Irodov Solutions → Oscillations and waves → Mechanical Oscillations 
4.2. A point moves along the x axis according to the law x = a sin^{2} (ωt  π/4). Find:
4.3. A particle performs harmonic oscillations along the x axis about the equilibrium position x = 0. The oscillation frequency is ω = 4.00 s^{1}. At a certain moment of time the particle has a coordinate x_{0} = 25.0 cm and its velocity is equal to v_{x0} = 100 cm/s. Find the coordinate x and the velocity v_{x} of the particle t = 2.40 s after that moment.
4.5. A point performs harmonic oscillations along a straight line with a period T = 0.60 s and an amplitude a = 10.0 cm. Find the mean velocity of the point averaged over the time interval during which it travels a distance a/2, starting from
4.7. A particle moves along the x axis according to the law x = a cos ωt. Find the distance that the particle covers during the time interval from t = 0 to t.
4.9. A particle performs harmonic oscillations along the x axis according to the law x = a cos ωt. 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 dP/dx depends on x. Here dP denotes the probability of the particle falling within an interval from x to x + dx. Plot dP/dx as a function of x.
4.12. The superposition of two harmonic oscillations of the same direction results in the oscillation of a point according to the law x = a cos 2.1t cos 50.0t, where t is expressed in seconds. Find the angular frequencies of the constituent oscillations and the period with which they beat.
4.15. Find the trajectory equation y(x) of a point if it moves according to the following laws:
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 ax); U_{0} and a are constants. Find the period of small oscillations that the particle performs about the equilibrium position.
4.18. 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 assumed to be constant and equal to F = 10 N.
4.19. 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 than that of the ball. The resistance of the liquid is to be neglected.
4.22. 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 = 3.2 mm, the density of the liquid is ρ = 1.00 g/cm^{3}. The resistance of the liquid is assumed to be negligible.
4.23. A nondeformed spring whose ends are fixed has a stiffness χ = 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. Neglecting the mass of the spring, find the period of small longitudinal oscillations of the body. The force of gravity is assumed to be absent.
4.25. 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 χ_{1} and χ_{2}, their masses are negligible.
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 cm, the coefficient of friction 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.
4.32. 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 the plank when the amplitude of oscillation of the plank becomes less than T = 1.0 s.
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 ωt), where y is the displacement from the initial position, ω = 11 s^{1}. Find:
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 χ. 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 elevator car if y(0) = 0 and y'(0) = 0. Consider the following two cases:
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 χ. The mass of the springs is negligible.
4.45. A uniform rod of mass m = 1.5 kg suspended by two identical threads l = 90 cm in length (Fig. 4.13) was turned through a small angle about the vertical axis passing through its middle point C. The threads deviated in the process through an angle α = 5.0°. Then the rod was released to start performing small oscillations. Find:
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 ω. Neglecting the friction find the period of small oscillations of the pendulum.
4.53. A smooth horizontal disc rotates about the vertical axis O (Fig. 4.15) with a constant angular velocity ω. A thin uniform rod AB 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 ω_{0} of these oscillations.
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 χ. 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 a thread tightly wound on the pulley carries a deadweight A. At a certain angle α it counterbalances a point mass m fixed at the rim of the pulley. Find the frequency of small oscillations of the arrangement.
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 mockup of a 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.
4.67. A point performs damped oscillations according to the law x = a_{0}e^{βt} sin ωt. Find:
4.70. A point performs damped oscillations with frequency ω = 25 s^{1}. Find the damping coefficient β if at the initial moment the velocity of the point is equal to zero and its displacement from the equilibrium position is η = 1.020 times less than the amplitude at that moment.
4.73. A mathematical pendulum oscillates in a medium for which the logarithmic damping decrement is equal to λ_{o} = 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 impossible?
4.74. A deadweight suspended from a weightless spring extends it by Δx = 9.8 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 λ = 3.1.
4.78. 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 logarithmic damping decrement is equal to λ = 1.00.
4.79. 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 equal to N = αφ, where α is a constant and φ is the angle of rotation from the equilibrium position. The resistance force acting on a unit area of the disc is equal to F_{1} = ηv, where η is a constant and v is the velocity of the given element of the disc relative to the liquid. Find the frequency of small oscillation.
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 magnetic field with induction 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 efold.
4.83. A ball of mass m can perform undamped harmonic oscillations about the point x = 0 with natural frequency ω_{0}. At the moment t = 0, when the ball was in equilibrium, a force F_{x} = F_{0} cos ωt coinciding with the x axis was applied to it. Find the law of forced oscillation x(t) for that ball.
