– Harmenie metion equatios asd its molutiea:
where $\omega_{4}$ is the natural escillation frequency.
– Damped escillation equatios and its solution:
where $\beta$ is the dampiag coelfieiest, o is the frequency of damped oscillations:
\[
=-\sqrt{\text { ब- }} \text {. }
\]
– Legarithmie damping decrement $\lambda$ asd quality factor $Q:$
\[
\lambda=\beta T, Q=\pi / \lambda \text {, }
\]
where $T=2 \pi / a$.
– Yorsed oscillation equation and its steady-atete soletios:
where
– Maximum shift amplitude oecars at
\[
\omega_{r e n}=\sqrt{\text { aj-2p }} \text {. }
\]
4.1. A point oscillates along the $x$ axis according to the law $x=$ – a cos $(a t-\pi / 4$ ). Draw the approximate plots
(a) of displacement $x$, velocity projection $v_{x}$ and acceleration projection $w_{x}$ as functions of time $t_{\text {; }}$
(b) velocity projection $v_{x}$ and acceleration projection $v_{x}$ as functions of the coordinate $x$.
4.2. A point moves along the $x$ axis according to the law $x=$ – $a \sin ^{2}(\omega t-\pi / 4)$. Find:
(a) the amplitude and period of oscillations; draw the plot $x(t)$;
(b) the velocity projection $v_{x}$ as a function of the coordiate $x$; draw the plot $v_{x}(x)$.
4.3. A particle performs harmonie ascillations along the $x$ axis about the equilibrium position $x=0$. The oscillation frequency is $\omega=4.00 \mathrm{~s}-1$. At a certain moment of time the particle has a coordinate $x_{0}=25.0 \mathrm{~cm}$ and its velocity is equal to $v_{x 4}=100 \mathrm{~cm} / \mathrm{s}$.
106
Find the coordinate $x$ and the velocity $v_{x}$ of the particle $t=2.40 \mathrm{~s}$ after that moment.
4.4. Find the angular frequency and the amplitude of harmonic oscillations of a particle if at distances $x_{1}$ and $x_{1}$ from the equilibrium position its velocity equals $v_{1}$ and $v_{2}$ respectively.
4.5. A point performs harmonic oscillations along a straight line with a period $T=0.60 \mathrm{~s}$ and an amplitude $a=10.0 \mathrm{~cm}$. Find the mean velocity of the point averaged over the time interval during which it travels a distance $a / 2$, starting from
(a) the extreme position;
(b) the equilibrium position.
4.6. At the moment $t=0$ a point starts oscillating along the $x$ axis according to the law $x=a$ sin ot. Find:
(a) the mean value of its velocity vector projection $\left(w_{n}\right)$;
(b) the modulus of the mean velocity vector $|(\mathbf{v})|$;
(c) the mean value of the velocity modulus (v) averaged over $3 / 8$ of the period after the start.
4.7. A particle moves along the $x$ axis according to the law $x=$ $=a \cos \omega t$. Find the distance that the particle covers during the time interval from $t=0$ to $t$.
4.8. At the moment $t=0$ a particle starts moving along the $x$ axis so that its velecity projection varies as $v_{x}=35 \cos n t \mathrm{~cm} / \mathrm{s}$, where $t$ is expresed in seconds. Find the distance that this partiele covers during $t=2.80 \mathrm{~s}$ after the start.
4.9. A particle performs harmonic oscillations along the $x$ axis sccording to the law $x=a$ cos of. Assuming the probability $P$ of the particle to fall within an interval from $-a$ to $+a$ to be equal to unity, find how the probability density $d P / d x$ depends on $x$. Here $\underset{P}{ }$ denotes the probability of the particle falling within an interval from $x$ to $x+d x$. Plot $d P / d x$ as a function of $x$.
4.10. Using graphical means, find an amplitude a of oscillations resulting from the superposition of the following oscillations of the same direction:
(a) $x_{1}=3.0 \cos (\omega t+\pi / 3), \quad x_{3}=8.0 \sin (\omega t+\pi / 6) ;$
(b) $x_{1}=3.0$ cos $\omega t, x_{2}=5.0 \cos (\omega t+\pi / 4), x_{3}=6.0 \mathrm{sin} \omega t$.
4.i1. A point participates simultaneously in two harmonic oscillations of the same direction: $x_{1}=a \cos$ ot and $x_{1}=a \cos 2 \omega t$. Find the maximum velocity of the point.
4.12. The superposition of two harmonic oscillations of the same direction results in the oscillation of a point according to the law $z=a \cos 2.1 t \cos 50.0 t$, where $t$ is expressed in seconds. Find the anzular frequencies of the constituent oscillations and the period with which they beat.
4.13. A point $A$ oseillates according to a certain harmonic law in the reference frame $\boldsymbol{K}^{\prime}$ which in its tum performs harmonic oscillations relative to the reference frame $K$. Both oscillations occur along the same direction. When the $\boldsymbol{K}^{\prime}$ frame oscillates at the frequency 20 or $24 \mathrm{~Hz}$, the beat frequency of the point $\boldsymbol{A}$ in the $K$ frame turns out to be equal to v. At what frequency of oscillation of the frame $\boldsymbol{K}^{\prime}$ will the beat frequency of the point $\boldsymbol{A}$ become equal to $2 \mathrm{v}$ ?
4.14. A point moves in the plane $x y$ according to the law $x=$ $=a \sin \omega t, y=b \cos \omega t$ where $a, b$, and $\omega$ are positive constants. Find:
(a) the trajectory equation $y(x)$ of the point and the direction of its motion along this trajectory;
(b) the acceleration $w$ of the point as a function of its radius vecter $r$ relative to the origin of coordinates.
4.15. Find the trajectory equation $y(x)$ of a point if it moves according to the following laws:
(a) $x=a$ sin $\omega t, y=a \sin 2 \mathrm{ut}$;
(b) $z=a \sin \omega t, y=a \cos 2 \omega t$.

Plot these trajectories.
4.16. A particle of mass $m$ is located in a unidimensional potential field where the potential energy of the particle depends on the coordinate $x$ as $U(x)=U_{0}(1-\cos a x) ; U_{0}$ and $a$ are constants. Find the period of small oscillations that the particle performs about the equilibrium position.
4.17. Solve the foregoing problem if the potential energy has the form $U(x)=a / x^{4}-b / x$, where $a$ and $b$ are positive constants.
4.18. Find the period of small oscillations in a vertical plane performed by a ball of mass $m-40 \mathrm{~g}$ fixed at the middle of a horizontally stretched string $l=1.0 \mathrm{~m}$ in length. The tension of the string is assumed to be constant and equal to $F=10 \mathrm{~N}$.
4.19. Determine the period of small secillations of a mathematical pendulum, that is a ball suspended by a thread $I-20 \mathrm{~cm}$ in length, if it is located in a liquid whose density is $\eta=3.0$ times less than that of the ball. The resistance of the liquid is to be neglected.
4.20. A ball is suspended by a thread of length $l$ at the point $O$ on the wall, forming a small angle $\alpha$ with the vertical (Fig. 4.1). Then
Fie. 4.t.
Fig. 4.2.
the thread with the ball was deviated through a small angle $\beta(\beta>\alpha)$ and set free. Assuming the collision of the ball against the wall to be perfectly elastic, find the oscillation period of sueh a pendulum.
168
4.21. A pendulum clock is mounted in an elevator car which starts going up with a constant acceleration $w$, with $w<\boldsymbol{g}$. At a height $h$ the acceleration of the car reverses, its magnitude remaining constant. How soon after the start of the motion will the clock show the right time again?
4.22. Calculate the peried of small escillations of a hydrometer (Fig. 4.2) which was slightly pushed down in the vertical direction. The mass of the hydrometer is $m=50 \mathrm{~g}$, the radius of its tube is $r=3.2 \mathrm{~mm}$, the density of the liquid is $p=1.00 \mathrm{~g} / \mathrm{cm}^{3}$. The resistance of the liquid is assumed to be negligible.
4.23. A non-deformed spring whose ends are fixed has astiffess $x=13 \mathrm{~N} / \mathrm{m}$. A small body of mass $m=25 \mathrm{~g}$ is attached at the point removed from one of the ends by $n=1 / 3$ of the spring’s length. Nezlecting the mass of the spring, find the period of small longitudinal oscillations of the body. The force of gravity is assumed to be absent.
Fig. 4.3.
4.24. Determine the period of small longitudinal oscillations of a body with mass $m$ in the system shown in Fig. 4.3. The stiflness values of the springs are $x_{4}$ and $x_{2}$. The frietion and the mases of the springs are neglizible.
4.25. Find the period of small vertical oscillations of a body with mass $m$ in the system illustrated in Fig. 4.4. The stiflness values of the springs are $x_{1}$ and $x_{2}$, their masses are negligible.
4.26. A small body of mass $m$ is fixed to the middle of a stretched string of length 21 . In the equilibrium position the string tension is equal to $T_{0}$. Find the angular frequency of small oscillations of the body in the transverse direction. The mass of the string is negligible, the gravitational field is absent.
Fig. 4.4.
ทie. 4.5.
4.27. Determine the period of oscillations of mercury of mase $m=200 \mathrm{~g}$ poured into a bent tube (Fig. 4.5) whose right arm forms an angle $\theta=30^{\circ}$ with the vertical. The cross-sectional area of the tube is $S=0.50 \mathrm{~cm}^{2}$. The viscosity of mercury is to be neglected.
4.28. 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 $l=20 \mathrm{~cm}$, the coefficient of Iriction between the rod and the wheels is $k=0.18$. Demonstrate that in this case the rod performs harmonic oscillations. Find the period of these oscillations.
4.29. Imagine a shaft going all the way through the Earth from pole to pole along its rotation axis. Assuming the Earth to be a homegeneous ball and neglecting the air drag, find:
(a) the equation of motion of a body falling down inte the shaft;
(b) how long does it take the body to reach the other end of the shaft:
(c) the velocity of the body at the Earth’s centre.
4.30. Find the period of small oscillations of a mathematical pendulum of length $P$ if its point of suspension $O$ moves relative to the Earth’s surface in an arbitrary direction with a constant acceleration w (Fig. 4.7). Calculate that period if $l=21 \mathrm{~cm}, w=g / 2$, and the angle between the vectors $w$ and $g$ equals $\beta-120^{\circ}$.
Fie. 4.7.
Fie. 4.8 .
4.31. In the arrangement shown in Fig. 4.8 the sleeve $M$ of mass $m=0.20 \mathrm{~kg}$ is fixed between two identical springs whose combined stiffness is equal to $\pi=20 \mathrm{~N} / \mathrm{m}$. The sleeve cas slide without frietion over a horizontal bar $A B$. The arrangement rotates with a constant angular velocity $\omega=4.4 \mathrm{rad} / \mathrm{s}$ about a vertical axis passing through the middle of the bar. Find the period of small escillations of the sleeve. At what values of $\theta$ will there be no escillations of the sleeve?
4.32. A plank with a bar placed on it performs horizontal harmonic oscillations with amplitude $a=10 \mathrm{~cm}$. Find the coefficieat of friction between the bar aad the plank if the former starts sliding along 120
the plank when the amplitude of oscillation of the plank becomes less than $T=1.0 \mathrm{~s}$.
4.39. Find the time dependence of the angle of deviation of a mathematical pendulum $80 \mathrm{~cm}$ in length if at the initial moment the pendulum
(a) was deviated through the angle $3.0^{\circ}$ and then set free without push;
(b) was in the equilibrium position and its lower end was imparted the horizontal velocity $0.22 \mathrm{~m} / \mathrm{s}$;
(c) was deviated through the angle $3.0^{\circ}$ and its lower end was imparted the velocity $0.22 \mathrm{~m} / \mathrm{s}$ directed toward the equilibrium position.
4.34. A body $A$ of mass $m_{1}=1.00 \mathrm{~kg}$ and a body $B$ of mass $m_{1}=$ $=4.10 \mathrm{~kg}$ are interconnected by a spring as shown in Fig. 4.9. The body $A$ performs free vertical harmonic oscillations with amplitude $a=1.6 \mathrm{~cm}$ and frequency $\omega=25 \mathrm{~s}^{-1}$. Neglecting the mass of the spring. find the maximum and minimum values of force that this system exerts on the bearing surface.
4.35. A plank with a body of mass $m$ placed on it starts moving straight up according to the law $y=a(1-\cos \omega \hat{)}$ ), where $y$ is the displacement from the initial position, $\omega=$ $=11 \mathrm{~s}^{-1}$, Find:
(a) the time dependence of the force that the body exerts on the plank if $a=4.0 \mathrm{~cm}$; plot this dependence;
(b) the miaimum amplitude of oscillation of the plank at which the body starts falling behiad the plank;
(c) the amplitude of escillation of the plank at which the body springs up to a height $h=50 \mathrm{~cm}$ relative to the initial position (at the moment $t=0$ ).
4.36. A body of mass $m$ was suspended by a non-stretehed spring. and then set free without push. The stiffness of the spring is $x$. Neglecting the mass of the spring, find:
(a) the low of motion $y(t)$, where $y$ is the displacement of the body from the equilibrium position;
(b) the maximum and minimum tensions of the spring in the process of motion.
4.37. A particle of mass $m$ moves due to the force $\mathbf{F}=-$ amr, where $\alpha$ is a positive constant, $r$ is the radius vector of the particle relative to the origin of coordinates. Find the trajectory of its motion if at the initial moment $\mathrm{r}=r_{0} \mathbf{l}$ and the velocity $\mathrm{v}=v_{\mathrm{d}}$. where $\mathrm{I}$ and $)$ are the unit vectors of the $x$ and $y$ axes.
4.38. A body of mass $m$ is suspended from a spring fixed to the ceiling of an elevator car. The stiffness of the spring is $\mathrm{K}$. At the moment $t=0$ the car starts going up with an acceleration w. Neglecting the mass of the spring. find the law of motion $y(t)$ of the body relative to the elevater car if $y(0)=0$ and $y(0)=0$. Consider the following two cases:
(a) $w=$ const;
(b) $w=a t$, where $a$ is a constant.
4.39. A body of mass $m=0.50 \mathrm{~kg}$ is suspended from a rubber cord with elasticity coeffeient $k=50 \mathrm{~N} / \mathrm{m}$. Find the maximum distance over which the body can be pulled down for the body’s oscillations to remain harmonic. What is the energy of oscillation in this case?
4.40. A body of mass $m$ fell from a height $h$ ente the pan of sopring balance (Fig. 4.10). The masses of the pan and the spring are negligible, the stiffness of the latter is $x$. Having stuck to the pan, the body starts performing harmonic escillations in the vertical direction. Find the amplitude and the energy of these oscillations.
Fie. 4.10.
Fie. 4.11.
4.41. Solve the foregoing problem for the case of the pan having a mass $M$. Find the oscillation amplitude in this case.
4.42. A partiele of mass $m$ meves in the plane $x y$ due to the force varying with velocity as $\mathbf{F}=a(\dot{\mathbf{y}} \mathbf{i}-\mathbf{j})$, where $a$ is a positive constant, $i$ and $)$ are the unit vectors of the $x$ and $y$ axes. At the initial moment $t=0$ the particle was located at the point $x=y=0$ and possessed a velocity $\mathrm{v}_{0}$ directed slong the unit vecter ). Find the law of motion $x(t), y(f)$ of the particle, and also the equation of its trajectory.
4.43. A pendulum is constructed as a light thin-walled sphere of radius $\boldsymbol{R}$ 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 centre of the sphere is equal to $L$. How many times will the small oscillations of such a pendulum change after the water freezes? The viscosity of water and the change of its volume on freezing are to be neglected.
4.44. 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 $\%$. The mass of the spring is negligible.
4.45. A uniform rod of mass $m=1.5 \mathrm{~kg}$ suspended by two identical threads $l=90 \mathrm{~cm}$ in length (Fig. 4.13) was turned through a
172
small angle about the vertical axis passing through its middle point $c$. The threads deviated in the process through an angle $\alpha=5.0^{\circ}$. Then the rod was released to start performing smalf oscillations. Find:
(a) the oscillation period;
(b) the rod’s oscillation energy.
Fis. 4.13.
Fig. 4.14.
4.46. An arrangement illustrated in Fig. 4.14 consists of a horizontal uniform dise $D$ of mass $m$ and radius $R$ and a thin rod $A O$ whose tornional coeffeient is equal to $k$. Find the amplitude and the energy of small torsional oscillations if at the initial moment the disc was deviated through an angle $\varphi_{0}$ from the equilibrium position and then imparted an angular velocity $\dot{\varphi}$.
4.47. A uniform rod of mass $m$ and length $t$ performs small oscillations about the horizontal axis passing through its upper end. Find the mean kinetie energy of the rod averaged over one oscillation period if at the initial moment it was defiected from the vertical by an angle $\theta_{0}$ and then imparted an angular velocity $\dot{\theta}_{c}$.
4.48. 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 angular velocity $a$. Neglecting the friction find the period of small oscillations of the peadulum.
4.49. A physical pendulum performs small oscillations about the horizontal axis with frequency $\omega_{1}=15.0 \mathrm{s-1}$. When a small body of mass $m=50 \mathrm{~g}$ is fixed to the pendulum at a distance $l=20 \mathrm{~cm}$ below the axis, the oscillation frequency becomes equal to $\omega_{1}=$ $=10.0 \mathrm{~s}^{-1}$. Find the moment of inertia of the pendulum relative to the oscillation axis.
4.50. Two physical pendulums perform small oscillations about the same horizontal axis with frequencies $\omega_{1}$ and $\omega_{2}$. Their moments of inertia relative to the given axis are equal to $I_{1}$ and $I_{\text {, respectively. }}$ In a state of stable equilibrium the pendulums were fastened rigidly together. What will be the frequency of small oscillations of the compound pendulam?
4.51. A uniform rod of length $l$ performs small oscillations about the horizontal axis $O O^{\prime}$ perpendicular to the rod and passing through one of its points. Find the distance between the centre of inertia of the rod and the axis $O O^{\prime}$ at which the oscillation period is the shortest. What is it equal to?
4.52. 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 reduced length of the given pendulum.
4.53. A smooth horizontal disc rotates about the vertical axis $O$ (Fig. 4.15) with a constant angular velocity $\omega$. A thin uniform $\operatorname{rod} A B$ of length $l$ performs small oscillations about the vertical axis $A$ fixed to the disc at a distance $a$ from the axis of the disc. Find the frequency $\omega_{0}$ of these oscillations.
Fig. 4.15.
Fig. 4.16.
4.54. 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 $m$, and the spring stiffness is $x$. The mass of the thread and the spring is negligible, the thread does not slide over the pulley, there is no friction in the axis of the pulley.
4.55. A uniform cylindrical pulley of mass $M$ and radius $R$ can freely rotate about the horizontal axis $O$ (Fig. 4.17). The free end of
Fig. 4.17.
Fig. 4.18.
a thread tightly wound on the pulley carries a deadweight $A$. At a certain angle $\alpha$ it counterbalances a point mass $m$ fixed at the rim
of the pulley. Find the frequency of small oscillations of the arrangement.
4.56. 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.
4.57. 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 absence of sliding.
4.58. Two cubes with masses $m_{1}$ and $m_{2}$ 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 simultaneously. Find the natural oscillation frequency of the system.
4.59. Two balls with masses $m_{1}=1.0 \mathrm{~kg}$ and $m_{2}=2.0 \mathrm{~kg}$ are slipped on a thin smooth horizontal rod (Fig. 4.19). The balls are
Fig. 4.19.
interconnected by a light spring of stiffness $x=24 \mathrm{~N} / \mathrm{m}$. The lefthand ball is imparted the initial velocity $v_{1}=12 \mathrm{~cm} / \mathrm{s}$. Find:
(a) the oscillation frequency of the system in the process of motion;
(b) the energy and the amplitude of oscillations.
4.60. 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 are equal to $I_{1}$ and $I_{2}$.
4.61. A mock-up of a $\mathrm{CO}_{2}$ 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 oscillations of two types, as shown by the arrows in Fig. 4.20. Knowing the masses of the atoms, find the ratio of frequencies of these oscillations.
Fig. 4.20.
Fig. 4.21.
4.62. 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 inte two equal parts, ench with volume $V_{8}$. The gas pressure is $p_{0}$. The piston was slighity displaced from the equilibrium position and released. Find its oscillation frequency, assuming the processes in the gas to be adiabatic and the friction negligible.
4.63. A small ball of mass $m-21 \mathrm{~g}$ suspended by an insulating thread at a height $h=12 \mathrm{~cm}$ from a large horizontal conducting plane performs small oscillations (Fig. 4.22). Alter a charge $q$ had been imparted to the ball, the oscillation period changed $n=2.0$ times. Find q.
Fie. 4.22.
Fie. 4.23 .
4.64. A small magnetic needle performs small oscillations about an axis perpendicular to the magnetic induction vector. On changing the magnetie induction the needle’s eseillation period decreased $\eta=5.0$ times. How much and in what way was the magnetic induction changed? The oscillation damping is assumed to be nezligible.
4.65. A loop (Fig. 4.23) is formed by two parallel conductors connected by a solenoid with inductance $L$ and a condweting rod of mase $m$ which can freely (without friction) slide over the conductors. The conductors are located in a horizontal plane in a uniform vertical magnetic field with induction $B$. The distance between the conductors is equal to $L$. At the moment $t=0$ the rod is imparted an initial velocity $v_{0}$ directed to the right. Find the law of its motion $x(t)$ if the electrie resistance of the loop is neglizible.
4.66. 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 sero initial velocity along the bars without losing contact with them. The whole system is located in a uniform magnetic field with induction $B$ perpendicular to the plane of the bars. Find the law of motion $x(f)$ of the connector.
4.67. A point performs damped oscillations according to the law $x=a, e^{-t h}$ sin et. Find:
(a) the oscillation amplitude and the velocity of the point at the moment $t=0$;
(b) the moments of time at which the point reaches the extreme positions.
4.68. A body performs torsional oscillations according to the law $\varphi=\uparrow e^{-t h} \cos$ ot. Find:
175
(a) the angular velecity $\dot{q}$ and the angular acceleration $\ddot{\varphi}$ of the body at the moment $t=0$;
(b) the moments of time at which the angular velocity becomes maximum.
4.69. A point performs damped oscillations with frequency o and damping coefficient $\beta$ according to the law (4.1b). Find the initial amplitude $a$, and the initial phase $a$ if at the moment $t=0$ the displacement of the point and its velocity projection are equal to
(a) $x(0)=0$ and $v_{x}(0)=\dot{x}_{4}$ :
(b) $x(0)=x_{4}$ and $y_{x}(0)=0$.
4.70. A point performs damped oscillations with frequency o = $=25 s^{-1}$. Find the damping coefficient $\beta$ if at the initial moment the velocity of the point is equal to aro and its displacement from the equilibrium position is $n=1.020$ times less than the amplitude at that moment.
4.71. A point performs damped oscillations with frequency $\Leftrightarrow$ and damping coelficient $\beta$. Find the velocity amplitude of the point as a function of time $t$ if at the moment $t=0$
(a) its displacement amplitude is equal to $a_{4}$ :
(b) the displacement of the point $x(0)=0$ and its velocity projection $v_{4}(0)-\dot{x}_{4}$.
4.72. There are two damped oscillations with the following periods $T$ and damping coefficients $\beta: T_{1}=0.10 \mathrm{~ms}, B_{2}=100 \mathrm{~s}^{-1}$ and $T_{2}=10 \mathrm{~ms}, \beta_{2}=10 \mathrm{~s}$. Which of them decays ?aster?
4.73. A mathematical pendulum oscillates in a medium for which the logarithmic damping decrement is equal to $\lambda_{0}-1.50$. What will be the logarithmic damping decrement if the resistance of the medium increases $n=2.00$ times? How many times has the resistance of the medium to be increased for the oscillations to become in possible?
4.74. A deadweight suspended from a weightless spring extends it by $\Delta x=9.8 \mathrm{~cm}$. What will be the oscillation period of the deadweight when it is pushed slightly in the vertical direction? The logarithmic damping decrement is equal to $\lambda=3.1$.
4.75. Find the quality factor of the oscillater whose displacement amplitude decreases $\eta=2.0$ times every $n=110$ escillations.
4.76. A particle was displaced from the equilibrium position by a distance $I^{-1.0 \mathrm{~cm}}$ and then left alone. What is the distance that the particle covers is the process of escillations till the complete stop, if the logarithmic damping decrement is equal to $\lambda-0.020$ ?
4.77. Find the quality factor of a mathematical pendulum $l$ $-50 \mathrm{~cm}$ long if during the time interval $\mathrm{r}-5.2 \mathrm{~m}$ in its total mechanical energy decreases $n=4.0 \cdot 10^{4}$ times.
4.78. A uniform dise of radius $R=13 \mathrm{~cm}$ can rotate about a horizontal axis perpendicular to its plane and passing through the edge of the dise. Find the period of small oscillations of that dise if the legarithmic damping decrement is equal to $\lambda=1.00$.
4.79. A thin uniform dise 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 equal to $N=a q$, where $\alpha$ is a constant and $\varphi$ is the angle of rotation from the equilibrium position. The resistance force acting on a unit area of the dise is equal to $F_{1}=\eta$, where $\eta$ is a constant and $y$ is the velocity of the given element of the disc relative to the liquid. Find the frequency of small oscillation.
4.80. A dise $\boldsymbol{A}$ of radius $\boldsymbol{A}$ suspended by an elastic thread between two stationary planes (Fig. 4.24) performs torsional oscillations about its axis $O O^{\prime}$. The moment of inertia of the disc relative to that axis is equal to $I$, the clearance between the dise and each of the planes is equal to $h$, with $h<R$. Find the viscosity of the gas surrounding the dise $A$ if the oscillation period of the dise equals $T$ and the logarithmic damping decrement, $\lambda$.
Vie. 4.24 .
Fic. 4.25.
4.81. A conductor in the shape of a square frame with side a suspended by an elastic thread is located in a uniform horizontal magnetie field with induetion $B$. In equilibrium the plane of the frame is parallel to the vector B (Fig. 4.25). Having been displaced from the equilibrium position, the frame performs small oscillations about a vertical axis passing through its centre. The moment of inertia of the frame relative to that axis is equal to $I$, its electric resistance is $R$. Neglecting the inductance of the frame, find the time interval after which the amplitude of the frame’s deviation angle decreases e-fold.
4.82. A bar of mass $m=0.50 \mathrm{~kg}$ lying on a horizontal plane with a friction coefficient $k=0.10$ is attached to the wall by means of – horizontal non-deformed spring. The stiffness of the spring is equal to $x=2.45 \mathrm{~N} / \mathrm{cm}$, its mass is negligible. The bar was displaced so that the spring was stretched by $x_{4}=3.0 \mathrm{~cm}$, and then released. Find:
(a) the period of oscillation of the bar:
(b) the total number of oscillations that the bar performs until it stops completely.
4.83. A ball of mass $m$ can perform undamped harmonic oseillations about the point $x=0$ with natural frequency $\omega_{0}$. At the moment $t=0$, when the ball was in equilibrium, a force $F_{s}=F_{0}$ cos ot coinciding with the $x$ axis was applied to it. Find the law of forced oscillation $x(f)$ for that ball.
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4.84. A particle of mass $m$ can perform undamped 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 $\tau$ seconds. Find the oscillation amplitude that the particle acquired after the action of the force ceased. Draw the approximate plot $x(f)$ of escillations. Investigate possible cases.
4.85. A ball of mass $m$ when suspended by a spring stretches the latter by $\Delta l$. Due to external vertical force varying according to harmenic law with amplitude $F$, the ball performs forced oscillations. The logarithmic damping decrement is equal to $\lambda$. Neglecting the mass of the spring, find the angular frequency of the external force at which the displacement amplitude of the ball is maximum. What is the magnitude of that amplitude?
4.86. The forced harmonic oscillations have equal displacement amplitudes at frequencies $\omega_{1}=400 \mathrm{~s}-1$ and $\omega_{2}=600 \mathrm{~s}-1$. Find the resonance frequency at which the displacement amplitude is maximum.
4.87. The velocity amplitude of a particle is equal to half the maximum value at the frequencies $\omega_{1}$ and $\omega_{2}$ of external harmonic force. Find:
(a) the frequency corresponding to the velocity resonance;
(b) the damping coefficient $\beta$ and the damped oscillation frequency a of the particle.
4.88. A certain resonance curve describes a mechanical oscillating system with logarithmic damping decrement $\lambda=1.60$. For this curve find the ratio of the maximum displacement amplitude to the displacement amplitude at a very low frequency.
4.89. Due to the external vertical force $F_{s}=F_{\text {, }}$ ces $\omega t$ a body suspended by a spring performs forced steady-state oscillations according to the law $x=4 \cos (\omega t-
abla)$. Find the work performed by the force $F$ during one escillaticn peried.
4.90. A ball of mass $m=50 \mathrm{~g}$ is suspended by a weightless spring with stiffness $x=20.0 \mathrm{~N} / \mathrm{m}$. Due to external vertical harmonic force with frequency $\oplus-25.0 \mathrm{~s}-1$ the ball performs steady-state oscillations with amplitude $a=1.3 \mathrm{~cm}$. In this case the displacement of the ball lags in phase behind the external force by $\varphi=\frac{3}{4} \pi$. Find:
(a) the quality factor of the given oscillator;
(b) the work performed by the external force during one oscillation period.
4.91. A ball of mass $m$ suspended by a weightless spring can perform vertical oscillations with damping coefficient $\beta$. The natural oseillation frequency is equal to $\omega_{6}$. Due to the extemal vertical force varying as $F=F$, ces et the ball performs steady-state harmonic oscillations. Find:
(a) the mean power (P), develeped by the force $F$, averaged over one oscillation period;
(b) the frequency $a$ of the force $F$ at which $(P)$ is maximum; what is $(P)_{\text {mox equal to }}$ equal
4.92. 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 $\eta$ times less than the aatural oscillation frequency $\omega_{0}$ of the ball. How much, in per cent, does the mean power $(P)$ developed by the force $F$ at the frequency of displacement resonance differ from the maximum mean power $(P)_{\text {man? Averaging is performed over one }}$ oscillation period.
4.93. A uniform horizontal dise fixed at its centre to an elastic vertical rod performs forced torsional oscillations due to the moment of forces $N_{1}=N_{m} \cos$ ot. The oscillations obey the law $\varphi=$ – $\varphi_{-} \cos (\omega t-a)$. Find:
(a) the work performed by friction forces acting on the dise during one oscillation period;
(b) the quality factor of the given escillater if the moment of inertia of the dise relative to the axis is equal to $\boldsymbol{I}$.