start to deviate from the straight-line fit as L decreases. To see this effect more clearly, plot T>T0 versus L, where T0 = 2p1L>g and g = 9.80 m>s 2 . (c) Use your graph of T>T0 versus L to estimate the angular amplitude of the pendulum (in degrees) for which the equation T = 2p1L>g is in error by 5%.

The Effective Force Constant of Two Springs. Two springs with the same unstretched length but different force constants k1 and k2 are attached to a block with mass m on a level, frictionless surface. Calculate the effective force constant keff in each of the three cases (a), (b), and (c) depicted in Fig. P14.92. (The effective force constant is defined by gFx = -keffx.) (d) An object with mass m, suspended from a uniform spring with a force constant k, vibrates with a frequency f1. When the spring is cut in half and the same object is suspended from one of the halves, the frequency is f2. What is the ratio f1>f2?

A Spring with Mass. The preceding problems in this chapter have assumed that the springs had negligible mass. But of course no spring is completely massless. To find the effect of the springs mass, consider a spring with mass M, equilibrium length L0, and spring constant k. When stretched or compressed to a length L, the potential energy is 1 2 kx2 , where x = L - L0. (a) Consider a spring, as described above, that has one end fixed and the other end moving with speed v. Assume that the speed of points along the length of the spring varies linearly with distance l from the fixed end. Assume also that the mass M of the spring is distributed uniformly along the length of the spring. Calculate the kinetic energy of the spring in terms of M and v. (Hint: Divide the spring into pieces of length dl; find the speed of each piece in terms of l, v, and L; find the mass of each piece in terms of dl, M, and L; and integrate from 0 to L. The result is not 12 Mv2, since not all of the spring moves with the same speed.) (b) Take the time derivative of the conservation of energy equation, Eq. (14.21), for a mass m moving on the end of a massless spring. By comparing your results to Eq. (14.8), which defines v, show that the angular frequency of oscillation is v = 1k>m. (c) Apply the procedure of part (b) to obtain the angular frequency of oscillation v of the spring considered in part (a). If the effective mass M of the spring is defined by v = 1k>M, what is M in terms of M?

If we model the vibrating system as a mass on a spring, what is the mass necessary to achieve the desired resonant frequency when the tip is not interacting with the surface? (a) 25 ng; (b) 100 ng; \(\text { (c) } 2.5 \mu \mathrm{g} \text {; (d) } 100 \mu \mathrm{g} \text {. }\)

In the model of Problem 14.94, what is the mechanical energy of the vibration when the tip is not interacting with the surface? (a) 1.2 * 10-18 J; (b) 1.2 * 10-16 J; (c) 1.2 * 10-9 J; (d) 5.0 * 10-8 J

By what percentage does the frequency of oscillation change if ksurf = 5 N>m? (a) 0.1%; (b) 0.2%; (c) 0.5%; (d) 1.0%.