| № |
Condition |
free/or 0.5$ |
| 1081 | 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 motion of a body falling down into 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. |
|
| 1082 | Find the period of small oscillations of a mathematical pendulum of length 1 if its point of suspension 0 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 cm, w = g/2, and the angle between the vectors w and g equals b = 120°. |
|
| 1083 | 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/m. The sleeve can slide without friction over a horizontal bar AB. The arrangement rotates with a constant angular velocity w = 4.4 rad/s about a vertical axis passing through the middle of the bar. Find the period of small oscillations of the sleeve. At what values of o will there be no oscillations of the sleeve? |
|
| 1084 | 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. |
|
| 1085 | 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 push; (b) was in the equilibrium position and its lower end was imparted the horizontal velocity 0.22 m/s; (c) was deviated through the angle 3.0° and its lower end was imparted the velocity 0.22 m/s directed toward the equilibrium position. |
|
| 1086 | A body A of mass m1 = 1.00 kg and a body B of mass m2 = 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 and frequency w = 25 s-1. Neglecting the mass of the spring, find the maximum and minimum values of force that this system exerts on the bearing surface. |
|
| 1087 | 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). |
|
| 1088 | A body of mass in was suspended by a non-stretched spring, and then set free without push. The stiffness of the spring is x. Neglecting the mass of the spring, find: (a) the law 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. |
|
| 1089 | A particle of mass in 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 motion if at the initial moment r = r0i and the velocity v = v0j, where i and j are the unit vectors of the x and y axes. |
|
| 1090 | 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. |
|
| 1091 | A body of mass m = 0.50 kg is suspended from a rubber cord with elasticity coefficient k = 50 N/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? |
|
| 1092 | A body of mass m fell from a height h onto the pan of a spring 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 oscillations in the vertical direction. Find the amplitude and the energy of these oscillations. |
|
| 1093 | Solve the foregoing problem for the case of the pan having a mass M. Find the oscillation amplitude in this case. |
|
| 1094 | 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 moment t = 0 the particle was located at the point x = y = 0 and possessed a velocity v0 directed along the unit vector j. Find the law of motion x (t) , y (t) of the particle, and also the equation of its trajectory. |
|
| 1095 | 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. 4.1.1). 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. |
|
| 1096 | 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. |
|
| 1097 | 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. |
|
| 1098 | 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 of small torsional oscillations if at the initial moment the disc was deviated through an angle f0, from the equilibrium position and then imparted an angular velocity f0'. |
|
| 1099 | A uniform rod of mass m and length L performs small oscillations about the horizontal axis passing through its upper end. Find the mean kinetic energy of the rod averaged over one oscillation period if at the initial moment it was deflected from the vertical by an angle a0 and then imparted an angular velocity a0'. |
|
| 1100 | 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. |
|
