|| For a particle in simple harmonic motion, show that $v_{\text{max}}=\left(\pi \text{/}2\right)v_{\text{avg}}$, where $v_{\text{avg}}$ is the average speed during one cycle of the motion.

1. || 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?

2. At what displacement, as a fraction of A, is the energy half kinetic and half potential?

|| The motion of a particle is given by $x\left(t\right)=(25\text{cm})\text{cos}\left(10t\right)$, where t is in s. What is the first time at which the kinetic energy is twice the potential energy?

|| A 100 g block attached to a spring with spring constant $2.5\text{N}\text{/}\text{m}$ oscillates horizontally on a frictionless table. Its velocity is $20\text{cm}\text{/}\text{s}$ when $x=-5.0\text{cm}$.

1. What is the amplitude of oscillation?

2. What is the block’s maximum acceleration?

3. What is the block’s position when the acceleration is maximum?

4. What is the speed of the block when $x=3.0\text{cm}?$

|| INT A 500 g mass on a horizontal spring oscillates with a period of 2.0 s and an amplitude of 75 cm. During one oscillation, at the instant the spring is at maximum extension, the mass explodes into two pieces. One piece is shot forward, away from the spring, at 2.0 m/s. The second piece, which remains attached to the spring, recoils at 3.0 m/s. After the explosion, what are the (a) period and (b) amplitude of the oscillation?

|| BIO An ultrasonic transducer, of the type used in medical ultrasound imaging, is a very thin disk $(m=0.10\text{g})$ driven back and forth in SHM at 1.0 MHz by an electromagnetic coil.

1. 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?

2. What is the disk’s maximum speed at this amplitude?

|| BIO 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 P15.46.

1. What is her mass if the spring constant is $240\text{N}\text{/}\text{m}?$

2. What is her speed when the spring’s length is 1.2 m?

FIGURE P15.46

|| 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:

Use the best-fit line of an appropriate graph to determine the spring constant.

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

||| A 200 g block hangs from a spring with spring constant $10\text{N}\text{/}\text{m}$. At $t=0\text{s}$ the block is 20 cm below the equilibrium point and moving upward with a speed of $100\text{cm}\text{/}\text{s}$. What are the block’s

1. Oscillation frequency?

2. Distance from equilibrium when the speed is $50\text{cm}\text{/}\text{s}?$

3. Distance from equilibrium at $t=1.0\text{s}?$

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

|| INT A 500 g wood block on a frictionless table is attached to a horizontal spring. A 50 g dart is shot into the face of the block opposite the spring, where it sticks. Afterward, the spring oscillates with a period of 1.5 s and an amplitude of 20 cm. How fast was the dart moving when it hit the block?

|| INT Scientists are measuring the properties of a newly discovered elastic material. They create a 1.5-m-long, 1.6-mm-diameter cord, attach an 850 g mass to the lower end, then pull the mass down 2.5 mm and release it. Their high-speed video camera records 36 oscillations in 2.0 s. What is Young’s modulus of the material?

||| INT A mass hanging from a spring oscillates with a period of 0.35 s. Suppose the mass and spring are swung in a horizontal circle, with the free end of the spring at the pivot. What rotation frequency, in rpm, will cause the spring’s length to stretch by 15%?

|| 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.

1. What is the spring constant of each spring if the empty car bounces up and down 2.0 times each second?

2. What will be the car’s oscillation frequency while carrying four 70 kg passengers?

|| INT The two blocks in FIGURE P15.55 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?

FIGURE P15.55

|| INT A 1.00 kg block is attached to a horizontal spring with spring constant $2500\text{N}\text{/}\text{m}$. 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 sticks. What was the bullet’s speed if the subsequent oscillations have an amplitude of 10.0 cm?

| BIO 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?

||| BIO It has recently become possible to “weigh” DNA molecules by measuring the influence of their mass on a nano-oscillator. FIGURE P15.58 shows a thin rectangular cantilever etched out of silicon (density $2300\text{kg}\text{/}{\text{m}}^3)$ with a small gold dot (not visible) at the end. If pulled down and released, the end of the cantilever vibrates with SHM, 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.

FIGURE P15.58

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 MHz. Attachment of a DNA molecule caused the frequency to decrease by 50 Hz. What was the mass of the DNA?

|| BIO Orangutans can move by brachiation, swinging like a pendulum beneath successive handholds. If an orangutan has arms that are 0.90 m long and repeatedly swings to a $20\text{°}$ angle, taking one swing after another, estimate its speed of forward motion in m/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.

|| 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.

1. How long is the lamp chain?

2. What maximum speed does the lamp have if its maximum angle from vertical is $3.0\text{°}?$

|| INT The pendulum shown in FIGURE P15.61 is pulled to a $10\text{°}$ angle on the left side and released.

1. What is the period of this pendulum?

2. What is the pendulum’s maximum angle on the right side?

FIGURE P15.61

|| INT A uniform rod of mass $M$ and length $L$ swings as a pendulum on a pivot at distance $L\text{/}4$ from one end of the rod. Find an expression for the frequency $f$ of small-angle oscillations.

|| INT A 500 g air-track glider attached to a spring with spring constant $10\text{N}\text{/}\text{m}$ 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 $120\text{cm}\text{/}\text{s}$. It collides with and sticks to the 500 g glider. What are the amplitude and period of the subsequent oscillations?

|| A 25-cm-long massless spring is suspended from a pivot. A ball attached to the spring’s lower end is initially at rest. The ball is gently pushed slightly to one side and then swings as a pendulum at the end of the spring with a period of 1.8 s. The ball is stopped, then pulled down slightly and allowed to oscillate vertically. What is the period of oscillation?

|| FIGURE P15.65 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.

FIGURE P15.65

|| INT A molecular bond can be modeled as a spring between two atoms that vibrate with simple harmonic motion. FIGURE P15.66 shows an SHM approximation for the potential energy of an HCl molecule. Because the chlorine atom is so much more massive than the hydrogen atom, it is reasonable to assume that the hydrogen atom $(\text{m}=1.67\times {10}^{-27}\text{kg})$ vibrates back and forth while the chlorine atom remains at rest. Use the graph to estimate the vibrational frequency of the HCl molecule.

FIGURE P15.66

|| INT 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.

1. At what point in the cycle does the penny first lose contact with the piston?

2. What is the maximum frequency for which the penny just barely remains in place for the full cycle?

|| 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?

||| A $15\text{-}\text{cm}\text{-}\text{long}\text{,}200\text{g}$ 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?

|| The 15 g head of a bobble-head doll oscillates in SHM at a frequency of 4.0 Hz.

1. What is the spring constant of the spring on which the head is mounted?

2. The amplitude of the head’s oscillations decreases to 0.5 cm in 4.0 s. What is the head’s damping constant?

|| 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?

||| 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?

|| 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 initial amplitude in 50 s. How many oscillations will have been completed when the amplitude is 30% of its initial value?

|| Prove that the expression for $x\left(t\right)$ in Equation 15.55 is a solution to the equation of motion for a damped oscillator, Equation 15.54, if and only if the angular frequency $\omega$ is given by the expression in Equation 15.56.

|| A block on a frictionless table is connected as shown in FIGURE P15.75 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.

FIGURE P15.75

||| Suppose a large spherical object, such as a planet, with radius R and mass M has a narrow tunnel passing diametrically through it. A particle of mass m is inside the tunnel at a distance $x\le R$ from the center. It can be shown that the net gravitational force on the particle is due entirely to the sphere of mass with radius $r\le x;$ there is no net gravitational force from the mass in the spherical shell with $r>x$.

1. Find an expression for the gravitational force on the particle, assuming the object has uniform density. Your expression will be in terms of x, R, m, M, and any necessary constants.

2. You should have found that the gravitational force is a linear restoring force. Consequently, in the absence of air resistance, objects in the tunnel will oscillate with SHM. Suppose an intrepid astronaut exploring a 150-km-diameter, $3.5\times {10}^{18}\text{kg}$ asteroid discovers a tunnel through the center. If she jumps into the hole, how long will it take her to fall all the way through the asteroid and emerge on the other side?

||| A solid sphere of mass $M$ and radius $R$ is suspended from a thin rod, as shown in FIGURE CP15.77. The sphere can swing back and forth at the bottom of the rod. Find an expression for the frequency $f$ of small-angle oscillations.

FIGURE CP15.77

||| CALC A uniform rod of length L oscillates as a pendulum about a pivot that is a distance x from the center.

1. For what value of x, in terms of L, is the oscillation period a minimum?

2. What is the minimum oscillation period of a 15 kg, 1.0-m-long steel bar?

||| 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?

||| The Timoshenko oscillator is an ingenious device to measure coefficients of kinetic friction. FIGURE CP15.80 shows a plate of mass m supported by two rollers a distance L apart. The rollers spin at a high speed in opposite directions, as shown, so that they are slipping against the bottom of the plate with coefficient of kinetic friction ${\mu }_{\text{k}}$. Let x measure the position of the plate’s center of mass relative to the midpoint between the rollers. Newton’s second law for the plate leads to a SHM equation for x and predicts the oscillation frequency in terms of m, L, and ${\mu }_{\text{k}}$. Suppose a 250 g plate oscillates with a 1.5 s period on two spinning rollers 12 cm apart. What is the coefficient of kinetic friction between the plate and the rollers?

FIGURE CP15.80

||| FIGURE CP15.81 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.

FIGURE CP15.81

||| The greenhouse-gas carbon dioxide molecule $\text{CO}{}_2$ strongly absorbs infrared radiation when its vibrational normal modes are excited by light at the normal-mode frequencies. $\text{CO}{}_2$ is a linear triatomic molecule, as shown in FIGURE CP15.82, with oxygen atoms of mass $m_{\text{O}}$ bonded to a central carbon atom of mass $m_{\text{C}.}$ You know from chemistry that the atomic masses of carbon and oxygen are, respectively, 12 and 16. Assume that the bond is an ideal spring with spring constant k. There are two normal modes of this system for which oscillations take place along the axis. (You can ignore additional bending modes.) In this problem, you will find the normal modes and then use experimental data to determine the bond spring constant.

FIGURE CP15.82

1. Let $x_1$, $x_2$, and $x_3$ be the atoms’ positions measured from their equilibrium positions. First, use Hooke’s law to write the net force on each atom. Pay close attention to signs! For each oxygen, the net force equals $m_{\text{O}}{d}^{2}x/d{t}^2$. Carbon has a different mass, so its net force is $m_{\text{C}}{d}^{2}x/d{t}^2$. Define ${\alpha }^2=k/m_{\text{O}}$ and ${\beta }^2=k/m_{\text{C}}$, then write three equations for the second derivatives of the three coordinates.

The main task is to find three new variables whose differential equation is the SHM equation, meaning that they oscillate at a single frequency that you can identify. The first is $w=x_1+(m_{\text{C}}/m_{\text{O}})x_2+x_3$. Notice that the ratio $m_{\text{C}}/m_{\text{O}}={\alpha }^2/{\beta }^2$. If you calculate $d^{2}w/d{t}^2$, you’ll find that it equals zero. (Check this to verify that your work is correct to this point.) This seems odd, but notice that multiplying w by $m_{\text{O}}$ gives an expression for the center of mass. So what you’ve learned is that the center of mass doesn’t accelerate. It could be drifting through space, but for this problem you can let the center of mass be at rest at the origin. So the solution to $d^{2}w/d{t}^2=0$ is $w=0$.

2. A second new variable (call it u) can be formed from only $x_1$ and $x_3$, the oxygen coordinates. You’re looking for a combination such that $d^{2}u/d{t}^2=-\text{constant}\times u$. Determine u and find an expression for its angular frequency (call it $\omega$) in terms of k and the masses. The solution to the equation for u is $u=2A\text{cos}(\omega t+{\varphi }_{\text{a}})$.

3. The third variable (call it v) is $v=x_1-c{x}_2+x_3$, where c is a constant you need to determine so that the equation for v is the SHM equation. Determine v and find an expression for its angular frequency (call it $\text{Ω}$) in terms of k and the masses. The solution to the equation for v is $v=2B\text{cos}(\text{Ω}t+{\varphi }_{\text{b}})$.

4. Use the definitions of u, v, and w to solve for $x_1$, $x_2$, and $x_3$ in terms of u, v, and w. Using $M=2m_{\text{O}}+m_{\text{C}}$ as the total mass of the molecule will keep the expressions simple. This is a general solution for the positions of the three masses.

5. Let $B=0$ and ${\varphi }_{\text{a}}={\varphi }_{\text{b}}=0$. This gives a solution that oscillates at frequency $\omega$. Find expressions for $x_1$, $x_2$, and $x_3$. You should find that this normal mode at frequency $\omega$ consists of the carbon atom sitting at rest while the oxygen atoms oscillate in and out in opposite directions. This is called the symmetric stretch mode.

6. Now let $A=0$ and ${\varphi }_{\text{a}}={\varphi }_{\text{b}}=0$. This gives a solution that oscillates at frequency $\text{Ω}$. Find expressions for $x_1$, $x_2$, and $x_3$. This normal mode at frequency $\text{Ω}$ consists of the carbon atom moving in one direction while the two oxygen atoms move in the opposite direction with a different amplitude. This is called the antisymmetric stretch mode. Notice that the center of mass remains stationary.

7. The symmetric stretch frequency is known to be $4.00\times {10}^{13}\text{Hz}$. What is the spring constant of the $\text{C}-\text{O}$ bond? Use $1\text{ u}=1\text{ atomic mass unit}=1.66\times {10}^{-27}\text{kg}$ to find the atomic masses in SI units. Interestingly, the spring constant is similar to that of springs you might use in the lab.

8. Use the frequency of the symmetric stretch to predict the frequency of the antisymmetric stretch. The measured frequency is $7.05\times {10}^{13}\text{Hz,}$ so your prediction is close but not perfect. The reason is that the bonds are not ideal springs but have a slight amount of anharmonicity. Nonetheless, you’ve learned a great deal about the $\text{CO}{}_2$ molecule from a simple model of oscillating masses.