Section 14.2 Simple Harmonic Motion and Circular MotionAn

Problem 1E When a guitar string plays the note “A,” the string vibrates at 440 Hz. What is the period of the vibration?

Problem 2CQ A pendulum on Planet X, where the value of g is unknown, oscillates with a period of 2.0 s. What is the period of this pendulum if: a. Its mass is doubled? b. Its length is doubled? c. Its oscillation amplitude is doubled? Note: You do not know the values of m, L, or g, so do not assume any specific values.

Problem 2E An air-track glider attached to a spring oscillates between the 10 cm mark and the 60 cm mark on the track. The glider completes 10 oscillations in 33 s. What are the (a) period, (b) frequency, (c) amplitude, and (d) maximum speed of the glider?

Figure CP14.80 shows a \(200\ g\) uniform rod pivoted at one end. The other end is attached to a horizontal spring. The spring is neither stretched nor compressed when the rod hangs straight down. What is the rod’s oscillation period? You can assume that the rod’s angle from vertical is always small Equation Transcription: Text Transcription: 200 g

Problem 79CP a. A mass m oscillating on a spring has period T. Suppose the mass changes very slightly from m to m + ?m, where ?m ? m. Find an expression for ?T, the small change in the period. Your expression should involve T, m, and ?m but not the spring constant. ________________ b. Suppose the period is 2.000 s and the mass increases by 0.1 %. What is the new period?

FIGURE Q14.3 shows a position- versus-time graph for a particle in SHM. What are (a) the amplitude \(A\), (b) the angular frequency \(\omega\), and (c) the phase constant \(\phi_{0}\) ? Explain. FIGURE Q14.3 Equation Transcription: Text Transcription: A watt phi_0

Problem 3E An air-track glider is attached to a spring. The glider is pulled to the right and released from rest at t = 0 s. It then oscillates with a period of 2.0 s and a maximum speed of 40 cm/s. a. What is the amplitude of the oscillation? b. What is the glider’s position at t = 0.25 s?

Equation 14.25 states that \(\frac{1}{2} k A^{2}=\frac{1}{2} m\left(v_{\max }\right)^{2}\). What does this mean? Write a couple of sentences explaining how to interpret this equation. Equation Transcription: Text Transcription: 1/2 kA^2=1/2 m(v_max)^2

Problem 1CQ A block oscillating on a spring has period T = 2 s. What is the period if: a. The block’s mass is doubled? Explain. Note that you do not know the value of either m or k, so do not assume any particular values for them. The required analysis involves thinking about ratios. ________________ b. The value of the spring constant is quadrupled? ________________ c. The oscillation amplitude is doubled while in and k are unchanged?

What are the (a) amplitude, (b) frequency, and (c) phase constant of the oscillation shown in Figure EX14.4?

Problem 5CQ A block oscillating on a spring has an amplitude of 20 cm. What will the amplitude be if the total energy is doubled? Explain.

What are the (a) amplitude, (b) frequency, and (c) phase constant of the oscillation shown in Figure EX14.5?

Problem 6CQ A block oscillating on a spring has a maximum speed of 20 cm/s. What will the block’s maximum speed be if the total energy is doubled? Explain.

Problem 6E Section 14.2 Simple Harmonic Motion and Circular Motion An object in simple harmonic motion has an amplitude of 4.0 cm, a frequency of 2.0 Hz, and a phase constant of 2? /3 rad. Draw a position graph showing two cycles of the motion.

Problem 7E Section 14.2 Simple Harmonic Motion and Circular Motion An object in simple harmonic motion has an amplitude of 8.0 cm, a frequency of 0.25 Hz, and a phase constant of ??/2 rad. Draw a position graph showing two cycles of the motion.

FIGURE Q14.8 shows a velocity-versus-time graph for a particle in SHM. a. What is the phase constant \(\phi_{0}\)? Explain. b. What is the phase of the particle at each of the three numbered points on the graph? Equation Transcription: Text Transcription: SHM phi_0

FIGURE Q14.9 shows the potential-energy diagram and the total energy line of a particle oscillating on a spring. a. What is the spring's equilibrium length? b. Where are the turning points of the motion? Explain. c. What is the particle's maximum kinetic energy? d. What will be the turning points if the particle's total energy is doubled?

Problem 8E Section 14.2 Simple Harmonic Motion and Circular Motion An object in simple harmonic motion has amplitude 4.0 cm and frequency 4.0 Hz, and at t = 0 s it passes through the equilibrium point moving to the right. Write the function x (t ) that describes the object’s position.

Problem 9E Section 14.2 Simple Harmonic Motion and Circular Motion An object in simple harmonic motion has amplitude 8.0 cm and frequency 0.50 Hz. At t = 0 s it has its most negative position. Write the function x (t ) that describes the object’s position.

Problem 10CQ Suppose the damping constant b of an oscillator increases. a. Is the medium more resistive or less resistive? ________________ b. Do the oscillations damp out more quickly or less quickly? ________________ c. Is the time constant ? increased or decreased?

Problem 11CQ a. Describe the difference between ? and T. Don’t just name them; say what is different about the physical concepts they represent. ________________ b. Describe the difference between ? and t 12 .

Problem 10E Section 14.2 Simple Harmonic Motion and Circular Motion An air-track glider attached to a spring oscillates with a period of 1.5 s. At t = 0 s the glider is 5.00 cm left of the equilibrium position and moving to the right at 36.3 cm/s. a. What is the phase constant? ________________ b. What is the phase at t = 0 s, 0.5 s, 1.0 s, and 1.5 s?

Problem 11E A block attached to a spring with unknown spring constant oscillates with a period of 2.00 s. What is the period if a. The mass is doubled? b. The mass is halved? c. The amplitude is doubled? d. The spring constant is doubled? Parts a to d are independent questions, each referring to the initial situation.

Problem 12CQ What is the difference between the driving frequency and the natural frequency of an oscillator?

Problem 12E A 200 g air-track glider is attached to a spring. The glider is pushed 10.0 cm against the spring, then released. A student with a stopwatch finds that 10 oscillations take 12.0 s. What is the spring constant?

Problem 14E Section 14.3 Energy in Simple Harmonic Motion Section 14.4 The Dynamics of Simple Harmonic Motion The position of a 50 g oscillating mass is given by x (t ) = (2.0 cm) cos(10t ? ?/4), where t is in s. Determine: a. The amplitude. b. The period. c. The spring constant. d. The phase constant. e. The initial conditions. f. The maximum speed. g. The total energy. h. The velocity at t = 0.40 s.

Problem 16E A spring is hanging from the ceiling. Attaching a 500 g mass to the spring causes it to stretch 20.0 cm in order to come to equilibrium. a. What is the spring constant? b. From equilibrium, the mass is pulled down 10.0 cm and released. What is the period of oscillation? c. What is the maximum speed of the mass? At what position or positions does it have this speed?

A \(1.0 \mathrm{~kg}\) block is attached to a spring with spring constant \(16 \mathrm{~N} / \mathrm{m}\). While the block is sitting at rest, a student hits it with a hammer and almost instantaneously gives it a speed of \(40 \mathrm{~cm} / \mathrm{s}\). What are a. The amplitude of the subsequent oscillations? b. The block's speed at the point where \(x=\frac{1}{2} A\) ? Equation Transcription: Text Transcription: 1.0 kg 16 N/m 40 cm/s x = 1/2 A

FIGURE Q14.7 shows a position-versus-time graph for a particle in SHM. a. What is the phase constant \(\phi_{0}\) ? Explain. b. What is the phase of the particle at each of the three numbered points on the graph? Equation Transcription: Text Transcription: SHM phi_0

Problem 17E A spring with spring constant 15.0 N/m hangs from the ceiling. A ball is suspended from the spring and allowed to come to rest. It is then pulled down 6.00 cm and released. If the ball makes 30 oscillations in 20.0 s, what are its (a) mass and (b) maximum speed?

A \(200 \mathrm{~g}\) mass attached to a horizontal spring oscillates at a frequency of \(2.0 \mathrm{~Hz}\). At \(t=0 \mathrm{~s}\), the mass is at \(x=5.0 \mathrm{~cm}\) and has \(v_{x}=-30 \mathrm{~cm} / \mathrm{s}\). Determine: a. The period. b. The angular frequency. c. The amplitude. d. The phase constant. e. The maximum speed. f. The maximum acceleration. g. The total energy. h. The position at \(t=0.40 \mathrm{~s}\). Equation Transcription: Text Transcription: 200 g 2.0 Hz t=0 s x=5.0 cm v_x=?30 cm/s t=0.40 s

Problem 18E A spring is hung from the ceiling. When a coffee mug is attached to its end, the spring stretches 2.0 cm before reaching its new equilibrium length. The mug is then pulled down slightly and released. What is the frequency of oscillation?

Problem 22E Section 14.6 The Pendulum What is the length of a pendulum whose period on the moon matches the period of a 2.0-m-long pendulum on the earth?

Problem 20E A 200 g ball is tied to a string. It is pulled to an angle of 8.00° and released to swing as a pendulum. A student with a stopwatch finds that 10 oscillations take 12.0 s. How long is the string?

Problem 23E Astronauts on the first trip to Mars take along a pendulum that has a period on earth of 1.50 s. The period on Mars turns out to be 2.45 s. What is the Martian free-fall acceleration?

Problem 19E A mass on a string of unknown length oscillates as a pendulum with a period of 4.00 s. What is the period if a. The mass is doubled? b. The string length is doubled? c. The string length is halved? d. The amplitude is halved? Parts a to d are independent questions, each referring to the initial situation.

Problem 21E Section 14.6 The Pendulum What is the period of a 1.0-m-long pendulum on (a) the earth and (b) Venus?

Problem 24E Section 14.6 The Pendulum A uniform steel bar swings from a pivot at one end with a period of 1.2 s. How long is the bar?

Problem 25E A 2.0 g spider is dangling at the end of a silk thread. You can make the spider bounce up and down on the thread by tapping lightly on his feet with a pencil. You soon discover that you can give the spider the largest amplitude on his little bungee cord if you tap exactly once every second. What is the spring constant of the silk thread?

Problem 26E The amplitude of an oscillator decreases to 36.8% of its initial value in 10.0 s. What is the value of the time constant?

Problem 27E Sketch a position graph from t = 0 s to t = 10 s of a damped oscillator having a frequency of 1.0 Hz and a time constant of 4.0 s.

Problem 28E In a science museum, a 110 kg brass pendulum bob swings at the end of a 15.0-m-long wire. The pendulum is started at exactly 8:00 A.M. every morning by pulling it 1.5 m to the side and releasing it. Because of its compact shape and smooth surface, the pendulum’s damping constant is only 0.010 kg/s. At exactly 12:00 noon, how many oscillations will the pendulum have completed and what is its amplitude?

Figure P14.30 is the velocity-versus-time graph of a particle in simple harmonic motion. a. What is the amplitude of the oscillation? b. What is the phase constant? c. What is the position at \(t = 0\ s\)? Equation Transcription: Text Transcription: t = 0 s

Problem 29E Vision is blurred if the head is vibrated at 29 Hz because the vibrations are resonant with the natural frequency of the eyeball in its socket. If the mass of the eyeball is 7.5 g, a typical value, what is the effective spring constant of the musculature that holds the eyeball in the socket?

FIGURE P14.31 is the position-versus-time graph of a particle in simple harmonic motion. a. What is the phase constant? b. What is the velocity at \(t=0 \mathrm{~s}\)? c. What is \(v_{\max }\)? Equation Transcription: Text Transcription: t = 0 s v_max

The two graphs in Figure P14.32 are for two different vertical mass-spring systems. If both systems have the same mass, what is the ratio \(k_{\mathrm{A}} / k_{\mathrm{B}}\) of their spring constants? Equation Transcription: Text Transcription: k_A / k_B

Problem 33P An object in SHM oscillates with a period of 4.0 s and an amplitude of 10 cm. How long does the object take to move from x = 0.0 cm to x = 6.0 cm?

Problem 34P A 1.0 kg block oscillates on a spring with spring constant 20 N/m. At t = 0 s the block is 20 cm to the right of the equilibrium position and moving to the left at a speed of 100 cm/s. Determine (a) the period and (b) the amplitude.

Astronauts in space cannot weigh themselves by standing on a bathroom scale. Instead, they determine their mass by oscillating on a large spring. Suppose an astronaut attaches one end of a large spring to her belt and the other end to a hook on the wall of the space capsule. A fellow astronaut then pulls her away from the wall and releases her. The spring’s length as a function of time is shown in Figure P14.35. a. What is her mass if the spring constant is \(240\ N/m\)? b. What is her speed when the spring’s length is \(1.2\ m\)? Equation Transcription: Text Transcription: 240 N/m 1.2 m

a. When the displacement of a mass on a spring is \(\frac{1}{2} A\), what fraction of the energy is kinetic energy and what fraction is potential energy? b. At what displacement, as a fraction of \(A\), is the energy half kinetic and half potential? Equation Transcription: Text Transcription: 1/2A A

Problem 38P For a particle in simple harmonic motion, show that v max = (? /2) v avg where v avg is the average speed during one cycle of the motion.

Problem 36P The motion of a particle is given by x (t ) ? (25 cm)cos( 10t ), where t is in s. At what time is the kinetic energy twice the potential energy?

Problem 40P A block on a spring is pulled to the right and released at t = 0 s. It passes x = 3.00 cm at t = 0.685 s, and it passes x = ?3.00 cm at t = 0.886 s. a. What is the angular frequency? ________________ b. What is the amplitude? Hint: cos (? ? 9 ) = ?cos ?.

Problem 39P A 100 g block attached to a spring with spring constant 2.5 N/m oscillates horizontally on a frictionless table. Its velocity is 20 cm/s when x = ?5.0 cm. a. What is the amplitude of oscillation? ________________ b. What is the block’s maximum acceleration? ________________ c. What is the block’s position when the acceleration is maximum? ________________ d. What is the speed of the block when x = 3.0 cm?

Problem 42P An ultrasonic transducer, of the type used in medical ultra sound imaging, is a very thin disk (m = 0.10 g) driven back and forth in SHM at 1.0 MHz by an electromagnetic coil. a. The maximum restoring force that can be applied to the disk without breaking it is 40,000 N. What is the maximum oscillation amplitude that won't rupture the disk? ________________ b. What is the disk’s maximum speed at this amplitude?

Problem 41P A 300 g oscillator has a speed of 95.4 cm/s when its displacement is 3.0 cm and 71.4 cm/s when its displacement is 6.0 cm. What is the oscillator’s maximum speed?

Problem 43P A 5.0 kg block hangs from a spring with spring constant 2000 N/m. The block is pulled down 5.0 cm from the equilibrium position and given an initial velocity of 1.0 m/s back toward equilibrium. What are the (a) frequency, (b) amplitude, and (c) total mechanical energy of the motion?

Problem 44P Your lab instructor has asked you to measure a spring constant using a dynamic method—letting it oscillate—rather than a static method of stretching it. You and your lab partner suspend the spring from a hook, hang different masses on the lower end, and start them oscillating. One of you uses a meter stick to measure the amplitude, the other uses a stopwatch to time 10 oscillations. Your data are as follows: Mass (g) Amplitude (cm) Time (s) 100 6.5 7.8 150 5.5 9.8 200 6.0 10.9 250 3.5 12.4 Use the best-fit line of an appropriate graph to determine the spring constant.

Problem 45P A 200 g block hangs from a spring with spring constant 10 N/m. At t = 0 s the block is 20 cm below the equilibrium point and moving upward with a speed of 100 cm/s. What are the block’s a. Oscillation frequency? ________________ b. Distance from equilibrium when the speed is 50 cm/s? ________________ c. Distance from equilibrium at t = 1.0 s?

The two blocks in Figure P14.49 oscillate on a frictionless surface with a period of \(1.5\ s\). The upper block just begins to slip when the amplitude is increased to \(40\ cm\). What is the coefficient of static friction between the two blocks? Equation Transcription: Text Transcription: 1.5 s 40 cm

Problem 47P While grocery shopping, you put several apples in the spring scale in the produce department. The scale reads 20 N, and you use your ruler (which you always carry with you) to discover that the pan goes down 9.0 cm when the apples are added. If you tap the bottom of the apple-filled pan to make it bounce up and down a little, what is its oscillation frequency? Ignore the mass of the pan.

Problem 48P A compact car has a mass of 1200 kg. Assume that the car has one spring on each wheel, that the springs are identical, and that the mass is equally distributed over the four springs. a. What is the spring constant of each spring if the empty car bounces up and down 2.0 times each second? ________________ b. What will be the car’s oscillation frequency while carrying four 70 kg passengers?

Problem 46P A spring with spring constant k is suspended vertically from a support and a mass m is attached. The mass is held at the point where the spring is not stretched. Then the mass is released and begins to oscillate. The lowest point in the oscillation is 20 cm below the point where the mass was released. What is the oscillation frequency?

Problem 51P It is said that Galileo discovered a basic principle of the pendulum—that the period is independent of the amplitude—by using his pulse to time the period of swinging lamps in the cathedral as they swayed in the breeze. Suppose that one oscillation of a swinging lamp takes 5.5 s. a. How long is the lamp chain? b. What maximum speed does the lamp have if its maximum angle from vertical is 3.0°?

It has recently become possible to "weigh" DNA molecules by measuring the influence of their mass on a nano-oscillator. FIGURE P14.50 shows a thin rectangular cantilever etched out of silicon (density \(2300 \mathrm{~kg} / \mathrm{m}^{3}\)) with a small gold dot at the end. If pulled down and released, the end of the cantilever vibrates with simple harmonic motion, moving up and down like a diving board after a jump. When bathed with DNA molecules whose ends have been modified to bind with gold, one or more molecules may attach to the gold dot. The addition of their mass causes a very slight-but measurable-decrease in the oscillation frequency. A vibrating cantilever of mass \(M\) can be modeled as a block of mass \(\frac{1}{3} M\) attached to a spring. (The factor of \(\frac{1}{3}\) arises from the moment of inertia of a bar pivoted at one end.) Neither the mass nor the spring constant can be determined very accurately- perhaps to only two significant figures-but the oscillation frequency can be measured with very high precision simply by counting the oscillations. In one experiment, the cantilever was initially vibrating at exactly \(12 \mathrm{MHz}\) Attachment of a DNA molecule caused the frequency to decrease by \(50 \mathrm{~Hz}\). What was the mass of the DNA? Equation Transcription: Text Transcription: 2300 kg/m^3 M 1/3M 12 MHz 50 Hz

Problem 52P A 100 g mass on a 1.0-m-long string is pulled 8.0° to one side and released. How long does it take for the pendulum to reach 4.0° on the opposite side?

Orangutans can move by brachiation, swinging like a pendulum beneath successive handholds. If an orangutan has arms that are \(0.90 \mathrm{~m}\) long and repeatedly swings to a \(20^{\circ}\) angle, taking one swing after another, estimate its speed of forward motion in \(\mathrm{m} / \mathrm{s}\). While this is somewhat beyond the range of validity of the small angle approximation, the standard results for a pendulum are adequate for making an estimate. Equation Transcription: 20o Text Transcription: 20 degree 0.90 m m/s

Problem 55P A 15-cm-long, 200 g rod is pivoted at one end. A 20 g ball of clay is stuck on the other end. What is the period if the rod and clay swing as a pendulum?

Problem 54P Show that Equation 14.51 for the angular frequency of a physical pendulum gives Equation 14.48 when applied to a simple pendulum of a mass on a string.

Problem 56P A uniform rod of mass M and length L swings as a pendulum on a pivot at distance L /A from one end of the rod. Find an expression for the frequency f of small-angle oscillations.

A solid sphere of mass \(M\) and radius \(R\) is suspended from a thin rod, as shown in Figure P14.57. The sphere can swing back and forth at the bottom of the rod. Find an expression for the frequency f of small-angle oscillations. Equation Transcription: Text Transcription: M R

Problem 58P A geologist needs to determine the local value of g. Unfortunately, his only tools are a meter stick, a saw, and a stopwatch. He starts by hanging the meter stick from one end and measuring its frequency as it swings. He then saws off 20 cm—using the centimeter markings—and measures the frequency again. After two more cuts, these are his data: Length (cm) Frequency (Hz) 100 0.61 80 0.67 60 0.79 40 0.96 Use the best-fit line of an appropriate graph to determine the local value of g.

Problem 60P A 500 g air-track glider attached to a spring with spring constant 10 N/m is sitting at rest on a frictionless air track. A 250 g glider is pushed toward it from the far end of the track at a speed of 120 cm/s. It collides with and sticks to the 500 g glider. What are the amplitude and period of the subsequent oscillations?

Problem 59P Interestingly, there have been several studies using cadavers to determine the moments of inertia of human body parts, information that is important in biomechanics. In one study, the center of mass of a 5.0 kg lower leg was found to be 18 cm from the knee. When the leg was allowed to pivot at the knee and swing freely as a pendulum, the oscillation frequency was 1.6 Hz. What was the moment of inertia of the lower leg about the knee joint?

A molecular bond can be modeled as a spring between two atoms that vibrate with simple harmonic motion. FIGURE P14.63 shows an SHM approximation for the potential energy of an \(\mathrm{HCl}\) molecule. For \(E<4 \times 10^{-19} \mathrm{~J}\) it is a good approximation to the more accurate \(\mathrm{HCl}\) potential-energy curve that was shown in Figure 10.31. Because the chlorine atom is so much more massive than the hydrogen atom, it is reasonable to assume that the hydrogen atom \(\left(m=1.67 \times 10^{-27} \mathrm{~kg}\right)\) vibrates back and forth while the chlorine atom remains at rest. Use the graph to estimate the vibrational frequency of the \(\mathrm{HCl}\) molecule. Equation Transcription: Text Transcription: HCl E<4 times 10^?19 J (m=1.67 times 10^?27 kg)

Problem 61P A 200 g block attached to a horizontal spring is oscillating with an amplitude of 2.0 cm and a frequency of 2.0 Hz. Just as it passes through the equilibrium point, moving to the right, a sharp blow directed to the left exerts a 20 N force for 1.0 ms. What are the new (a) frequency and (b) amplitude?

Problem 64P An ice cube can slide around the inside of a vertical circular hoop of radius R. It undergoes small-amplitude oscillations if displaced slightly from the equilibrium position at the lowest point. Find an expression for the period of these small-amplitude oscillations.

Figure P14.62 is a top view of an object of mass \(m\) connected between two stretched rubber bands of length \(L\). The object rests on a frictionless surface. At equilibrium, the tension in each rubber band is \(T\). Find an expression for the frequency of oscillations perpendicular to the rubber bands. Assume the amplitude is sufficiently small that the magnitude of the tension in the rubber bands is essentially unchanged as the mass oscillates. Equation Transcription: Text Transcription: m T L

Problem 65P A penny rides on top of a piston as it undergoes vertical simple harmonic motion with an amplitude of 4.0 cm. If the frequency is low, the penny rides up and down without difficulty. If the frequency is steadily increased, there comes a point at which the penny leaves the surface. a. At what point in the cycle does the penny first lose contact with the piston? ________________ b. What is the maximum frequency for which the penny just barely remains in place for the full cycle?

Problem 66P On your first trip to Planet X you happen to take along a 200 g mass, a 40-cm-long spring, a meter stick, and a stopwatch. You’re curious about the free-fall acceleration on Planet X, where ordinary tasks seem easier than on earth, but you can’t find this information in your Visitor’s Guide. One night you suspend the spring from the ceiling in your room and hang the mass from it. You find that the mass stretches the spring by 31.2 cm. You then pull the mass down 10.0 cm and release it. With the stopwatch you find that 10 oscillations take 14.5 s. Based on this information, what is g ?

Problem 67P The 15 g head of a bobble-head doll oscillates in SHM at a frequency of 4.0 Hz. a. What is the spring constant of the spring on which the head is mounted? ________________ b. The amplitude of the head’s oscillations decreases to 0.5 cm in 4.0 s. What is the head’s damping constant?

Problem 68P An oscillator with a mass of 500 g and a period of 0.50 s has an amplitude that decreases by 2.0% during each complete oscillation. If the initial amplitude is 10 cm, what will be the amplitude after 25 oscillations?

Problem 69P A spring with spring constant 15.0 N/m hangs from the ceiling. A 500 g ball is attached to the spring and allowed to come to rest. It is then pulled down 6.0 cm and released. What is the time constant if the ball’s amplitude has decreased to 3.0 cm after 30 oscillations?

Problem 71P A 200 g oscillator in a vacuum chamber has a frequency of 2.0 Hz. When air is admitted, the oscillation decreases to 60% of its initiai amplitude in 50 s. How many oscillations will have been completed when the amplitude is 30% of its initial value?

Problem 70P A 250 g air-track glider is attached to a spring with spring constant 4.0 N/m. The damping constant due to air resistance is 0.015 kg/s. The glider is pulled out 20 cm from equilibrium and released. How many oscillations will it make during the time in which the amplitude decays to e?1 of its initial value?

PProve that the expression for \(x(t)\) in Equation 14.55 is a solution to the equation of motion for a damped oscillator, Equation 14.54, if and only if the angular frequency \(\omega\) is given by the expression in Equation 14.56. Equation Transcription: Text Transcription: x(t) watt

A block on a frictionless table is connected as shown in FIGURE P14.73 to two springs having spring constants \(k_{1}\) and \(k_{2}\). Show that the block's oscillation frequency is given by \(f=\sqrt{f_{1}^{2}+f_{2}^{2}}\) where \(f_{1}\) and \(f_{2}\) are the frequencies at which it would oscillate if attached to spring 1 or spring 2 alone. Equation Transcription: Text Transcription: k_1 k_2 f= square root f_1^2+f_2^2 f_1 f_2

Problem 76CP 76 A 1.00 kg block is attached to a horizontal spring with spring constant 2500 N/m. The block is at rest on a frictionless surface. A 10 g bullet is fired into the block, in the face opposite the spring, and slicks. What was the bullet’s speed if the subsequent oscillations have an amplitude of 10.0 cm?

A block on a frictionless table is connected as shown in FIGURE P14.74 to two springs having spring constants \(k_{1}\) and \(k_{2}\). Find an expression for the block's oscillation frequency \(f\) in terms of the frequencies \(f_{1}\) and \(f_{2}\) at which it would oscillate if attached to spring 1 or spring 2 alone. Equation Transcription: Text Transcription: k_1 k_2 F f_1 f_2

Problem 77CP A spring is standing upright on a table with its bottom end fastened to the table. A block is dropped from a height 3.0 cm above the top of the spring. The block sticks to the top end of the spring and then oscillates with an amplitude of 10 cm. What is the oscillation frequency?

Problem 75CP A block hangs in equilibrium from a vertical spring. When a second identical block is added, the original block sags by 5.0 cm. What is the oscillation frequency of the two-block system?

Problem 78CP The analysis of a simple pendulum assumed that the mass was a particle, with no size. A realistic pendulum is a small, uniform sphere of mass M and radius R at the end of a massless string, with L being the distance from the pivot to me center of the sphere. a. Find an expression for the period of mis pendulum. ________________ b. Suppose M = 25 g, R = 1.0 cm, and L = 1.0 m, typical values for a real pendulum. What is the ratio T real /T simple, where T real is your expression from part a and T simple is the expression derived in this chapter?