# Irodov Solutions → Oscillations and waves → Mechanical Oscillations

 4.2. A point moves along the x axis according to the law x = a sin2 (ωt - π/4). Find: (a) the amplitude and period of oscillations; draw the plot x(t); (b) the velocity projection vx as a function of the coordinate x; draw the plot vx(x). Free solution >> 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 x0 = 25.0 cm and its velocity is equal to vx0 = 100 cm/s. Find the coordinate x and the velocity vx of the particle t = 2.40 s after that moment. Free solution >> 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 (a) the extreme position; (b) the equilibrium position. Free solution >> 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. Free solution >> 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. Free solution >> 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. Free solution >> 4.15. Find the trajectory equation y(x) of a point if it moves according to the following laws: (a) x = a sin ωt, y = a sin 2ωt; (b) x = a sin ωt, y = a cos 2ωt. Plot these trajectories. Free solution >> 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) = U0(1 - cos ax); U0 and a are constants. Find the period of small oscillations that the particle performs about the equilibrium position. Free solution >> 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. Free solution >> 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. Free solution >> 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/cm3. The resistance of the liquid is assumed to be negligible. Free solution >> 4.23. A non-deformed 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. Free solution >> 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 χ1 and χ2, their masses are negligible. Free solution >> 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. Free solution >> 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. Free solution >> 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: (a) the time dependence of the force that the body exerts on the plank if a = 4.0 cm; plot this dependence; (b) the minimum amplitude of oscillation of the plank at which the body start falling behind the plank; (c) the amplitude of oscillation of the plank at which the body springs up to a height h = 50 cm relative to the initial position (at the moment t = 0). Free solution >> 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: (a) w = const; (b) w = αt, where α is a constant. Free solution >> 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. Free solution >> 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: (a) the oscillation period; (b) the rod's oscillation energy. Free solution >> 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. Free solution >> 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 ω0 of these oscillations. Free solution >> 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. Free solution >> 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. Free solution >> 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 I1 and I2. Free solution >> 4.61. 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 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. Free solution >> 4.67. A point performs damped oscillations according to the law x = a0e-βt sin ωt. 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. Free solution >> 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. Free solution >> 4.73. 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 increases n = 2.00 times? How many times has the resistance of the medium to be increased for the oscillations to become impossible? Free solution >> 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. Free solution >> 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. Free solution >> 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 F1 = 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. Free solution >> 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 e-fold. Free solution >> 4.83. A ball of mass m can perform undamped harmonic oscillations about the point x = 0 with natural frequency ω0. At the moment t = 0, when the ball was in equilibrium, a force Fx = F0 cos ωt coinciding with the x axis was applied to it. Find the law of forced oscillation x(t) for that ball. Free solution >>