A physical pendulum performs small oscillations about the horizontal axis with frequency co = 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 axis, the oscillation frequency becomes equal to w2 = 10.0 s-1. Find the moment• of inertia of the pendulum relative to the oscillation axis.
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 w. Neglecting the friction find the period of small
oscillations of the pendulum.
An arrangement illustrated in Fig 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 φ, from the equilibrium position and then imparted an angular velocity φo.
A uniform rod of mass m = 1.5 kg suspended by two identical threads 1 = 90 cm in length (Fig) 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 a = 5.0°.Then the rod was released to start performing small oscillations.Find:
(a) the oscillation period;
(b) the rod's oscillation energy.
A pendulum is constructed as a light thin-walled sphere of radius R filled up with water and suspended at the point 0 from a light rigid rod (Fig). The distance between the point 0 and the centre of the sphere is equal to 1. 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.
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.
Solve the foregoing problem for the case of the pan having a mass M. Find the oscillation amplitude in this case.
A body of mass m fell from a height h onto the pan of a spring balance (Fig.)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.
A particle of mass in moves due to the force F = - αmr,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 = roi and the velocity v = voj, where I and j are the unit vectors of the x and y axes.
A mathematical pendulum oscillates in a medium for which the logarithmic damping decrement is equal to 20 = 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?
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.
A plank with a body of mass m placed on it starts moving straight up according to the law y = a (1 — cos wt), where y is the displacement from the initial position w = 11 /s. 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 starts 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).
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;
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 scillation of the plank becomes less than T = 1.0 s.
In the arrangement shown in Fig 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?.
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
A uniform rod is placed on two spinning wheels as shown in Fig. The axes of the wheels are separated by a distance 1= 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.
Determine the period of oscillations of mercury of mass = 200 g poured into a bent tube (Fig.) whose right arm forms an angle 0 = 30° with the vertical. The cross-sectional area of the tube is S = 0.50 cm2. The viscosity of mercury is to be neglected.
A small body of mass in 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 the body in the transverse direction. The mass of the string is negligible,the gravitational field is absent.
Find the period of small vertical oscillations of a body with mass m in the system illustrated in Fig.The stiffness values of the springs are xi and x2, their masses are negligible.
Determine the period of small longitudinal oscillations of a body with mass m in the system shown in Fig. The stiffness values of the springs are xi and x2. The friction and the masses of the springs are negligible.
Calculate the period of small oscillations of a hydrometer (Fig) 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 p = 1.00 g/cm3. The resistance of the liquid is assumed to be negligible.
ball is suspended by a thread of length l at the point 0 on
the wall, forming a small angle a with the vertical (Fig. 4.1). Then
Fig. 4.1, Fig. 4.2.
the thread with the ball was deviated through a small angle 13(l > a)
and set free. Assuming the collision of the ball against the wall to
be perfectly elastic, find the oscillation period of such a pendulum
Determine the period of small oscillations of a mathematical pendulum, that is a ball suspended by a thread 1 = 20 cm in length,if it is located in a liquid whose density is = 3.0 times less than that of the ball. The resistance of the liquid is to be neglected.
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 1 = 1.0 m in length. The tension of the string is assumed to be constant and equal to F = 10 N.
Solve the foregoing problem if the potential energy has the form U (x) = al/2 — b/x, where a and b are positive constants.
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.
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 wt, y = a cos 2wt.
Plot these trajectories.
A point moves in the plane xy according to the law x = a sin wt y = b cos wt, where a, b, and w 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 vector r relative to the origin of coordinates.
A point A oscillates according to a certain harmonic law in the reference frame K' which in its turn performs harmonic oscillations relative to the reference frame K. Both oscillations occur along the same direction. When the K' frame oscillates at the frequency 20 or 24 Hz, the beat frequency of the point A in the K frame turns out to be equal to v. At what frequency of oscillation of the frame K' will the beat frequency of the point A become equal to 2v?
A point A oscillates according to a certain harmonic law in the reference frame K' which in its turn performs harmonic oscillations relative to the reference frame K. Both oscillations occur along the same direction. When the K' frame oscillates at the frequency 20 or 24 Hz, the beat frequency of the point A in the K frame turns
H2O2 can be obtained when following reacts with H2SO4 except with
The volume strength of 3.57 M solution of hydrogen peroxide is
The boiling point of water is exceptionally high due to
In alkaline medium, which elements can produce hydrogen?
2-ethyl anthraquinol when oxidised in air produces
The term hydride gap refers to which region of periodic table?
When 1 mole of PbS reacts completely with 
An orange coloured solution of K2Cr2O7 acidified with H2SO4 and treated with a substance X gives a blue coloured solution of CrO5. The substance X is
Water gas is a mixture of
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.
A point participates simultaneously in two harmonic oscillations of the same direction: x1 = a cos wt and x2 = a cos 2wt.Find the maximum velocity of the point.
Using graphical means, find an amplitude a of oscillations resulting from the superposition of the following oscillations of the same direction:
(a) x1 =3 .0 cos (wt -1- π/3), x 2 =8 .0sin (wt π/6);
For a first order reaction A -> P, the temperature (T) dependent rate constant (k) was found to follow the equation
log k = – (2000) 1/T + 6.0. The pre-exponential factor A and the activation energy Ea, respectively, are
For the elementary reaction M -> N, the rate of disappearance of M increases by a factor of 8 upon doubling the concentration of M. The order of the reaction with respect to M is
Consider a reaction aG + bH -> Products. When concentration of both the reactants G and H is doubled, the rate increases by eight times. However, when concentration of G is doubled keeping the concentration of H fixed, the rate is doubled. The overall order of the reaction is
A positron is emitted from 
. The ratio of the atomic mass and atomic number of the resulting nuclide is
The half life for decay of a radio active isotope is 14 hrs. Determine the fraction of radio isotope remaining undecayed after 30 hours
Half life of a chemical reaction at a particular concentration is 50 minutes. When the concentration of the reactant is doubled, the half life becomes 100 minutes, then the order of reaction is
The concentration of a reactant in a solution falls from 0.2 M to 0.1 M in 2 hours and to 0.05 M in 4 hours. The order of reaction is