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Multiple Choice

A mass-spring system with an angular frequency ω = 8π rad/s oscillates back and forth. (a) Assuming it starts from rest, how much time passes before the mass has a speed of 0 again? (b) How many full cycles does the system complete in 60s?

A 4-kg mass on a spring is released 5 m away from equilibrium position and takes 1.5 s to reach its equilibrium position. (a) Find the spring's force constant. (b) Find the object's max speed.

What is the equation for the position of a mass moving on the end of a spring which is stretched 8.8cm from equilibrium and then released from rest, and whose period is 0.66s? What will be the object's position after 1.4s?

An air-track glider attached to a spring oscillates between the 50 cm mark and the 62 cm mark. It completes seven oscillations in 10 s. What is the maximum speed of the glider as it oscillates?

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Multiple Choice

A certain astronaut oscillates back and forth on a chair attached to springs. The spring constants are such that when her mass is known to be

55 . 0 kg,

she oscillates with a period of

2 . 85 s .

After some time in space, a repetition of the measurement yields a period of

2 . 75 s .

What is the astronaut's mass now?

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Multiple Choice

A 250 g air-track glider attached to a spring oscillates with maximum speed of

0.97 \frac{m}{s}

. If the spring has a spring constant of

450 \frac{N}{m}

, at what distance from the equilibrium position will the glider have a speed of

0.43 \frac{m}{s}

Carbon dioxide is a linear molecule. The carbon–oxygen bonds in this molecule act very much like springs. Figure 14–45 shows one possible way the oxygen atoms in this molecule can oscillate: the oxygen atoms oscillate symmetrically in and out, while the central carbon atom remains at rest. Hence each oxygen atom acts like a simple harmonic oscillator with a mass equal to the mass of an oxygen atom. It is observed that this oscillation occurs with a frequency of ƒ = 2.83 x 10¹³ Hz. What is the spring constant of the C O bond?

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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 (b) from x = 0.090 m to x = -0.090 m?

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Textbook Question

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. (b) from x = 0.160 m to x = 0.

(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 person's vocal cords to vibrate through one complete cycle, and what is the angular frequency of the cords?

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Textbook Question

A 0.500-kg mass on a spring has velocity as a function of time given by vx(t) = -(3.60 cm/s) sin[(4.71 rad/s)t - (pi/2)]. What are (a) the period; (b) the amplitude; (c) the maximum acceleration of the mass; (d) the force constant of the spring?

725

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Textbook Question

A 0.500-kg mass on a spring has velocity as a function of time given by vx(t) = -(3.60 cm/s) sin[(4.71 rad/s)t - (pi/2)]. What are (a) the period; (b) the amplitude; (c) the maximum acceleration of the mass; (d) the force constant of the spring?

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Textbook Question

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?

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?

812

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Textbook Question

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?

1628

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Textbook Question

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.

650

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Textbook Question

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.

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 spring's force constant.

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Textbook Question

A 2.40-kg ball is attached to an unknown spring and allowed to oscillate. Figure E14.7

shows a graph of the ball's position x as a function of time t. What are the oscillation's (a) period, (b) frequency, (c) angular frequency, and (d) amplitude? (e) What is the force constant of the spring?

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Textbook Question

A 2.40-kg ball is attached to an unknown spring and allowed to oscillate. Figure E14.7

shows a graph of the ball's position x as a function of time t. What are the oscillation's (a) period, (b) frequency, (c) angular frequency, and (d) amplitude? (e) What is the force constant of the spring?

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Textbook Question

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 bird's wings, (b) the frequency of the wings' vibration, and (c) the angular frequency of the bird's wing beats.

473

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Textbook Question

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 ?

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Textbook Question

The displacement of an oscillating object as a function of time is shown in Fig. E14.4

. What is (a) the frequency? (b) the amplitude? (c) the period? (d) the angular frequency of this motion?

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Textbook Question

The displacement of an oscillating object as a function of time is shown in Fig. E14.4

. What is (a) the frequency? (b) the amplitude? (c) the period? (d) the angular frequency of this motion?

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Textbook Question

(II) Consider two objects, A and B, both undergoing SHM, but with different frequencies, as described by the equations 𝓍ₐ = (2.0 m) sin (4.0 t) and 𝓍₆ = (5.0 m) sin (3.0 t) , where t is in seconds. After t = 0, find the next three times t at which both objects simultaneously pass through the origin.

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Textbook Question

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. b. What is her speed when the spring's length is 1.2 m?

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Textbook Question

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 c/m when 𝓍 = ─5.0 cm c. What is the block's position when the acceleration is maximum?

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Textbook Question

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 kg/m³) 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. A vibrating cantilever of mass M can be modeled as a block of mass ⅓M attached to a spring. (The factor of ⅓ 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?

388

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Textbook Question

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%?

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Textbook Question

A 1.0 kg block is attached to a spring with spring constant 16 N/m. While the block is sitting at rest, a student hits it with a hammer and almost instantaneously gives it a speed of 40 cm/s. What are b. The block's speed at the point where 𝓍 = ½A?

1005

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Textbook Question

The position of a 50 g oscillating mass is given by 𝓍(t) = (2.0 cm) cos (10 t ─ π/4), where t is in s. Determine: h. The velocity at t = 0.40 s.

718

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Textbook Question

The position of a 50 g oscillating mass is given by 𝓍(t) = (2.0 cm) cos (10 t ─ π/4), where t is in s. Determine: e. The initial conditions.

363

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Textbook Question

A 200 g mass attached to a horizontal spring oscillates at a frequency of 2.0 Hz. At t = 0 s, the mass is at x = 5.0 cm and has vₓ = ─30 cm/s. Determine: h. The position at t = 0.40 s.

484

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Textbook Question

A 200 g mass attached to a horizontal spring oscillates at a frequency of 2.0 Hz. At t = 0 s, the mass is at x = 5.0 cm and has vₓ = ─30 cm/s. Determine: e. The maximum speed.

2028

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Textbook Question

A block attached to a spring with unknown spring constant oscillates with a period of 2.0 s. What is the period if c. The amplitude is doubled?

447

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Textbook Question

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 sticks. What was the bullet's speed if the subsequent oscillations have an amplitude of 10.0 cm?

520

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Textbook Question

A block attached to a spring with unknown spring constant oscillates with a period of 2.0 s. What is the period if d. The spring constant is doubled? Parts a to d are independent questions, each referring to the initial situation.

449

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Textbook Question

A 200 g air-track glider is attached to a spring. The glider is pushed in 10 cm and released. A student with a stopwatch finds that 10 oscillations take 12.0 s. What is the spring constant?

407

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Textbook Question

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?

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?

955

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Textbook Question

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. b. What is the glider's position at t = 0.25 s?

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Textbook Question

A 200 g mass attached to a horizontal spring oscillates at a frequency of 2.0 Hz. At t = 0 s, the mass is at x = 5.0 cm and has vₓ = ─30 cm/s. Determine: g. The total energy.

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Textbook Question

(II) A small fly of mass 0.28 g is caught in a spider’s web. The web oscillates predominantly with a frequency of 4.0 Hz.

(b) At what frequency would you expect the web to oscillate if an insect of mass 0.46 g is trapped?

219

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Textbook Question

(II) A tuning fork oscillates at a frequency of 441 Hz and the tip of each prong moves 1.8 mm to either side of center. Calculate

(a) the maximum speed ..

219

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Textbook Question

(II) A tuning fork oscillates at a frequency of 441 Hz and the tip of each prong moves 1.8 mm to either side of center. Calculate

(b) the maximum acceleration of the tip of a prong.

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Textbook Question

(II) A 0.25-kg mass at the end of a spring oscillates 3.2 times per second with an amplitude of 0.15 m. Determine

(b) the speed when it is 0.10 m from equilibrium.

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Textbook Question

(II) At t = 0, an 885-g mass at rest on the end of a horizontal spring (k = 184 N/m) is struck by a hammer which gives it an initial speed of 2.12 m/s. Determine

(a) the period and frequency of the motion,

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Textbook Question

(II) At t = 0, an 885-g mass at rest on the end of a horizontal spring (k = 184 N/m) is struck by a hammer which gives it an initial speed of 2.12 m/s. Determine

(b) the amplitude,

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Textbook Question

(II) A 0.25-kg mass at the end of a spring oscillates 3.2 times per second with an amplitude of 0.15 m. Determine

(d) the equation describing the motion of the mass, assuming that at t = 0, 𝓍 was a maximum.

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Textbook Question

(III) A glider on an air track is connected by springs to either end of the track (Fig. 14–41). Both springs have the same spring constant, k, and the glider has mass M.

(a) Determine the frequency of the oscillation, assuming no damping, if k = 125 N/m and M = 215 g.

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Textbook Question

(III) A mass m is at rest on the end of a spring of spring constant k. At t = 0 it is given an impulse J by a hammer. Write the formula for the subsequent motion in terms of m, k, J, and t.

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Textbook Question

(II) At t = 0, an 885-g mass at rest on the end of a horizontal spring (k = 184 N/m) is struck by a hammer which gives it an initial speed of 2.12 m/s. Determine

(c) the maximum acceleration,

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Textbook Question

(II) At t = 0, an 885-g mass at rest on the end of a horizontal spring (k = 184 N/m) is struck by a hammer which gives it an initial speed of 2.12 m/s. Determine

(d) the position as a function of time,

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Textbook Question

(II) The graph of displacement vs. time for a small mass m at the end of a spring is shown in Fig. 14–30. At t = 0 , 𝓍 = 0.43 cm.

(a) If m= 7.7 g , find the spring constant, k.

(b) Write the equation for displacement 𝓍 as a function of time.

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Textbook Question

An oxygen atom at a particular site within a DNA molecule can be made to execute simple harmonic motion when illuminated by infrared light. The oxygen atom is bound with a spring-like chemical bond to a phosphorus atom, which is rigidly attached to the DNA backbone. The oscillation of the oxygen atom occurs with frequency ƒ = 3.7 x 10¹³ Hz. If the oxygen atom at this site is chemically replaced with a sulfur atom, the spring constant of the bond is unchanged (sulfur is just below oxygen in the Periodic Table). Predict the frequency after the sulfur substitution.

191

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Textbook Question

(II) Determine the phase constant ϕ in Eq. 14–4 if, at t = 0 , the oscillating mass is at

(c) 𝓍 = A ,

182

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Textbook Question

(II) Determine the phase constant ϕ in Eq. 14–4 if, at t = 0 , the oscillating mass is at

(a) 𝓍 = ― A

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