- 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 resistance, (a) Show that the ensuing motion is periodic
- 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) where x is in centimeters and t is in seconds. At t =
- 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) The frequency and period of the motion, (b) The
- (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 object is pulled down an additional 18.0 cm and
- 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 amplitude of its motion is 2.00 cm, and the
- The initial position, velocity, and acceleration of an object moving in simple harmonic motion are xi, vi, and ai; the angular frequency of oscillation is w. (a) Show that the position and velocity
- 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 (c) The angular frequency in radians per second.
- 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 (like a hacksaw blade) that lies in a vertical
- 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 period of 2.60 s. Find the force constant of the
- 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 maximum velocity and acceleration of the piston when
- 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) The maximum value of its speed and
- 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 released from rest at x = ─3.00 cm. (That 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 proceeds to move without friction. The next time the
- 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 motionless (relative to the elevator car) as the car
- 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. When the block is halfway between its equilibrium
- 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 2.00 J, find (a) The force constant of the spring
- 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 N/m and compresses 3.16 cm as the car is brought to
- 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) The mechanical energy of the system, (b)
- 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 cm. Find (a) The total energy of the system and
- 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 rest when it is pulled 0.200 m from its equilibrium
- 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 maximum acceleration, and (d) the period.
- 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 cord is 11.0 m. She reaches the bottom of her
- 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 0 < t < 1.75 s, just when is the system’s
- While riding behind a car traveling at 3.00 m/s, you notice that one of the car€™s tires has a small hemispherical bump on its rim, as in Figure P15.25.(a) Explain why the bump, from your
- Consider the simplified single-piston engine in Figure P15.26. If the wheel rotates with constant angular speed, explain why the piston rod oscillates in simple harmonic motion.
- 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 is 12.0 s. (a) How tall is the tower? (b)
- 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 length of a second’s pendulum is 0.992 7 m at
- 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 of wind sets a light swinging in a vertical
- 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 period and length of the pendulum.
- 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 maximum speed, (b) The maximum angular acceleration,
- 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 upward at 5.00 m/s2? (b) What is its period if the
- 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 from equilibrium, the particle moves in simple
- 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 displacements and three lengths, each time clocking the
- 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 and the pivot is located 0.350 m from the center
- 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 end of the rod and is set into oscillation. (a)
- 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 the axis passing through its pivot point is ICM
- 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 wire supports the stick as the stick turns in a
- 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 a rim of radius 0.500 cm. What are (a) The
- 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. Proceed as follows: Differentiate the expression
- 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°. What is the value of b/2m?
- 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 by the damping coefficient b = 3.00 N ∙ s/m.
- 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 crib mattress can be modeled as a light spring
- 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 constant of the spring is 20.0 N/m, determine
- 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 15.36.
- 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 force of frequency 10.0 Hz, resulting in a
- 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. At what frequency will the force make the object
- 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 you put it into a side pocket of your suit coat. The
- 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 stop, as the vertical oscillations of the ground
- 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) (a) Determine the tensions in the rod at the pivot and at the
- 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 Figure P15.52a. A second object, m2 = 7.00 kg, is
- 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 it, as shown in Figure P15.53, and the coefficient
- 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 shown in Figure P15.53, and the coefficient of static
- 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 vibrational frequency of D2? Assume that 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 displacements from equilibrium perpendicular to the length of
- 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 form a simple pendulum of length Li, measured from
- 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 not go slack. Your little brother can make a pest
- 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 the frequency of vibration of the system for small
- 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 20.0 m/s and is moving to the left. (a) Determine
- A horizontal plank of mass m and length L is pivoted at one end. The plank€™s other end is supported by a spring of force constant k (Fig P15.61). The moment of inertia of the plank about
- 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 amplitude of 2.00 m. A 6.00-kg object is dropped
- 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 undergoes simple harmonic motion, and determine its (a)
- 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 horizontal spring. The string changes from
- 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 spaced ridges are worn into the road. What is so
- 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 force constant of the spring is k and 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 perpendicular to the length of the rubber bands.
- 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 its motion isT = 2π √M + (ms/3)/kA
- 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 center of the small disk is located at the edge of
- 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 s/m. (a) How long does it takes for the amplitude
- 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 frictionless table after it is displaced from
- 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, having density 6. A passing shark tugs on the slack
- 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 released. Predict the subsequent angular
- 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. You can make a goblet sing by wiping your moistened
- 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 pulled toward the center of the Earth only by the
- 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 describes this pulse if it is traveling in the positive x
- 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 m/s. (a) Sketch y(x, t) at t = 0. (b) Sketch y(x, t)
- 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 (a) The direction of the wave motion, and (b) The speed of the
- 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 a P wave that reaches point B by traveling
- 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 have traveled over the same path at speeds of 4.50
- 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 travel every 12.0 s. Calculate the wave speed.
- 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 travels 425 cm along the rope in 10.0 s. What is the
- 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 along the wire.
- 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 amplitude, wavelength, frequency, and speed of 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 move in 10.0 s? Does it move in the positive or
- 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 transverse acceleration of a point on the string.
- 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 at the origin and point B be the first point
- 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) amplitude, (b) angular frequency, (c) angular
- (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 seconds. (b) Determine the period of vibration from
- (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 characteristics: A = 8.00 cm, A = 80.0 cm, f = 3.00
- 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. The transverse position of an element of the
- 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 acceleration at t = 0.200 s for the point on the
- 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 particle on the string at x = 0 has a transverse
- 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 the string is at the origin. Find (a) The
- 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 executes simple harmonic motion. (b) Determine the
- 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 makes four trips down and back along the cord in 0.800
- 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. What is the required tension?
- 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 this string.
- 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 equation in SI units of the form y = A sin (kx
- 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 large mass suspended from it. Assume a wire has a