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physics
oscillations mechanical waves

Questions and Answers of Oscillations Mechanical Waves

Find the trajectory equation y (x) of a point if it moves according to the following laws: (a) x = a sin wt, y = a sin 2wt; (b) x = a sin cot, y = a cos 2wt. Plot these trajectories.
A particle of mass m is located in a one-dimensional potential field where the potential energy of the particle depends on the coordinate x as U (x) = Uo (1 – cos ax); Uo and a are constants. Find
Solve the foregoing problem if the potential energy has the form U (x) = a/x2 – b/x, where a and b are positive constants.
Find the period of small oscillations in a vertical plane performed by a ball of mass m = 40 g fixed at the middle of a horizontally stretched string l = 1.0 m in length. The tension of the string is
Determine the period of small oscillations of a mathematical pendulum, that is a ball suspended by a thread l = 20 cm in length, if it is located in a liquid whose density is η = 3.0 times less
A ball is suspended by a thread of length l at the point O on the wall, forming a small angle a with the vertical (Fig. 4.1). Then the thread with the ball was deviated through a small angle β
A pendulum clock is mounted in an elevator car which starts going up with a constant acceleration w, with w = g. At a height h the acceleration of the car reverses, its magnitude remaining constant.
Calculate the period of small oscillations of a hydrometer (Fig. 4.2) which was slightly pushed down in the vertical direction. The mass of the hydrometer is m = 50 g, the radius of its tube is r =
A non-deformed spring whose ends are fixed has a stiffness × = 13 N/m. A small body of mass m = 25 g is attached at the point removed from one of the ends by η = 1/3 of the spring's length.
Determine the period of small longitudinal oscillations 'of a body with mass m in the system shown in Fig. 4.3. The stiffness values of the springs are Ã—1 and Ã—2- The
Find the period of small vertical oscillations of a body with mass m in the system illustrated in Fig. 4.4. The stiffness values of the springs are x1 and x2, their masses are negligible.
A small body of mass m is fixed to the middle of a stretched string of length 2l. In the equilibrium position the string tension is equal to To. Find the angular frequency of small oscillations of
Determine the period of oscillations of mercury of mass m = 200 g poured into a bent tube (Fig. 4.5) whose right arm forms an angle 0 = 30° with the vertical. The cross-sectional area of the tube
A uniform rod is placed on two spinning wheels as shown in Fig. 4.6. The axes of the wheels are separated by a distance l = 20 cm, the coefficient of friction between the rod and the wheels is k =
Imagine a shaft going all the way through the Earth from pole to pole along its rotation axis. Assuming the Earth to be a homogeneous ball and neglecting the air drag, find: (a) The equation of
Find the period of small oscillations of a mathematical pendulum of length l if its point of suspension O moves relative to the Earth's surface in an arbitrary direction with a constant acceleration
In the arrangement shown in Fig. 4.8 the sleeve M of mass m = 0.20 kg is fixed between two identical springs whose combined stiffness is equal to x = 20 N/re. The sleeve can slide without friction
A plank with a bar placed on it performs horizontal harmonic oscillations with amplitude a = 10 cm. Find the coefficient of friction between the bar and the plank if the former starts sliding along
Find the time dependence of the angle of deviation of a mathematical pendulum 80 cm in length if at the initial moment the pendulum (a) Was deviated through the angle 3.0° and then set free without
A body A of mass m1 = 1.00 kg and a body B of mass m = 4.10 kg are interconnected by a spring as shown in Fig. 4.9. The body A performs free vertical harmonic oscillations with amplitude a = 1.6 cm
A plank with a body of mass m placed on it starts moving straight up according to the law y = a (1 = cos cot), where y is the displacement from the initial position, w = 11 s-1. Find: (a) The time
A body of mass m was suspended by a non-stretched spring, and then sot free without push. The stiffness of the spring is Neglecting the mass of the spring, find: (a) The law of motion y (t), where y
A particle of mass m moves due to the force F = – amr, where a is a positive constant, r is the radius vector of the particle relative to the origin of coordinates. Find the trajectory of its
A body of mass m is suspended from a spring fixed to the ceiling of an elevator car. The stiffness of the spring is ×. At the moment t = 0 the car starts going up with an acceleration w. Neglecting
A body of mass m = 0.50 kg is suspended from a rubber cord with elasticity coefficient k = 50 N/re. Find the maximum distance over which the body can be pulled down for the body's oscillations to
A body of mass m = 0.50 kg is suspended from a rubber cord with elasticity coefficient k = 50 N/re. Find the maximum distance over which the body can be pulled down for the body's oscillations to
Solve the foregoing problem for the case of the pan having a mass M. Find the oscillation amplitude in this case.
A particle of mass m moves in the plane xy due to the force varying with velocity as F = a (yi = xj), where a is a positive constant, i and j are the unit vectors of the x and y axes. At the initial
A pendulum is constructed as a light thin-walled sphere of radius R filled up with water and suspended at the point O from a light rigid rod (Fig. 4.11). The distance between the point O and the
Find the frequency of small oscillations of a thin uniform vertical rod of mass m and length l hinged at the point O (Fig. 4.12). The combined stiffness of the springs is equal to x. The mass of the
A uniform rod of mass m = 1.5 kg suspended by two identical threads l= 90 cm in length (Fig. 4.t3) was turned through a small angle about the vertical axis passing through its middle pointC. The
An arrangement illustrated in Fig. 4.14 consists of a horizontal uniform disc D of mass m and radius R and a thin rod AO whose torsional coefficient is equal to k. Find the amplitude and the energy
A physical pendulum is positioned so that its centre of gravity is above the suspension point. From that position the pendulum started moving toward the stable equilibrium and passed it with an
A physical pendulum is positioned so that its centre of gravity is above the suspension point. From that position the pendulum started moving toward the stable equilibrium and passed it with an
A physical pendulum performs small oscillations about the horizontal axis with frequency w1 = 15.0 S–1. When a small body of mass m = 50 g is fixed to the pendulum at a distance l = 20 cm below the
Two physical pendulums perform small oscillations about the same horizontal axis with frequencies w1 and w2. Their moments of inertia relative to the given axis are equal to I1 and I2 respectively.
A uniform rod of length l performs small oscillations about the horizontal axis 00' perpendicular to the rod and passing through one of its points. Find the distance between the centre of inertia of
A thin uniform plate shaped as an equilateral triangle with a height h performs small oscillations about the horizontal axis coinciding with one of its sides. Find the oscillation period and the
A smooth horizontal disc rotates about the vertical axis O (Fig. 4.15) with a constant angular velocity (o. A thin uniform rod AB of length l performs small oscillations about the vertical axis A
Find the frequency of small oscillations of the arrangement illustrated in Fig. 4.16. The radius of the pulley is R, its moment of inertia relative to the rotation axis is I, the mass of the body is
A uniform cylindrical pulley of mass M and radius R can freely rotate about the horizontal axis O (Fig. 4.17). The free end a thread tightly wound on the pulley carries deadweight A. At a certain
A solid uniform cylinder of radius r rolls without sliding along the inside surface of a cylinder of radius R, performing small oscillations. Find their period.
A solid uniform cylinder of mass m performs small oscillations due to the action of two springs whose combined stiffness is equal to x (Fig. 4.18). Find the period of these oscillations in the
Two cubes with masses m1 and m2 were interconnected by a weightless spring of stiffness x and placed on a smooth horizontal surface. Then the cubes were drawn closer to each other and released
Two balls with masses m1 = 1.0 kg and m2 = 2.0 kg are slipped on a thin smooth horizontal rod (Fig. 4.19). The balls are interconnected by a light spring of stiffness x = 24 N/m. The left-hand ball
Find the period of small torsional oscillations of a system consisting of two discs slipped on a thin rod with torsional coefficient k. The moments of inertia of the discs relative to the rod's axis
A mock-up of a CO2 molecule consists of three balls interconnected by identical light springs and placed along a straight line in the state of equilibrium. Such a system can freely perform
In a cylinder filled up with ideal gas and closed from both ends there is a piston of mass m and cross-sectional area S (Fig. 4.21). In equilibrium the piston divides the cylinder into two equal
A small ball of mass m = 21g suspended by an insulating thread at a height h = 12 cm from a large horizontal conducting plane performs small oscillations (Fig. 4.22). After a charge q had been
A small magnetic needle performs small oscillations about an axis perpendicular to the magnetic induction vector. On changing the magnetic induction the needle's oscillation period decreased η =
A loop (Fig. 4.23) is formed by two parallel conductors connected by a solenoid with inductance L and a conducting rod of mass m which can freely (without friction) slide over the conductors. The
A coil of inductance L connects the upper ends of two vertical copper bars separated by a distance l. A horizontal conducting connector of mass m starts falling with zero initial velocity along the
A point performs damped oscillations according to the law x = aoe – βt sin wt. Find: (a) The oscillation amplitude and the velocity of the point at the moment t = 0; (b) The moments of
A body performs torsional oscillations according to the law φ = φoe – βt cos cot. Find: (a) The angular velocity φ and the angular acceleration φ of the body at the moment
A point performs damped oscillations with frequency w and damping coefficient β according to the law (4.1b). Find the initial amplitude a o and the initial phase (z if at the moment t = 0 the
A point performs damped oscillations with frequency w = 25 s–1. Find the damping coefficient if at the initial moment the velocity of the point is equal to zero and its displacement from the
A point performs damped oscillations with frequency w and damping coefficient β. Find the velocity amplitude of the point as a function of time t if at the moment t = 0. (a) Its displacement
There are two damped oscillations with the following periods T and damping coefficients β: T1 = 0.10 ms, β1 = 100 s–1 and T2 = 10 ms, β2 = 10 s–1 which of them decays faster?
A mathematical pendulum oscillates in a medium for which the logarithmic damping decrement is equal to λo = 1.50. What will be the logarithmic damping decrement if the resistance of the medium
A deadweight suspended from a weightless spring extends it by Ax =- 9.8 cm. What will be the oscillation period of the dead-weight when it is pushed slightly in the vertical direction? The
Find the quality factor of the oscillator whose displacement amplitude decreases η = 2.0 times every n = 110 oscillations.
A particle was displaced from the equilibrium position by a distance l = 1.0 cm and then left alone. What is the distance that the particle covers in the process of oscillations till the complete
Find the quality factor of a mathematical pendulum l = 50 cm long if during the time interval v = 5.2 min its total mechanical energy decreases η = 4.0 ∙ 104 times.
A uniform disc of radius R = 13 cm can rotate about a horizontal axis perpendicular to its plane and passing through the edge of the disc. Find the period of small oscillations of that disc if the
A thin uniform disc of mass m and radius R suspended by an elastic thread in the horizontal plane performs torsional oscillations in a liquid. The moment of elastic forces emerging in the thread is
A disc A of radius R suspended by an elastic thread between two stationary planes (Fig. 4.24) performs torsional oscillations about its axis OO'. The moment of inertia of the disc relative to that
A conductor in the shape of a square frame with side a suspended by an elastic thread is located in a uniform horizontal magnetic field with induction B. In equilibrium the plane of the frame is
A bar of mass m = 0.50 kg lying on a horizontal plane with a friction coefficient k = 0.10 is attached to the wall by means of a horizontal non-deformed spring. The stiffness of the spring is equal
A ball of mass m can perform un-damped harmonic oscillations about the point x = 0 with natural frequency coo. At the moment t = 0, when the ball was in equilibrium, a force Fx = Fo cos cot
A particle of mass m can perform un-damped harmonic oscillations due to an electric force with coefficient k. When the particle was in equilibrium, a permanent force F was applied to it for τ
A ball of mass m when suspended by a spring stretches the latter by Al. Due to external vertical force varying according to a harmonic law with amplitude F o the ball performs forced oscillations.
The forced harmonic oscillations have equal displacement amplitudes at frequencies wl = 400 s–1 and w2 = 600 s–1. Find the resonance frequency at which the displacement amplitude is maximum?
The velocity amplitude of a particle is equal to half the maximum value at the frequencies wl and w2 of external harmonic force. Find: (a) The frequency corresponding to the velocity resonance;
A certain resonance curve describes a mechanical oscillating system with logarithmic damping decrement λ = 1.60. For this curve find the ratio of the maximum displacement amplitude to the
Due to the external vertical force Fx = F 0 cos cot a body suspended by a spring performs forced steady-state oscillations according to the law x = a cos (wt – φ). Find the work performed by
A ball of mass m = 50 g is suspended by a weightless spring with stiffness x = 20.0 N/m. Due to external vertical harmonic force with frequency w = 25.0 s–1 the ball performs steady-state
A ball of mass m suspended by a weightless spring can perform vertical oscillations with damping coefficient β. The natural oscillation frequency is equal to w0. Due to the external vertical
An external harmonic force F whose frequency can be varied, with amplitude maintained constant, acts in a vertical direction on a ball suspended by a weightless spring, the damping coefficient is q
A uniform horizontal disc fixed at its centre to an elastic vertical rod performs forced torsional oscillations due to the moment of forces Nz = Nm cos cot. The oscillations obey the law φ =
An object of mass m oscillates on the end of a spring with spring constant k. Derive a formula for the time it takes the spring to stretch from its equilibrium position to the point of maximum
An object of mass m oscillates at the end of a spring with spring constant k and amplitude A. Derive a formula for the speed of the object when it is at a distance d from the equilibrium position.
A block of mass m is connected to a spring with spring constant k, and oscillates on a horizontal, frictionless surface. The other end of the spring is fixed to a wall. If the amplitude of
A particle that hangs from a spring oscillates with an angular frequency w. The spring-particle system is suspended from the ceiling of an elevator car and hangs motionless (relative to the elevator
A large block, with a second block sitting on top, is connected to a spring and executes horizontal simple harmonic motion as it slides across a frictionless surface with an angular frequency w. The
A simple pendulum consists of a ball of mass M hanging from a uniform string of mass m, with m
A uniform rope of mass m and length L is suspended vertically. Derive a formula for the time it takes a transverse wave pulse to travel the length of the rope.
A uniform cord has a mass m and a length L. The cord passes over a pulley and supports an object of mass M as shown in the figure. Derive a formula for the speed of a wave pulse traveling along the
A block of mass M, supported by a string, rests on an incline making an angle θ with the horizontal. The string's length is L and its mass is m
A stationary train emits a whistle at a frequency f. The whistle sounds higher or lowers in pitch depending on whether the moving train is approaching or receding. Derive a formula for the difference
What is the maximum acceleration of a platform that oscillates at amplitude 2.20 cm and frequency 6.60 Hz?
A particle with a mass of 1.00 x 10-20 kg is oscillating with simple harmonic motion with a period of 1.00 x 10-5 s and a maximum speed of 1.00 x 103 m/s. Calculate(a) The angular frequency and(b)
In an electric shaver, the blade moves back and forth over a distance of 2.0 mm in simple harmonic motion with frequency 120Hz find?(a) The amplitude,(b) The maximum blade speed, and(c) The magnitude
A 0.12 kg body undergoes simple harmonic motion of amplitude 8.5 cm and period 0.20 s.(a) What is the magnitude of the maximum force acting on it?(b) If the oscillations are produced by a spring,
An object undergoing simple harmonic motion takes 0.25s to travel from one point of zero velocity to the next such point. The distance between those points is 36 cm. Calculate the(a) Period,(b)
An automobile can be considered to be mounted on four identical springs as far as vertical oscillations are concerned. The springs of a certain car are adjusted so that the oscillations have a
An oscillator consists of a block of mass 0.500 kg connected to a spring. When set into oscillation with amplitude 35.0 cm, the oscillator repeats its motion every 0.500 s. Find the(a) Period,(b)
An oscillating block-spring system takes 0.75 s to begin repeating its motion. Find(a) The period,(b) The frequency in hertz, and(c) The angular frequency in radians per second.
A loudspeaker produces a musical sound by means of the oscillation of a diaphragm whose amplitude is limited to 1.00μm.(a) At what frequency is the magnitude a of the diaphragm's acceleration equal
What is the phase constant for the harmonic oscillator with the position function x(t) given in Figure. If the position function has the form x = xm cos (wt + Φ? The vertical axis scale is set by xs
The function x = (6.0 m) cos [(3π rad/s)t + π/3 rad] gives the simple harmonic motion of a body. At t = 2.0 s, what are the(a) Displacement,(b) Velocity,(c) Acceleration, and(d) Phase of the

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