Mastering Physics Solutions Chapter 13 Oscillations About Equilibrium
Chapter 13 Oscillations About Equilibrium Q.1CQ
A basketball player dribbles a ball with a steady period of T seconds. Is the motion of the ball periodic? Is it simple harmonic? Explain.
Solution:
The motion of a particle, which is repeated in position and phase after a certain interval of time. is defined as periodic motion The periodic motion, in which a particle moves to and fr0 about a fixed point such that its acceleration is always directly proportional to its displacement from its mean position. is defined as simple harmonic motion. Since the player dribbles the ball with constant period, the motion is periodic. On the other hand. the position and velocity of the ball do not vary sinusoidally with time, instead it varies linearly. So the motion is not a simple harmonic motion.
Chapter 13 Oscillations About Equilibrium Q.1P
A small cart on a 5.0-m-long air track moves with a speed of 0.85 m/s. Bumpers at either end of the track cause the cart to reverse direction and maintain the same speed. Find the period and frequency of this motion.
Solution:
Chapter 13 Oscillations About Equilibrium Q.2CQ
A person rides on a Ferris wheel that rotates with constant angular speed If the Sun is directly overhead. does the person’s shadow on the ground undergo periodic motion? Does it undergo simple harmonic motion? Explain.
Solution:
The persons shadow undergoes periodic motioni with the same period as the period of the Ferris wheel’s rotation In fact, if we take into account the connection between uniform circular motion and simple harmonic motion, we can say that the shadow exhibits simple harmonic motion as it moves back and forth on the ground
Chapter 13 Oscillations About Equilibrium Q.2P
A person in a rocking chair completes 12 cycles in 21 s. What are the period and frequency of the rocking?
Solution:
Chapter 13 Oscillations About Equilibrium Q.3CQ
An air-track cart bounces back and forth between the two ends of an air track. Is this motion periodic? Is it simple harmonic? Explain.
Solution:
Chapter 13 Oscillations About Equilibrium Q.3P
While fishing for catfish, a fisherman suddenly notices that the bobber (a floating device) attached to his line is bobbing up and down with a frequency of 2.6 Hz. What is the period of the bobber’s motion?
Solution:
Chapter 13 Oscillations About Equilibrium Q.4CQ
If a mass m and a mass 2m oscillate on identical springs with identical amplitudes, they both have the same maximum kinetic energy. How can this be? Shouldn’t the larger mass have more kinetic energy? Explain.
Solution:
Chapter 13 Oscillations About Equilibrium Q.4P
If you dribble a basketball with a frequency of 1.77 Hz, how long does it take for you to complete 12 dribbles?
Solution:
Chapter 13 Oscillations About Equilibrium Q.5CQ
An object oscillating with simple harmonic motion completes a cycle in a time T. If the object’s amplitude is doubled, the time required for one cycle is still T, even though the object covers twice the distance. How can this be? Explain.
Solution:
Chapter 13 Oscillations About Equilibrium Q.5P
You take your pulse and observe 74 heartbeats in a minute. What are the period and frequency of your heartbeat?
Solution:
Chapter 13 Oscillations About Equilibrium Q.6CQ
The position of an object undergoing simple harmonic motion is given by x =A cos(Bt). Explain the physical significance of the constants A and B. What is the frequency of this object’s motion?
Solution:
Chapter 13 Oscillations About Equilibrium Q.6P
(a) Your heart beats wi th a frequency of 1.45 Hz. How many beats occur in a minute? (b) If the frequency of your heartbeat increases, will the number of beats in a minute increase, decrease, or stay the same? (c) How many beats occur in a minute if the frequency increases to 1.55 Hz?
Solution:
Chapter 13 Oscillations About Equilibrium Q.7CQ
The velocity of an object undergoing simple harmonic motion is given by v = −C sin(Dt). Explain the physical significance of the constants C and D. What are the amplitude and period of this object’s motion?
Solution:
Chapter 13 Oscillations About Equilibrium Q.7P
You rev your car’s engine to 2700 rpm (rev/min). (a) What are the period and frequency of the engine? (b) If you change the period of the engine to 0.044 s, how many rpms is it doing?
Solution:
Frequency is number of cycles (or revolution) per second. Time is period is the time take to complete one cycle.
Given that
Frequency of car’s engine = 2700 rpm
Chapter 13 Oscillations About Equilibrium Q.8CQ
The pendulum bob in Figure leaks sand onto the strip chart. What effect does this loss of sand have on the period of the pendulum? Explain.
Solution:
The period of a pendulum is independent of the mass of its bob. Therefore, the period should be unaffected.
Chapter 13 Oscillations About Equilibrium Q.8P
A mass moves back and forth in simple harmonic motion with amplitude A and period T.(a) In terms of A,through what distance does the mass move in the time T? (b) Through what distance does it move in the time 5T/2?
Solution:
Chapter 13 Oscillations About Equilibrium Q.9CQ
Soldiers on the march are often ordered to break cadence in their step when crossing a bridge. Why is this a good idea?
Solution:
The soldiers are marching on the bridge. If the natural frequency of the bridge is equal to the frequency of the soldiers, then the bridge vibrates with greater amplitude because of resonance. This may cause the bridge to collapse at once. Because of this, all soldiers are ordered to break cadence in their steps when crossing a bridge.
Chapter 13 Oscillations About Equilibrium Q.9P
A mass moves back and forth in simple harmonic motion with amplitude A and period T. (a) In terms of T, how long does it take for the mass to move through a total distance of 2A? (b) How long does it take for the mass to move through a total distance of 3A?
Solution:
Chapter 13 Oscillations About Equilibrium Q.10P
The position of a mass oscillating on a spring is given by x = (3.2 cm) cos[2πt/(0.58 s)]. (a) What is the period of this motion? (b) What is the first time the mass is at the position x = 0?
Solution:
Chapter 13 Oscillations About Equilibrium Q.11P
The position of a mass oscillating on a spring is given by x = (7.8 cm) cos[2πt/(0.68 s)] (a) What is the frequency of this motion? (b) When is the mass first at the position x = –7.8 cm?
Solution:
Chapter 13 Oscillations About Equilibrium Q.12
Solution:
Chapter 13 Oscillations About Equilibrium Q.13P
Amass on a spring osculates with simple harmonic motion of amplitude A about the equilibrium position x = 0. Its maximum speed is vmax and its maximum acceleration is amax. (a) What is the speed of the mass at x =0? (b) What is the acceleration of the mass at x =0? (c) What is the speed of the mass at x = A? (d) What is the acceleration of the mass at x = A?
Solution:
Chapter 13 Oscillations About Equilibrium Q.14P
A mass oscillates on a spring with a period of 0.73 s and an amplitude of 5.4 cm. Write an equation giving x as a function of time, assuming the mass starts at x = A at time t = 0.
Solution:
Chapter 13 Oscillations About Equilibrium Q.15P
An atom in a molecule oscillates about its equilibrium position with a frequency of 2.00 × 1014 Hz and a maximum displacement of 3.50 nm. (a) Write an expression giving x as a function of time for this atom, assuming that x = A at t = 0. (b) If, instead, we assume that x = 0 at t =0, would your expression for position versus time use a sine function or a cosine function? Explain.
Solution:
Chapter 13 Oscillations About Equilibrium Q.16P
A mass oscillates on a spring with a period T and an amplitude 0.48 cm. The mass is at the equilibrium position x = 0 at t = 0, and is moving in the positive direction. Where is the mass at the times (a) t = T/8, (b) t = T/4, (c) t = T/2and (d) t = 3T/4? (e) Plot your results for parts (a) through (d) with the vertical axis representing position and the horizontal axis representing time.
Solution:
Chapter 13 Oscillations About Equilibrium Q.17P
The position of a mass on a spring is given by x = (6.5 cm) cos[2πt/(0.88 s)]. (a) What is the period, T, of this motion? (b) Where is the mass at t = 0.25 s? (c) Show that the mass is at the same location at 0.25 s + T seconds as it is at 0.25 s.
Solution:
Chapter 13 Oscillations About Equilibrium Q.18P
A mass attached to a spring oscillates with a period of 3.35 s. (a) If the mass starts from rest at x = 0.0440 m and time t = 0, where is it at time t = 6.37 s? (b) Is the mass moving in the positive or negative x direction at t =6.37 s? Explain.
Solution:
Chapter 13 Oscillations About Equilibrium Q.19P
An object moves with simple harmonic motion of period T and amplitude A. During one complete cycle, for what length of time is the position of the object greater than A/2?
Solution:
Chapter 13 Oscillations About Equilibrium Q.20P
An object moves with simple harmonic motion of period T and amplitude A. During one complete cycle, for what length of time is the speed of the object greater than vmax/2?
Solution:
Chapter 13 Oscillations About Equilibrium Q.21P
An object executing simple harmonic motion has a maximum speed vmax and a maximum acceleration amax. Find (a) the amplitude and (b) the period of this motion. Express your answers in terms of vmax and amax.
Solution:
Chapter 13 Oscillations About Equilibrium Q.22P
A ball rolls on a circular track of radius 0.62 m with a constant angular speed of 1.3 rad/s inthe counterclockwise direction. If the angular position of the ball at t = 0 is θ = 0, find the x component of the ball’s position at the times 2.5 s, 5.0 s, and 7.5 s. Let θ = 0 correspond to the positive x direction.
Solution:
Chapter 13 Oscillations About Equilibrium Q.23P
An object executing simple harmonic motion has a maximum speed of 4.3 m/s and a maximum acceleration of 0.65 m/s2. Find (a) the amplitude and (b) the period of this motion.
Solution:
Chapter 13 Oscillations About Equilibrium Q.24P
A child rocks back and forth on a porch swing with an amplitude of 0.204 m and a period of 2.80 s. Assuming the motion is approximately simple harmonic, find the child’s maximum speed.
Solution:
Chapter 13 Oscillations About Equilibrium Q.25P
A 30.0-g goldfinch lands on a slender branch, where it oscillates up and down with simple harmonic motion of amplitude 0.0335 m and period 1.65 s. (a) What is the maximum acceleration of the finch? Express your answer as a fraction of the acceleration of gravity, g. (b)What is the maximum speed of the goldfinch? (c) At the time when the goldfinch experiences its maximum acceleration, is its speed a maximum or a minimum? Explain.
Solution:
c) For the maximum acceleration, the velocity in a simple harmonic motion becomes zero.
Chapter 13 Oscillations About Equilibrium Q.26P
Solution:
Chapter 13 Oscillations About Equilibrium Q.27P
A vibrating structural beam in a spacecraft can cause problems if the frequency of vibration is fairly high. Even if the amplitude of vibration is only a fraction of a millimeter, the acceleration of the beam can be several times greater than the acceleration due to gravity. As an example, find the maximum acceleration of a beam that vibrates with an amplitude of 0.25 mm at the rate of 110 vibrations per second. Give your answer as a multiple of g.
Solution:
Chapter 13 Oscillations About Equilibrium Q.28P
A peg on a turntable moves with a constant tangential speed of 0.77 m/s in a circle of radius 0.23 m. The peg casts a shadow on a wall. Find the following quantities related to the motion of the shadow: (a) the period, (b) the amplitude, (c) the maximum speed, and (d) the maximum magnitude of the acceleration.
Solution:
Chapter 13 Oscillations About Equilibrium Q.29P
The pistons in an internal combustion engine undergo a motion that is approximately simple harmonic. If the amplitude of motion is 3.5 cm, and the engine runs at 1700 rev/min, find (a) the maximum acceleration of the pistons and (b) their maximum speed.
Solution:
Chapter 13 Oscillations About Equilibrium Q.30P
A 0.84-kg air cartis attached to a spring and allowed to oscillate. If the displacement of the air cart from equilibrium is x = (10.0 cm) cos[(2.00 s−1)t + π], find (a) the maximum kinetic energy of the cart and (b) the maximum force exerted on it by the spring.
Solution:
Chapter 13 Oscillations About Equilibrium Q.31P
Solution:
Chapter 13 Oscillations About Equilibrium Q.32P
If a mass m is attached to a given spring,. its period of oscillation is T. If two such springs are connected end to end and the same mass m is attached, (a) is the resulting period of oscillation greater than, less than, or equal to T? (b) Choose the best explanation from among the following:
I. Connecting two springs together makes the spring suffer, which means that less time is required for an oscillation.
II. The period of oscillation does not depend on the length of a spring, only on its force constant and the mass attached to it.
III. The longer spring stretches more easily, and hence takes longer to complete an oscillation.
Solution:
Chapter 13 Oscillations About Equilibrium Q.33P
An old car with worn-out shock absorbers oscillates with a given frequency when it hits a speed bump. If the driver adds a couple of passengers to the car and hits another speed bump, (a) is the car’s frequency of oscillation greater than, less than, or equal to what it was before? (b) Choose the best explanation from among the following:
I. Increasing the mass on a spring increases its period, and hence decreases its frequency.
II. The frequency depends on the force constant of the spring but is independent of the mass.
III. Adding mass makes the spring oscillate more rapidly, which increases the frequency.
Solution:
a) The car’s frequency of oscillation is less than to what it was before.
b) Increase in mass increases its period of oscillation of the car and hence frequency decreases. Option I is correct.
Chapter 13 Oscillations About Equilibrium Q.34P
Solution:
a) When both the blocks are set into oscillation, the period of the block 1 is equal to the period of block 2.
b) The two blocks experience the same restoring force for a given displacement from the equilibrium and hence they have equal periods of oscillation.
Chapter 13 Oscillations About Equilibrium Q.35P
Solution:
Chapter 13 Oscillations About Equilibrium Q.36P
A 0.46-kg mass attached to a spring undergoes simple harmonic motion with a period of 0.77 s. What is the force constant of the spring?
Solution:
Chapter 13 Oscillations About Equilibrium Q.37P
System A consists of a mass m attached to a spring with a force constant k;system B has a mass 2m attached to a spring with a force constant k;system C has a mass 3m attached to a spring with a force constant 6k; and system D has a mass m attached to a spring with a force constant 4k. Rank these systems in order of Increasing period of oscillation.
Solution:
Chapter 13 Oscillations About Equilibrium Q.38P
Solution:
Chapter 13 Oscillations About Equilibrium Q.39P
When a 0.50-kg mass is attached to a vertical spring, the spring stretches by 15 cm. How much mass must be attached to the spring to result in a 0.75-s period of oscillation?
Solution:
Chapter 13 Oscillations About Equilibrium Q.40P
A spring with a force constant of 69 N/m is attached to a 0.57-kg mass. Assuming that the amplitude of motion is 3.1 cm, determine the following quantities for this system: (a) ω, (b) vmax, (c) T.
Solution:
Chapter 13 Oscillations About Equilibrium Q.41P
Two people with a combined mass of 125 kg hop into an old car with worn-out shock absorbers. This causes the springs to compress by 8.00 cm. When the car hits a bump in the road, it oscillates up and down with a period of 1.65 s. Find (a) the total load supported by the springs and (b) the mass of the car.
Solution:
Chapter 13 Oscillations About Equilibrium Q.42P
A 0.85-kg mass attached to a vertical spring of force constant 150 N/m oscillates with a maximum speed of 0.35 m/s. Find the following quantities related to the motion of the mass: (a) the period, (b) the amplitude, (c) the maximum magnitude of the acceleration.
Solution:
Chapter 13 Oscillations About Equilibrium Q.43P
When a 0.213-kg mass is attached to a vertical spring, it causes the spring to stretch a distance d. If the mass is now displaced slightly from equilibrium, it is found to make 102 oscillations in 56.7 s. Find the stretch distance, d.
Solution:
Chapter 13 Oscillations About Equilibrium Q.44P
The springs of a 511-kg motorcycle have an effective force constant of 9130 N/m. (a) If a person sits on the motorcycle, does its period of oscillation increase, decrease, or stay the same? (b) By what percent and in what direction does the period of oscillation change when a 112-kg person rides the motorcycle?
Solution:
Chapter 13 Oscillations About Equilibrium Q.45P
If a mass m is attached to a given spring, its period of oscillation is T. If two such springs are connected end to end, and the same mass m is attached, (a) is its period greater than, less than, or the same as with a single spring? (b) Verify your answer to part (a) by calculating the new period, T’,in terms of the old period T.
Solution:
Chapter 13 Oscillations About Equilibrium Q.46P
How much work is required to stretch a spring 0.133 m if its force constant is 9.17 N/m?
Solution:
Chapter 13 Oscillations About Equilibrium Q.47P
A 0.321-kg mass is attached to a spring with a force constant of 13.3 N/m. If the mass is displaced 0.256 m from equilibrium and released, what is its speed when it is 0.128 m from equilibrium?
Solution:
Chapter 13 Oscillations About Equilibrium Q.48P
Find the total mechanical energy of the system described in the previous problem.
Solution:
Chapter 13 Oscillations About Equilibrium Q.49P
A 1.8-kg mass attached to aspring oscillates with an amplitude of 7.1 cm and a frequency of 2.6 Hz. What is its energy of motion?
Solution:
Chapter 13 Oscillations About Equilibrium Q.50P
A 0.40-kg mass is attached to a spring with a force constant of 26 N/m and released from rest a distance of 3.2 cm from the equilibrium position of the spring. (a) Give a strategy that allows you to find the speed of the mass when it is halfway to the equilibrium position. (b) Use your strategy to find this speed.
Solution:
Chapter 13 Oscillations About Equilibrium Q.51P
(a) What is the maximum speed of the mass in the previous problem? (b) How far is the mass from the equilibrium position when its speed is half the maximum speed?
Solution:
Chapter 13 Oscillations About Equilibrium Q.52P
A bunch of grapes is placed in a spring scale at a supermarket. The grapes oscillate up and down with a period of 0.48 s, and the spring in the scale has a force constant of 650 N/m. What are (a) the mass and (b) the weight of the grapes?
Solution:
a) The time period of a spring mass system is directly proportional to the square root of mass and inversely proportional to the square root of force constant of the spring. The time period is given by relation
Chapter 13 Oscillations About Equilibrium Q.53P
What is the maximum speed of the grapes in the previous problem if their amplitude of oscillation is 2.3 cm?
Solution:
Chapter 13 Oscillations About Equilibrium Q.54P
A 0.505-kg block slides on a frictionless horizontal surface with a speed of 1.18 m/s. The block encounters an unstretched spring and compresses it 23.2 cm before coming to rest. (a) What is the force constant of this spring? (b) For what length of time is the block in contact with the spring before it comes to rest? (c) If the force constant of the spring is increased, docs the time required to stop the block increase, decrease, or stay the same? Explain.
Solution:
Chapter 13 Oscillations About Equilibrium Q.55P
A 2.25-g bullet embeds itself in a 1.50-kg block, which is attached to a spring of force constant 785 N/m. If the maximum compression of the spring is 5.88 cm, find (a) the initialspeed of the bullet and (b) the time for the bullet-block system to come to rest.
Solution:
Chapter 13 Oscillations About Equilibrium Q.56P
Solution:
Chapter 13 Oscillations About Equilibrium Q.57P
A grandfather clock keeps correct time at sea level. If the clock is taken to the top of a nearby mountain, (a) would you expect it to keep correct time, run slow, or run fast? (b) Choose the best explanation from among the following:
I. Gravity is weaker at the top of the mountain, leading to a greater period oi oscillation.
II. The length of the pendulum is unchanged, and therefore its period remains the same.
III. The extra gravity from the mountain causes the period to decrease.
Solution:
Chapter 13 Oscillations About Equilibrium Q.58P
A pendulum of length L has a period T. How long must the pendulum be if its period is to be 2T?
Solution:
Chapter 13 Oscillations About Equilibrium Q.59P
Solution:
Chapter 13 Oscillations About Equilibrium Q.60P
A simple pendulum of length 2.5 m makes 5.0 complete swings in 16 s. What is the acceleration of gravity at the location of the pendulum?
Solution:
Chapter 13 Oscillations About Equilibrium Q.61P
A large pendulum with a 200-lb gold-plated bob 12 inches in diameter is on display in the lobby of the United Nations building. The penduliun has a length of 75 ft. It is used to show the rotation of the Earth—for this reason it is referred to as a Foucault pendulum, What is the least amount of time it takes for the bob to swing from a position of maximum displacement to the equilibrium position of the pendulum? (Assume that the acceleration due to gravity is g = 9.81 m /s2 at the UN building.)
Solution:
Chapter 13 Oscillations About Equilibrium Q.62P
Find the length of a simple pendulum that has a period oi 1.00 s. Assume that the acceleration of gravity is g = 9.81 m/s2.
Solution:
Chapter 13 Oscillations About Equilibrium Q.63P
If the pendulum in the previous problem were to be taken to the Moon, where the acceleration of gravity is g/6, (a) would its period increase, decrease, or stay the same? (b) Check your result in part (a) by calculating the period of the pendulum on the Moon.
Solution:
Chapter 13 Oscillations About Equilibrium Q.64P
A hula hoop hangs from a peg. Find the period of the hoop as it gently rocks back and forth on the peg. (For a hoop with axis at the rim I = 2mR2, where R is the radius of the hoop.)
Solution:
Chapter 13 Oscillations About Equilibrium Q.65P
Solution:
Chapter 13 Oscillations About Equilibrium Q.66P
Consider a meterstick that oscillates back and forth about a pivot point at one of its ends. (a) Is the period of a simple pendulum of length L =1.00 m greater than, less than, or the same as the period of the meterstick? Explain. (b) Find the length L of a simple pendulum that has a period equal to the period of the meterstick.
Solution:
Chapter 13 Oscillations About Equilibrium Q.67P
On the construction site for a new skyscraper, a uniform beam of steel is suspended from one end. If the beam swings back and forth with a period of 2.00 s, what is its length?
Solution:
Chapter 13 Oscillations About Equilibrium Q.68P
(a) Find the period of a child’s leg as it swings about the hip joint. Assume the leg is 0.55 m long and can be treated as a uniform rod. (b) Estimate the child’s walking speed.
Solution:
Chapter 13 Oscillations About Equilibrium Q.69P
Suspended from the ceiling of an elevator is a simple pendulum of length L. What is the period of this pendulum if the elevator (a) accelerates upward with an acceleration a, or (b) accelerates downward with an acceleration whose magnitude is greater than zero but less than g? Give your answer in terms of L, g,and a.
Solution:
Chapter 13 Oscillations About Equilibrium Q.70GP
An object undergoes simple harmonicmotion with a period T. In the time 3T/2 the object moves through a total distance of 12D. In terms of D, what is the object’s amplitude of motion?
Solution:
Chapter 13 Oscillations About Equilibrium Q.71GP
A mass on a string moves with simple harmonic motion. If the period of motion is doubled, with the force constant and the amplitude remaining the same, by what multiplicative factor do the following quantities change: (a) angular frequency, (b) frequency, (c) maximum speed, (d) maximum acceleration, (e) total mechanical energy?
Solution:
Chapter 13 Oscillations About Equilibrium Q.72GP
If the amplitude of a simple harmonic oscillator is doubled, by what multiplicative factor do the following quantities change: (a) angular frequency, (b) frequency, (c) period, (d) maximum. speed, (e) maximum acceleration, (f) total mechanical energy?
Solution:
Chapter 13 Oscillations About Equilibrium Q.73GP
A mass m is suspended from the ceiling of an elevator by a spring of force constant k. When the elevator is at rest, the period of the mass is T. Does the period increase, decrease, or remain the same when the elevator (a) moves upward with constant speed or (b) moves upward with constant acceleration?
Solution:
Chapter 13 Oscillations About Equilibrium Q.74GP
A pendulum of length L is suspended from the ceiling of an elevator. When the elevator is at rest, the period of the pendulum is T. Does the period increase, decrease, or remain the same when the elevator (a) moves upward with constant speed or (b) moves upward with constant acceleration?
Solution:
Chapter 13 Oscillations About Equilibrium Q.75GP
A 1.8-kg mass is attached to a spring with a force constant of 59 N/m. If the mass is released with a speed of 0.25 m/s at a distance of 8.4 cm from the equilibrium position of the spring, what is its speed when it is halfway to the equilibrium position?
Solution:
We can find the speed of the mass attached to the spring at any point using conservation of energy. Initially the mass has both potential energy as well as kinetic energy. Now our interest is to know the speed of the block when it is halfway from its equilibrium point.
According to conservation of energy
Chapter 13 Oscillations About Equilibrium Q.76GP
An astronaut uses a Body Mass Measurement Device (BMMD) to determine her mass. What is the astronaut’s mass, given that the force constant of the BMMD is 2600 N/m and the period of oscillation is 0.85 s? (See the discussion on page 427 for more details on the BMMD.)
Solution:
Chapter 13 Oscillations About Equilibrium Q.77GP
A typical atom in a solid might oscillate with a frequency of 1012 Hz andan amplitude of 0.10 angstrom (10−11 m). Find the maximum acceleration of the atom and compare it with the acceleration of gravity.
Solution:
Chapter 13 Oscillations About Equilibrium Q.78GP
Sunspots vary in number as a function of time, exhibiting an approximately 11-year cycle. Galileo made the first European observations of sunspots in 1610, and daily observations were begun in Zurich in 1749. At the present time we are well into the 23rd observed cycle. What is the frequency of the sunspot cycle? Give your answer in Hz.
Solution:
Chapter 13 Oscillations About Equilibrium Q.79GP
Solution:
Chapter 13 Oscillations About Equilibrium Q.80GP
An object undergoing simple harmonic motion with a period T is at the position x = 0 at the time t =0. At the time t = 0.25T the position of the object is positive. State whether x is positive, negative, or zero at the following times: (a) t = 1.5T, (b) t = 2T, (c) t = 2.25T, and (d) t = 6.75T.
Solution:
Chapter 13 Oscillations About Equilibrium Q.81GP
The maximum speed of a 3.1-kg mass attached to aspring is 0.68 m/s, and the maximum force exerted on the mass is 11 N. (a) Whatis the amplitude of motion for this mass? (b) What is the force constant of the spring? (c) What is the frequency of this system?
Solution:
Chapter 13 Oscillations About Equilibrium Q.82GP
The acceleration of a block attached to a spring is given by a = −(0.302 m/s2) cos([2.41 rad/s]t). (a) What isthe frequency of the block’s motion? (b) What is the maximum, speed of the block? (c) What is the amplitude of the block’s motion?
Solution:
Chapter 13 Oscillations About Equilibrium Q.83GP
Solution:
Chapter 13 Oscillations About Equilibrium Q.84GP
A 9.50-g bullet, moving horizontally with an initial speed v0, embeds itself in a 1.45-kg pendulum bob that is initially at rest. The length of the pendulum is L = 0.745 m. After the collision, the pendulum swings to one side and comes to rest when it has gained a vertical height of 12.4 cm. (a) Is the kinetic energy of the bullet-bob system immediately after the collision greater than, less than, or the same as the kinetic energy of the system just before the collision? Explain. (b) Find the initial speed of the bullet. (c) How long does it take for the bullet-bob system to come to rest for the first time?
Solution:
Chapter 13 Oscillations About Equilibrium Q.85GP
A 1.44-g spider oscillates on its web, which has a damping constant of 3.30 × 10−5 kg/s. How long does it take for the spider’s amplitude of osculation to decrease by 10.0 percent?
Solution:
Chapter 13 Oscillations About Equilibrium Q.86GP
An object undergoes simple harmonic motion with a period T and amplitude A. In terms of T, how long does it take the object to travel from x = A to x = A/2?
Solution:
Chapter 13 Oscillations About Equilibrium Q.87GP
Find the period of oscillation of a disk of mass 0.32 kg and radius 0.15 m if it is pivoted about a small hole drilled near its rim.
Solution:
Chapter 13 Oscillations About Equilibrium Q.88GP
Calculate the ratio of the kinetic energy to the potential energy of a simple harmonic oscillator when its displacement is half its amplitude.
Solution:
Let an object of mass m be oscillating under simple harmonic motion with amplitude A. When it is displaced, half of the amplitude at this position speed of the object is
Chapter 13 Oscillations About Equilibrium Q.89GP
A 0.363-kg mass slides on a frictionless floor with a speed of 1.24 m/s. The mass strikes and compresses a spring with a force constant of 44.5 N/m. (a) How far docs the mass travel after contacting the spring before it comes to rest? (b) How long does it take for the spring to stop the mass?
Solution:
Chapter 13 Oscillations About Equilibrium Q.90GP
A large rectangular barge floating on a lake oscillates up and down with a period of 4.5 s. Find the damping constant for the barge, given that its mass is 2.44 × 105 kg and that its amplitude of oscillation decreases by a factor of 2.0 in 5.0 minutes.
Solution:
Chapter 13 Oscillations About Equilibrium Q.91GP
Solution:
Chapter 13 Oscillations About Equilibrium Q.92GP
Solution:
3.31N
Chapter 13 Oscillations About Equilibrium Q.93GP
A 0.45-kg crow lands on a slender branch and bobs up and down with a period of 1.5 s. An eagle flies up to the same branch, scaring the crow away, and lands. The eagle now bobs up and down with a period of 4.8 s. Treating the branch as an ideal spring, find (a) the effective force constant of the branch and (b) the mass of the eagle.
Solution:
Chapter 13 Oscillations About Equilibrium Q.94GP
Solution:
Chapter 13 Oscillations About Equilibrium Q.95GP
Solution:
Chapter 13 Oscillations About Equilibrium Q.96GP
When a mass m is attached to a vertical spring with a force constant k,it stretches the spring by the amount L. Calculate (a) the period of this mass and (b) the period of a simple pendulum of length L.
Solution:
Chapter 13 Oscillations About Equilibrium Q.97GP
Solution:
The total mechanical energy of an object in simple harmonic motion is equal to the sum of kinetic and potential energies.
For a mass undergoing simple harmonic motion to a spring with force constant(k)
Chapter 13 Oscillations About Equilibrium Q.98GP
Solution:
Chapter 13 Oscillations About Equilibrium Q.99GP
Solution:
Chapter 13 Oscillations About Equilibrium Q.100PP
Insects are ectothermic, which means their body temperature is largely determined by the temperature of their surroundings. This can have a number of interesting consequences. For example, the wing coloration in some butterfly species is determined by the ambient temperature, as is the body color of several species of dragonfly. In addition, the wing beat frequency of beetles taking flight varies with temperature due to changes in the resonant frequency of their thorax.
The origin of such temperature effects can be traced back to the fact that molecules have higher speeds and greater energy as temperature is increased (see Chapters 16 and 17). Thus, for example, molecules that collide and react as part of the metabolic process will do so more rapidly when the reactions are occurring at a higher temperature. As a result, development rates, heart rates, wing beats, and other processes all occur more rapidly.
One of the most interesting thermal effects is the temperature dependence of chirp rate in certain insects. This behavior has been observed in cone-headed grasshoppers, as well as several types of cricket. A particularly accurate connection between chirp rate and temperature is found in the snowy tree cricket (Oecanthus fultoni Walker), which chirps at a rate that follows the expression N = T − 39, where N is the number of chirps in 13 seconds, and T is the numerical value of the temperature in degrees Fahrenheit. This formula, which is known as Dolbear’s law, is plotted in Figure (green line) along with data points (blue dots) for the snowy tree cricket.
Solution:
Chapter 13 Oscillations About Equilibrium Q.101PP
Insects are ectothermic, which means their body temperature is largely determined by the temperature of their surroundings. This can have a number of interesting consequences. For example, the wing coloration in some butterfly species is determined by the ambient temperature, as is the body color of several species of dragonfly. In addition, the wing beat frequency of beetles taking flight varies with temperature due to changes in the resonant frequency of their thorax.
The origin of such temperature effects can be traced back to the fact that molecules have higher speeds and greater energy as temperature is increased (see Chapters 16 and 17). Thus, for example, molecules that collide and react as part of the metabolic process will do so more rapidly when the reactions are occurring at a higher temperature. As a result, development rates, heart rates, wing beats, and other processes all occur more rapidly.
One of the most interesting thermal effects is the temperature dependence of chirp rate in certain insects. This behavior has been observed in cone-headed grasshoppers, as well as several types of cricket. A particularly accurate connection between chirp rate and temperature is found in the snowy tree cricket (Oecanthus fultoni Walker), which chirps at a rate that follows the expression N = T − 39, where N is the number of chirps in 13 seconds, and T is the numerical value of the temperature in degrees Fahrenheit. This formula, which is known as Dolbear’s law, is plotted in Figure (green line) along with data points (blue dots) for the snowy tree cricket.
Solution:
Chapter 13 Oscillations About Equilibrium Q.102PP
Insects are ectothermic, which means their body temperature is largely determined by the temperature of their surroundings. This can have a number of interesting consequences. For example, the wing coloration in some butterfly species is determined by the ambient temperature, as is the body color of several species of dragonfly. In addition, the wing beat frequency of beetles taking flight varies with temperature due to changes in the resonant frequency of their thorax.
The origin of such temperature effects can be traced back to the fact that molecules have higher speeds and greater energy as temperature is increased (see Chapters 16 and 17). Thus, for example, molecules that collide and react as part of the metabolic process will do so more rapidly when the reactions are occurring at a higher temperature. As a result, development rates, heart rates, wing beats, and other processes all occur more rapidly.
One of the most interesting thermal effects is the temperature dependence of chirp rate in certain insects. This behavior has been observed in cone-headed grasshoppers, as well as several types of cricket. A particularly accurate connection between chirp rate and temperature is found in the snowy tree cricket (Oecanthus fultoni Walker), which chirps at a rate that follows the expression N = T − 39, where N is the number of chirps in 13 seconds, and T is the numerical value of the temperature in degrees Fahrenheit. This formula, which is known as Dolbear’s law, is plotted in Figure (green line) along with data points (blue dots) for the snowy tree cricket.
Solution:
Chapter 13 Oscillations About Equilibrium Q.103PP
Insects are ectothermic, which means their body temperature is largely determined by the temperature of their surroundings. This can have a number of interesting consequences. For example, the wing coloration in some butterfly species is determined by the ambient temperature, as is the body color of several species of dragonfly. In addition, the wing beat frequency of beetles taking flight varies with temperature due to changes in the resonant frequency of their thorax.
The origin of such temperature effects can be traced back to the fact that molecules have higher speeds and greater energy as temperature is increased (see Chapters 16 and 17). Thus, for example, molecules that collide and react as part of the metabolic process will do so more rapidly when the reactions are occurring at a higher temperature. As a result, development rates, heart rates, wing beats, and other processes all occur more rapidly.
One of the most interesting thermal effects is the temperature dependence of chirp rate in certain insects. This behavior has been observed in cone-headed grasshoppers, as well as several types of cricket. A particularly accurate connection between chirp rate and temperature is found in the snowy tree cricket (Oecanthus fultoni Walker), which chirps at a rate that follows the expression N = T − 39, where N is the number of chirps in 13 seconds, and T is the numerical value of the temperature in degrees Fahrenheit. This formula, which is known as Dolbear’s law, is plotted in Figure (green line) along with data points (blue dots) for the snowy tree cricket.
Solution:
Chapter 13 Oscillations About Equilibrium Q.104IP
Suppose we can change the plane’s period of oscillation, while keeping its amplitude of motion equal to 30.0 m. (a) If we want to reduce the maximum acceleration of the plane, should we increase or decrease the period? Explain. (b) Find the period that results in a maximum acceleration of 1.0g.
Solution:
Chapter 13 Oscillations About Equilibrium Q.105IP
Suppose the force constant of the spring is doubled, but the mass and speed of the block are still 0.980 kg and 1.32 m/s, respectively. (a) By what multiplicative factor do you expect the maximum compression of the spring to change? Explain. (b) Find the new maximum compression of the spring. (c) Find the time required for the mass to come to rest after contacting the spring.
Solution:
Chapter 13 Oscillations About Equilibrium Q.106IP
If the block’s initial speed is increased, does the total time the block is in contact wi th the spring increase, decrease, or stay the same? (b) Find the total time of contact for v0 = 1.65 m/s, m = 0.980 kg, and k = 245 N/m.
Solution:
