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physics
mechanics

Questions and Answers of Mechanics

The thickness of a rectangular steel girder equals h. Using the equation of Problem 1.301, find the deflection λ, caused by the weight of the girder in two cases:(a) One end of the girder is
A steel plate of thickness h has the shape of a square whose side equals l, with h
Determine the relationship between the torque N and the torsion angle φ for (a) The tube whose wall thickness hr is considerably less than the tube radius; (b) For the solid rod of circular
Calculate the torque N twisting a steel tube of length l = 3.0 m through an angle φ = 2.0° about its axis, if the inside and outside diameters of the tube are equal to d1 = 30 mm and d2 = 50 mm.
Find the maximum power which can be transmitted by means of a steel shaft rotating about its axis with an angular velocity ω = 120 rad/s, if its length l = 200 cm, radius r = 1.50 cm, and the
A uniform ring of mass m, with the outside radius r2, is fitted tightly on a shaft of radius r1. The shaft is rotated about its axis with a constant angular acceleration β. Find the moment of
Find the elastic deformation energy of a steel rod of mass m = 3.1 kg stretched to a tensile strain ε = 1.0.10-3.
A steel cylindrical rod of length l and radius r is suspended by its end from the ceiling. (a) Find the elastic deformation energy U of the rod. (b) Define U in terms of tensile strain Δl/l
What work has to be performed to make a hoop out of a steel hand of length l = 2.0 m, width h = 6.0 cm, and thickness δ = 2.0 mm? The process is assumed to proceed within the elasticity range of
Find the elastic deformation energy of a steel rod whose one end is fixed and the other is twisted through an angle φ = 6.0°. The length of the rod is equal to l = 1.0 m, and the radius to r =
Find how the volume density of the elastic deformation energy is distributed in a steel rod depending on the distance r from its axis. The length of the rod is equal to l, the torsion angle to φ.
Find the volume density of the elastic deformation energy in fresh water at the depth of h = 1000 m.
Ideal fluid flows along a flat tube of constant cross-section, located in a horizontal plane and bent as shown in Fig. 1.80 (top view). The flow is steady. Are the pressures and velocities of the
Two manometric tubes are mounted on a horizontal pipe of varying cross-section at the sections S1 and S2 (Fig). Find the volume of water flowing across the pipe's section per unit time if the
A Pitot tube (Fig. 1.82) is mounted along the axis of a gas pipeline whose cross-sectional area is equal to S. Assuming the viscosity to be negligible, find the volume of gas flowing across the
A wide vessel with a small hole in the bottom is filled with water and kerosene. Neglecting the viscosity, find the velocity of the water flow, if the thickness of the water layer is equal to h1 = 30
A wide cylindrical vessel 50 cm in height is filled with water and rests on a table. Assuming the viscosity to be negligible, find at what height from the bottom of the vessel a small hole should be
A bent tube is lowered into a water stream as shown in Fig. 1.83. The velocity of the stream relative to the tube is equal to v = 2.5 m/s. The closed upper end of the tube located at the height ho =
The horizontal bottom of a wide vessel with an ideal fluid has a round orifice of radius R1 over which a round closed cylinder is mounted, whose radius R2 > R1 (Fig. 1.84). The clearance between the
What work should be done in order to squeeze all water from a horizontally located cylinder (Fig. 1.85) during the time t by means of a constant force acting on the piston? The volume of water in the
A cylindrical vessel of height h and base area S is filled with water. An orifice of area s
A horizontally oriented tube AB of length l rotates with a constant angular velocity ω about a stationary vertical axis OO' passing through the end A (Fig. 1.86). The tube is filled with an ideal
Demonstrate that in the case of a steady flow of an ideal fluid Eq. {1.7a) turns into Bernoulli equation.
On the opposite sides of a wide vertical vessel filled with water two identical holes are opened, each having the cross-sectional area S = 0.50 cm2. The height difference between them is equal to
The side wail of a wide vertical cylindrical vessel of height h = 75 cm has a narrow vertical slit running all the way down to the bottom of the vessel. The length of the slit is l = 50 cm and the
Water flows out of a big tank along a tube bent at right angles: the inside radius of the tube is equal to r = 0.50 cm (Fig. 1.87). The length of the horizontal section of the tube is equal to l = 22
A side wall of a wide open tank is provided with a narrowing tube (Fig. 1.88) through which water flows out. The cross-sectional area of the tube decreases from S = 3.0 cm2 to s = 1.0 cm2. The water
A cylindrical vessel with water is rotated about its vertical axis with a constant angular velocity ω. Find: (a) The shape of the free surface of the water; (b) The water pressure
A thin horizontal disc of radius R = 10 cm is located with-in a cylindrical cavity filled with oil whose viscosity η = 0.08 P (Fig. 1.89). The clearance between the disc and the horizontal planes
A long cylinder of radius R1 is displaced along its axis with a constant velocity vo inside a stationary co-axial cylinder of radius R2. The space between the cylinders is filled with viscous liquid.
A fluid with viscosity η fills the space between two long co-axial cylinders of radii R1 and R2, with R1
A tube of length l and radius R carries a steady flow of fluid whose density is p and viscosity r1. The fluid flow velocity depends on the distance r from the axis of the tube as v = vo (i –
In the arrangement shown in Fig. 1.90 a viscous liquid whose density is p = 1.0 g/cm3 flows along a tube out of a wide tank A. Find the velocity of the liquid flow, if h1 = 10 cm, h2 = 20 cm, and h3
The cross-sectional radius of a pipeline decreases gradually as r = roe -ax, where a = 0.50 m-1, x is the distance from the pipeline inlet. Find the ratio of Reynolds numbers for two cross-sections
When a sphere of radius r1 = 1.2 mm moves in glycerin, the laminar flow is observed if the velocity of the sphere does not exceed v1 = 23 cm/s. At what minimum velocity v2 of a sphere of radius r2 =
A lead sphere is steadily sinking in glycerin whose viscosity is equal to η = 13.9 P. What is the maximum diameter of the sphere at which the flow around that sphere still remains laminar? It is
A steel ball of diameter d = 3.0 mm starts sinking with zero initial velocity in olive oil whose viscosity is η = 0.90 P. How soon after the beginning of motion will the velocity of the ball
A rod moves lengthwise with a constant velocity v relative to the inertial reference frame K. At what value of v will the length of the rod in this frame be η = 0.5% less than its proper length?
In a triangle the proper length of each side equals a. Find the perimeter of this triangle in the reference frame moving relative to it with a constant velocity V along one of its (a) Bisectors;
Find the proper length of a rod if in the laboratory frame of reference its velocity is v = c/2, the length l = 1.00 m, and the angle between the rod and its direction of motion is θ = 45°.
A stationary upright cone has a taper angle θ = 45°, and the area of the lateral surface So = 4.0 m2. Find: (a) Its taper angle; (b) Its lateral surface area, in the reference frame moving
With what velocity (relative to the reference frame K) did the clock move, if during the time interval t = 5.0 s, measured by the clock of the frame K, it became slow by Δt = 0.10 s?
A rod flies with constant velocity past a mark which is stationary in the reference frame K. In the frame K it takes Δt = 20 ns for the rod to fly past the mark. In the reference frame fixed to
The proper lifetime of an unstable particle is equal to Δto = 10 ns. Find the distance this particle will traverse till its decay in the laboratory frame of reference, where its lifetime is
In the reference frame K a muon moving with a velocity v = 0.990c travelled a distance l = 3.0 km from its birthplace to the point where it decayed. Find: (a) The proper lifetime of this muon; (b)
Two particles moving in a laboratory frame of reference along the same straight line with the same velocity v = (3/4)c strike against a stationary target with the time interval Δt = 50 ns. Find
A rod moves along a ruler with a constant velocity. When the positions of both ends of the rod are marked simultaneously in the reference frame fixed to the ruler, the difference of readings on the
Two rods of the same proper length lo move toward each other parallel to a common horizontal axis. In the reference frame fixed to one of the rods the time interval between the moments, when the
Two unstable particles move in the reference frame K along a straight line in the same direction with a velocity v = 0.990c. The distance between them in this reference frame is equal to l = 120 m.
A rod AB oriented along the x axis of the reference frame K moves in the positive direction of the x axis with a constant velocity v. The point A is the forward end of the rod, and the point B its
The rod A'B' moves with a constant velocity v relative to the rod AB (Fig). Both rods have the same proper length lo and at the ends of each of them clocks are mounted, which are synchronized
There are two groups of mutually synchronized clocks K and K' moving relative to each other with a velocity v as shown in Fig. The moment when the clock A' gets opposite the clock A is taken for the
The reference frame K' moves in the positive direction of the x axis of the frame K with a relative velocity V. Suppose that at the moment when the origins of coordinates O and O' coincide, the clock
At two points of the reference frame K two events occurred separated by a time interval Δt. Demonstrate that if these events obey the cause-and-effect relationship in the frame K (e.g. a shot
The space-time diagram of Fig shows three events A, B, and C which occurred on the x axis of some inertial reference frame. Find:(a) The time interval between the events A and B in the reference
The velocity components of a particle moving in the xy plane of the reference frame K are equal to vx and vy. Find the velocity v' of this particle in the frame K' which moves with the velocity V
Two particles move toward each other with velocities v1 = 0.50c and v2 = 0.75c relative to a laboratory frame of reference. Find:(a) The approach velocity of the particles in the laboratory frame of
Two rods having the same proper length lo move lengthwise toward each other parallel to a common axis with the same velocity v relative to the laboratory frame of reference. What is the length of
Two relativistic particles move at right angles to each other in a laboratory frame of reference, one with the velocity v1 and the other with the velocity v2. Find their relative velocity.
An unstable particle moves in the reference frame K' along its y' axis with a velocity v'. In its turn, the frame K' moves relative to the frame K in the positive direction of its x axis with a
A particle moves in the frame K with a velocity v at an angle θ to the x axis. Find the corresponding angle in the frame K' moving with a velocity V relative to the frame K in the positive
The rod AB oriented parallel to the x' axis of the reference frame K' moves in this frame with a velocity v' along its y' axis. In its turn, the frame K' moves with a velocity V relative to the frame
The frame K' moves with a constant velocity V relative to the frame K. Find the acceleration w' of a particle in the frame K', if in the frame K this particle moves with a velocity v and acceleration
An imaginary space rocket launched from the Earth moves with an acceleration w' = 10g which is the same in every instantaneous co-moving inertial reference frame. The boost stage lasted τ = 1.0
From the conditions of the foregoing problem determine the boost time τo in the reference frame fixed to the rocket. Remember that this time is defined by the formulawhere dt is the time in the
How many times does the relativistic mass of a particle whose velocity dithers from the velocity of light by 0.010% exceed its rest mass?
The density of a stationary body is equal to Po. Find the velocity (relative to the body) of the reference frame in which the density of the body is η = 25% greater than Po.
A proton moves with a momentum p = 10.0 GeV/c, where c is the velocity of light. How much (in per cent) does the proton velocity dither from the velocity of light?
Find the velocity at which the relativistic momentum of a particle exceeds its Newtonian momentum η = 2 times.
What work has to be performed in order to increase the velocity of a particle of rest mass mo from 0.60 c to 0.80 c? Compare the result obtained with the value calculated from the classical formula.
Find the velocity at which the kinetic energy of a particle equals its rest energy.
At what values of the ratio of the kinetic energy to rest energy can the velocity of a particle be calculated from the classical formula with the relative error less than ε = 0.010?
Find how the momentum of a particle of rest mass mo depends on its kinetic energy. Calculate the momentum of a proton whose kinetic energy equals 500 MeV.
A beam of relativistic particles with kinetic energy T strikes against an absorbing target. The beam current equals I, the charge and rest mass of each particle are equal to e and mo respectively.
A sphere moves with a relativistic velocity v through a gas whose unit volume contains n slowly moving particles, each of mass m. Find the pressure p exerted by the gas on a spherical surface element
A particle of rest mass mo starts moving at a moment t = 0 due to a constant force F. Find the time dependence of the particle's velocity and of the distance covered.
A particle of rest mass mo moves along the x axis of the frame K in accordance with the law x = √a2+c 2t2, where a is a constant, c is the velocity of light, and t is time. Find the force
Proceeding from the fundamental equation of relativistic dynamics, find: (a) Under what circumstances the acceleration of a particle coincides in direction with the force F acting on it; (b) The
A relativistic particle with momentum p and total energy E moves along the x axis of the frame K. Demonstrate that in the frame K' moving with a constant velocity V relative to the frame K in the
The photon energy in the frame K is equal to ε. Making use of the transformation formulas cited in the foregoing problem, find the energy ε' of this photon in the frame K' moving with a
Demonstrate that the quantity E2- p2c2 for a particle is an invariant, i.e. it has the same magnitude in all inertial reference frames. What is the magnitude of this invariant?
A neutron with kinetic energy T = 2moc2, where mo is its rest mass, strikes another, stationary, neutron. Determine: (a) The combined kinetic energy T of both neutrons in the frame of their centre
A particle of rest mass mo with kinetic energy T strikes a stationary particle of the same rest mass. Find the rest mass and the velocity of the compound particle formed as a result of the collision.
How high must be the kinetic energy of a proton striking another, stationary, proton for their combined kinetic energy in the frame of the centre of inertia to be equal to the total kinetic energy of
A stationary particle of rest mass mo disintegrates into three particles with rest masses m1, m2, and m3. Find the maximum total energy that, for example, the particle m1 may possess.
A relativistic rocket emits a gas jet with non-relativistic velocity u constant relative to the rocket. Find how the velocity v of the rocket depends on its rest mass m if the initial rest mass of
Does a car speedometer measure speed, velocity, or both?
Can an object have a varying speed if its velocity is constant? If yes, give examples?
When an object moves with constant velocity, does its average velocity during any time interval differ from its instantaneous velocity at any instant?
In drag racing, is it possible for the car with the greatest speed crossing the finish line to lose the race? Explain.
If on object has a greater speed than a second object, does the first necessarily have a greater acceleration? Explain, using examples.
Compare the acceleration of a motorcycle that accelerates from 80 km/h to 90 km/h with the acceleration of a bicycle that accelerates from rest to 10 km/h in the same time.
Can an object have a northward velocity and a southward acceleration? Explain.
Can the velocity of an object be negative when its acceleration is positive? What about vice versa?
Give an example where both the velocity and acceleration are negative.
Two cars emerge side by side from a tunnel. Car A is traveling with a speed of 60 km/h and has an acceleration of 40 km/h/min. Car B has a speed of 40 km/h and has an acceleration of 60 km/h/min.
Can an object be increasing in speed as its acceleration decreases? If so, give an example. If not, explain.
A baseball player hits a foul ball straight up into the air. It leaves the bat with a speed of 120 km/h. In the absence of air resistance, how fast will the ball be travelling when the catcher
As a freely falling object speeds up, what is happening to its acceleration due to gravity-does it increase, decrease, or stay the same?
How would you estimate the maximum height you could throw a ball vertically upward? How would you estimate the maximum speed you could give it?

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