A ball dropped from a height of 4.00 m makes a perfectly elastic collision with the ground. Assuming no mechanical energy is lost due to air
In an engine, a piston oscillates with simple harmonic motion so that its position varies according to the expression x = (5.00 cm) cos (2t + /6)
The position of a particle is given by the expression x = (4.00 m) cos (3.00)t % )), where x is in meters and t is in seconds. Determine
(a) A hanging spring stretches by 35.0 cm when an object of mass 450 g is hung on it at rest. In this situation, we define its position as x = 0. The
A particle moving along the x axis in simple harmonic motion starts from its equilibrium position, the origin, at t = 0 and moves to the right. The
The initial position, velocity, and acceleration of an object moving in simple harmonic motion are xi, vi, and ai; the angular frequency of
A simple harmonic oscillator takes 12.0 s to undergo five complete vibrations. Find
(a) The period of its motion,
(b) The frequency in hertz, and
A vibration sensor, used in testing a washing machine, consists of a cube of aluminum 1.50 cm on edge mounted on one end of a strip of spring steel
A 7.00-kg object is hung from the bottom end of a vertical spring fastened to an overhead beam. The object is set into vertical oscillations having a
A piston in a gasoline engine is in simple harmonic motion. If the extremes of its position relative to its center point are ± 5.00 cm, find the
A 0.500-kg object attached to a spring with a force constant of 8.00 N/m vibrates in simple harmonic motion with an amplitude of 10.0 cm. Calculate
A 1.00-kg glider attached to a spring with a force constant of 25.0 N/m oscillates on a horizontal, frictionless air track. At t = 0 the glider is
A 1.00-kg object is attached to a horizontal spring. The spring is initially stretched by 0.100 m, and the object is released from rest there. It
A particle that hangs from a spring oscillates with an angular frequency '. The spring is suspended from the ceiling of an elevator car and hangs
A block of unknown mass is attached to a spring with a spring constant of 6.50 N/m and undergoes simple harmonic motion with an amplitude of 10.0 cm.
A 200-g block is attached to a horizontal spring and executes simple harmonic motion with a period of 0.250 s. If the total energy of the system is
An automobile having a mass of 1 000 kg is driven into a brick wall in a safety test. The bumper behaves like a spring of force constant 5.00, 106
A block–spring system oscillates with an amplitude of 3.50 cm. If the spring constant is 250 N/m and the mass of the block is 0.500 kg, determine
A 50.0-g object connected to a spring with a force constant of 35.0 N/m oscillates on a horizontal, frictionless surface with an amplitude of 4.00
A 2.00-kg object is attached to a spring and placed on a horizontal, smooth surface. A horizontal force of 20.0 N is required to hold the object at
The amplitude of a system moving in simple harmonic motion is doubled. Determine the change in (a) the total energy, (b) the maximum speed, (c) the
A 65.0-kg bungee jumper steps off a bridge with a light bungee cord tied to herself and to the bridge (Figure P15.22). The unstretched length of the
A particle executes simple harmonic motion with an amplitude of 3.00 cm. At what position does its speed equal half its maximum speed?
A cart attached to a spring with constant 3.24 N/m vibrates with position given by x = (5.00 cm) cos (3.60t rad/s).
(a) During the first cycle, for
While riding behind a car traveling at 3.00 m/s, you notice that one of the cars tires has a small hemispherical bump on its rim, as in
Consider the simplified single-piston engine in Figure P15.26. If the wheel rotates with constant angular speed, explain why the piston rod
A man enters a tall tower, needing to know its height. He notes that a long pendulum extends from the ceiling almost to the floor and that its period
A “second’s pendulum” is one that moves through its equilibrium position once each second. (The period of the pendulum is precisely 2 s.) The
A rigid steel frame above a street intersection supports standard traffic lights, each of which is hinged to hang immediately below the frame. A gust
The angular position of a pendulum is represented by the equation θ = (0.320 rad) cos wt, where is in radians and w = 4.43 rad/s. Determine the
A simple pendulum has a mass of 0.250 kg and a length of 1.00 m. It is displaced through an angle of 15.0° and then released. What are (a) the
A simple pendulum is 5.00 m long.
(a) What is the period of small oscillations for this pendulum if it is located in an elevator accelerating
A particle of mass m slides without friction inside a hemispherical bowl of radius R. Show that, if it starts from rest with a small displacement
A small object is attached to the end of a string to form a simple pendulum. The period of its harmonic motion is measured for small angular
A physical pendulum in the form of a planar body moves in simple harmonic motion with a frequency of 0.450 Hz. If the pendulum has a mass of 2.20 kg
A very light rigid rod with a length of 0.500 m extends straight out from one end of a meter stick. The stick is suspended from a pivot at the far
Consider the physical pendulum of Figure 15.18.
(a) If it’s moment of inertia about an axis passing through its center of mass and parallel to
A torsional pendulum is formed by taking a meter stick of mass 2.00 kg, and attaching to its center a wire. With its upper end clamped, the vertical
A clock balance wheel (Fig P15.39) has a period of oscillation of 0.250 s. The wheel is constructed so that its mass of 20.0 g is concentrated around
Show that the time rate of change of mechanical energy for a damped, undriven oscillator is given by dE/dt = ─bv2 and hence is always negative.
A pendulum with a length of 1.00 m is released from an initial angle of 15.0°. After 1 000 s, its amplitude has been reduced by friction to 5.50°.
Show that Equation 15.32 is a solution of Equation 15.31 provided that b2 < 4mk.
A 10.6-kg object oscillates at the end of a vertical spring that has a spring constant of 2.05, 104 N/m. The effect of air resistance is represented
The front of her sleeper wet from teething, a baby rejoices in the day by crowing and bouncing up and down in her crib. Her mass is 12.5 kg, and the
A 2.00-kg object attached to a spring moves without friction and is driven by an external force given by F = (3.00 N) sin (2) πt). If the force
Considering an undamped, forced oscillator (b = 0), show that Equation 15.35 is a solution of Equation 15.34, with an amplitude given by Equation
A weight of 40.0 N is suspended from a spring that has a force constant of 200 N/m. The system is undamped and is subjected to a harmonic driving
Damping is negligible for a 0.150-kg object hanging from a light 6.30-N/m spring. A sinusoidal force with an amplitude of 1.70 N drives the system.
You are a research biologist. You take your emergency pager along to a fine restaurant. You switch the small pager to vibrate instead of beep, and
Four people, each with a mass of 72.4 kg, are in a car with a mass of 1 130 kg. An earthquake strikes. The driver manages to pull off the road and
A small ball of mass M is attached to the end of a uniform rod of equal mass M and length L that is pivoted at the top (Fig. P15.51)
An object of mass m1 = 9.00 kg is in equilibrium while connected to a light spring of constant k = 100 N/m that is fastened to a wall as shown in
A large block P executes horizontal simple harmonic motion as it slides across a frictionless surface with a frequency f = 1.50 Hz. Block B rests on
A large block P executes horizontal simple harmonic motion as it slides across a frictionless surface with a frequency f. Block B rests on it, as
The mass of the deuterium molecule (D2) is twice that of the hydrogen molecule (H2). If the vibrational frequency of H2 is 1.30, 1014 Hz, what is the
A solid sphere (radius = R) rolls without slipping in a cylindrical trough (radius = 5R) as shown in Figure P15.56. Show that, for small
A light, cubical container of volume a3 is initially filled with a liquid of mass density 6. The cube is initially supported by a light string to
After a thrilling plunge, bungee-jumpers bounce freely on the bungee cord through many cycles (Fig. P15.22) After the first few cycles, the cord does
A pendulum of length L and mass M has a spring of force constant k connected to it at a distance h below its point of suspension (Fig. P15.59). Find
A particle with a mass of 0.500 kg is attached to a spring with a force constant of 50.0 N/m. At time t = 0 the particle has its maximum speed of
A horizontal plank of mass m and length L is pivoted at one end. The planks other end is supported by a spring of force constant k (Fig
A particle of mass 4.00 kg is attached to a spring with a force constant of 100 N/m. It is oscillating on a horizontal frictionless surface with an
A simple pendulum with a length of 2.23 m and a mass of 6.74 kg is given an initial speed of 2.06 m/s at its equilibrium position. Assume it
Review problem. One end of a light spring with force constant 100 N/m is attached to a vertical wall. A light string is tied to the other end of the
People who ride motorcycles and bicycles learn to look out for bumps in the road, and especially for wash boarding, a condition in which many equally
A block of mass M is connected to a spring of mass m and oscillates in simple harmonic motion on a horizontal, frictionless track (Fig. P15.66). The
A ball of mass m is connected to two rubber bands of length L, each under tension T, as in Figure P15.67. The ball is displaced by a small distance y
When a block of mass M, connected to the end of a spring of mass ms = 7.40 g and force constant k, is set into simple harmonic motion, the period of
A smaller disk of radius r and mass m is attached rigidly to the face of a second larger disk of radius R and mass M as shown in Figure P15.69. The
Consider a damped oscillator as illustrated in Figures 15.21 and 15.22. Assume the mass is 375 g, the spring constant is 100 N/m, and b = 0.100 N4
A block of mass m is connected to two springs of force constants k1 and k2 as shown in Figures P15.71a and P15.71b. In each case, the block moves on
A lobsterman’s buoy is a solid wooden cylinder of radius r and mass M. It is weighted at one end so that it floats upright in calm sea water,
Consider a bob on a light stiff rod, forming a simple pendulum of length L = 1.20 m. It is displaced from the vertical by an angle θmax and then
Your thumb squeaks on a plate you have just washed. Your sneakers often squeak on the gym floor. Car tires squeal when you start or stop abruptly.
Imagine that a hole is drilled through the center of the Earth to the other side. An object of mass m at a distance r from the center of the Earth is
At t = 0, a transverse pulse in a wire is described by the function Y = 6/X2 + 3 where x and y are in meters. Write the function y(x, t) that
Ocean waves with a crest-to-crest distance of 10.0 m can be described by the wave function
y(x, t) = (0.800 m) sin[0.628(x - vt)] where v = 1.20
A pulse moving along the x axis is described by y(x, t) = 5.00e - (x +5.00t) 2 where x is in meters and t is in seconds. Determine
Two points A and B on the surface of the Earth are at the same longitude and 60.0° apart in latitude. Suppose that an earthquake at point A creates
S and P waves, simultaneously radiated from the hypocenter of an earthquake, are received at a seismographic station 17.3 s apart. Assume the waves
For a certain transverse wave, the distance between two successive crests is 1.20 m, and eight crests pass a given point along the direction of
A sinusoidal wave is traveling along a rope. The oscillator that generates the wave completes 40.0 vibrations in 30.0 s. Also, a given maximum
When a particular wire is vibrating with a frequency of 4.00 Hz, a transverse wave of wavelength 60.0 cm is produced. Determine the speed of waves
A wave is described by y = (2.00 cm) sin (kx ─ wt), where k = 2.11 rad/m, w = 3.62 rad/s, x is in meters, and t is in seconds. Determine the
A sinusoidal wave on a string is described by y = (0.51 cm) sin (kx ─ wt), where k = 3.10 rad/cm and w = 9.30 rad/s. How far does a wave crest
Consider further the string shown in Figure 16.10 and treated in Example 16.3. Calculate
(a) The maximum transverse speed and
(b) The maximum
Consider the sinusoidal wave of Example 16.2, with the wave function
y = (15.0 cm) cos (0.157x ─ 50.3t).
At a certain instant, let point A be
A sinusoidal wave is described by y = (0.25 m) sin (0.30x ─ 40t) where x and y are in meters and t is in seconds. Determine for this wave the
(a) Plot y versus t at x = 0 for a sinusoidal wave of the form y = (15.0 cm) cos (0.157x - 50.3t), where x and y are in centimeters and t is in
(a) Write the expression for y as a function of x and t for a sinusoidal wave traveling along a rope in the negative x direction with the following
A sinusoidal wave traveling in the ─x direction (to the left) has an amplitude of 20.0 cm, a wavelength of 35.0 cm, and a frequency of 12.0 Hz.
A transverse wave on a string is described by the wave function
y = (0.120 m) sin [(πx/8) + 4πt]
(a) Determine the transverse speed and
A transverse sinusoidal wave on a string has a period T = 25.0 ms and travels in the negative x direction with a speed of 30.0 m/s. At t = 0, a
A sinusoidal wave of wavelength 2.00 m and amplitude 0.100 m travels on a string with a speed of 1.00 m/s to the right. Initially, the left end of
A wave on a string is described by the wave function y = (0.100 m) sin (0.50x ─ 20t).
(a) Show that a particle in the string at x = 2.00 m
A telephone cord is 4.00 m long. The cord has a mass of 0.200 kg. A transverse pulse is produced by plucking one end of the taut cord. The pulse
Transverse waves with a speed of 50.0 m/s are to be produced in a taut string. A 5.00-m length of string with a total mass of 0.060 0 kg is used.
A piano string having a mass per unit length equal to 5.00 x 10─3 kg/m is under a tension of 1 350 N. Find the speed of a wave traveling on
A transverse traveling wave on a taut wire has an amplitude of 0.200 mm and a frequency of 500 Hz. It travels with a speed of 196 m/s.
(a) Write an
An astronaut on the Moon wishes to measure the local value of the free-fall acceleration by timing pulses traveling down a wire that has an object of
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