Software laboratory complex for the simulation of laboratory work on the main sections of the course of mechanics for technical specialties.

Laboratory equipment is made in accordance with its real analogues. Each laboratory work includes brief guidelines and reference data necessary for the processing of experimental data.

Type of target computing device and supported platform: IBM – compatible PC running Microsoft Windows, Apple Macintosh PC running MacOS, mobile devices based on Android and iOS operating systems. Additionally, program execution is possible in a web browser environment with support for HTML5 technology and hardware support for 3D graphics (WebGL technology).

Graphics software uses OpenGL 2.0 components. The graphical user interface of the program is implemented in English and Russian.

Multi-platform support allows you to use the software on various computing devices, including interactive whiteboards, smartphones, tablet and desktop computers, which, in turn, increases the flexibility and mobility of the educational process, corresponding to the modern level of education informatization. The web version allows the integration of software into distance e-learning systems.

Laboratory complex includes laboratory works:

###### 1. Uniformly Accelerated Motion

**OBJECTIVE:** Measurement of instantaneous velocity as a function of distance covered.

**SUMMARY:** In the case of uniform acceleration, the instantaneous velocity increases as the distance covered becomes greater. The constant of proportionality between the square of the velocity and the distance covered can be used to calculate the acceleration. This will be investigated in an experiment involving a carriage rolling along a track. In order to measure the instantaneous velocity, a flag of known width attached to the wagon breaks the beam of a photoelectric sensor. The time for which the beam is broken is then measured by means of a digital counter.

###### 2. Motion with Uniform Acceleration

**OBJECTIVE:** Record and evaluate motion with uniform acceleration on a roller track.

**SUMMARY:** When uniformly accelerated motion takes place the velocity at any instant is linearly proportional to the time, while the relationship between distance and time is quadratic. These relationships are to be recorded in an experiment using a roller track with the combination of a spoked wheel employed as a pulley and a photoelectric light barrier.

###### 3. Laws of Collisions

**OBJECTIVE:** Investigate uni-dimensional collisions on an air track.

**SUMMARY:** One important consequence of Newton’s third law is the conservation of momentum in collisions between two bodies. One way of verifying this is to investigate collisions between two sliders on an air track. When all of the kinetic energy is conserved, we speak of elastic collisions. In cases where kinetic energy is only conserved for the common centre of gravity of the two bodies, we use the term inelastic collisions. In this experiment, the individual velocities of the sliders are determined from the times that photoelectric light barriers are interrupted and the momentum values are calculated from these speeds.

###### 4. Free Fall

**OBJECTIVE:** Determine the gravitational acceleration.

**SUMMARY:** In free fall the distance fallen h is proportional to the square of the time t taken to fall that distance. The coefficient of that proportionality can be used to calculate the gravitational acceleration g.

###### 5. Inclined Launch

**OBJECTIVE:** Plotting the «parabolic» trajectories point by point.

**SUMMARY:** The motion of a ball that is thrown upward at an angle to the horizontal in the earth’s gravitational field follows a parabolic curve whose height and width depend on the throwing angle and the initial velocity. The curve is measured point by point using a height scale with two pointers.

###### 6. Precession and Nutation of a Gyroscope

**OBJECTIVE:** Experimental investigation of precession and nutation of a gyroscope and determination of moment of inertia.

**SUMMARY:** A spinning disc exhibits motions known as precession and nutation in addition to its rotational motion, depending on whether there is an external force, and thereby an additional torque, acting upon its axle or if the axle of a disc spinning in an equilibrium state is then deflected from its equilibrium position. The period of precession is inversely proportional to the period of rotation while the period of nutation is directly proportional to the period of rotation. The way the period of precession depends on the period of rotation makes it possible to determine the moment of inertia of the rotating disc.

###### 7. Rotational Motion with Uniform Acceleration

**OBJECTIVE:** Confirm Newton’s equation of motion.

**SUMMARY:** For a body that rotates about a fixed axis with uniform acceleration, the angle of rotation φ increases in proportion to the square of the time t. From this proportionality factor it is possible to calculate the angular acceleration α, which in turn depends, according to Newton’s equation of motion, on the accelerating torque (turning moment) and the moment of inertia of the rigid body.

###### 8. Moment of Inertia of a Horizontal Rod

**OBJECTIVE:** Determine the moment of inertia of a horizontal rod with additional weights attached.

**SUMMARY:** The moment of inertia of a body about its axis of rotation depends on the distribution of its weight in relation to the axis. This is to be investigated for the case of a horizontal rod to which two additional weights are attached symmetrically about the axis of rotation. The rod is coupled to a torsion spring, and its period of oscillation increases as its moment of inertia, which is determined by the additional weights and their distance from the axis, is raised.

###### 9. Moment of Inertia of Various Test Bodies

**OBJECTIVE:** Determine the moment of inertia for various test bodies.

**SUMMARY:** A body’s moment of inertia around an axis of rotation depends on how the mass of the object is distributed with respect to the axis. This will be investigated for a dumbbell, which has two weights symmetrically aligned either side of the axis, for a circular wooden disc, a wooden sphere and both solid and hollow cylinders. The period of oscillation of the test bodies is dependent on the mass distribution and the effective radius of the object.

###### 10. Maxwell’s Wheel

**OBJECTIVE:** Confirm the conservation of energy with the help of Maxwell’s wheel.

**SUMMARY:** Maxwell’s wheel is suspended from threads at both ends of its axle in such a way that it can roll along the threads. In the course of its motion, potential energy is converted into kinetic energy and back again. The process of rolling up and down is repeated until the potential energy derived from the initial height of the wheel is entirely lost due to reflection losses and friction. In this experiment a photoelectric light barrier is set up at various different heights in such a way that the axle of Maxwell’s wheel repeatedly breaks the beam. From the times between these interruptions of the beam it is possible to establish the instantaneous speed of the wheel and thereby to calculate its kinetic energy.

###### 11. Hooke’s Law

**OBJECTIVE:** Confirm Hooke’s law for coil springs under tension.

**SUMMARY:** In any elastic body, extension and tension are proportional to one another. This relationship was discovered by Robert Hooke and is frequently demonstrated using a coil spring with weights suspended from it. The change in the length of the spring is proportional to the force of gravity F on the suspended weight. In this experiment, five different coil springs will be measured. Thanks to a suitable choice of wire diameter and coil diameter, the spring constants all span one order of magnitude. In each case, the validity of Hooke’s law will be demonstrated for forces in excess of the initial tension.

###### 12. First- and Second-Class Levers

**OBJECTIVE:** Verification of the law of the lever.

**SUMMARY:** The law of the lever follows from the equilibrium of moments, which works for all three classes of lever. It represents the physical basis for all kinds of mechanical transmission of force.

###### 13. Parallelogram of Forces

**OBJECTIVE:** Experimental investigation of the vector addition of forces.

**SUMMARY:** The vector addition of forces can be demonstrated in a clear and simple manner on the force table. The point of action of three individual forces in equilibrium is exactly in the middle of the table. Determine the magnitude of the individual forces from the suspended weights and, using a protractor, note the angle of each force vector (the direction of each force). The result of the experiment can be evaluated analytically or represented as a graph.

###### 14. Inclined Plane

**OBJECTIVE:** Determine the forces acting on an inclined plane.

**SUMMARY:** If a body needs to be propelled up an inclined plane, it is not the body’s full weight G which needs to be overcome, but only the component which acts parallel to the plane F1. The fact that this component is less than the weight is more pronounced the smaller the inclination α of the plane becomes.

###### 15. Static and Dynamic Friction

**OBJECTIVE:** Measurement of friction forces.

**SUMMARY:** In order to measure dynamic friction, a friction measuring apparatus is used. It is composed of movable friction strips, which are pulled from under a stationary rough body connected to a dynamometer at constant speed. In order to vary the effective weight (and therefore the normal force) of the stationary body, the angle of the track can be set to any angle.

###### 16. Bending of Flat Beams

**OBJECTIVE:** Measurement of deformation of flat beams supported at both ends and determination of modulus of elasticity.

**SUMMARY:** A flat, level beam’s resistance to deformation in the form of bending by an external force can be calculated mathematically if the degree of deformation is much smaller than the length of the beam. The deformation is proportional to the modulus of elasticity E of the material from which the beam is made. In this experiment, the deformation due to a known force is measured and the results are used to determine the modulus of elasticity for both steel and aluminium.

###### 17. Torsion on Cylindrical Rods

**OBJECTIVE:** Determination of torsional coefficients and shear modulus.

**SUMMARY:** In order for solid bodies to be deformed, an external force needs to be applied. This acts against the body’s own resistance to deformation, which is dependent on the material from which the body is made, as well as its geometry and the direction of the applied force. The deformation is reversible and proportional to the applied force as long as that force is not too great. One example which is often investigated is torsion applied to a uniform cylindrical rod which is fixed at one end. The resistance of the rod to deformation can be numerically analysed and determined by building a set-up which is capable of oscillating involving the rod itself and a pendulum disc and then measuring the period of the oscillation.

###### 18. Falling Sphere Viscosimeter

**OBJECTIVE:** Determine the dynamic viscosity of an aqueous solution of glycerine.

**SUMMARY:** Dynamic viscosity, the coefficient of proportionality between velocity gradient and sheer stress in a liquid, characterises how difficult it is for an object to flow through the liquid. This can be measured using a falling sphere viscosimeter of a type designed by Hoppler. It is also possible to make temperature-independent measurements in conjunction with a circulation thermostat. Measurements are made in an experiment involving an aqueous solution of glycerine. This allows the way that viscosity depends on temperature to be described by the Andrade equation.

###### 19. Surface Tension

**OBJECTIVE:** Measure the surface tension by the «breakaway» method.

**SUMMARY:** To determine the surface tension of a liquid, a blade is immersed horizontally in the liquid and is slowly pulled out upwards while measuring the pulling force. The lamella of liquid that forms at the blade «breaks away» when the force exceeds a certain value. From this force and the length of the blade one can calculate the surface tension.

###### 20. Archimedes’ Principle

**OBJECTIVE:** Determining buoyant updraught as a function of immersion depth.

**SUMMARY:** Archimedes’ principle states that a body immersed in a fluid experiences an upward force (updraught or force of buoyancy) FG. The magnitude of this force is equal to the weight of the displaced fluid. For a regularly shaped immersed body, the updraught is proportional to the depth h to which the body is immersed as long as this is smaller than the height H of the body itself.

###### 21. Harmonic Oscillation of a String Pendulum

**OBJECTIVE:** Measuring the period of oscillation of a string pendulum with bobs of various masses.

**SUMMARY:** The period of oscillation T for a string pendulum is dependent on the length of the pendulum L, but does not depend on the mass of the bob m. This is to be verified by a series of measurements in which the period of oscillation of such a pendulum is measured by means of a photoelectric sensor connected to a digital counter.

###### 22. Elliptical Oscillation of a String Pendulum

**OBJECTIVE:** Description of elliptical oscillations of a string pendulum as the superimposition of two components perpendicular to one another.

**SUMMARY:** Depending on the initial conditions, a suitable suspended string pendulum will oscillate in such a way that the bob’s motion describes an ellipse for small pendulum deflections. If the motion is resolved into two perpendicular components, there will be a phase difference between those components. This experiment will investigate the relationship by measuring the oscillations with the help of two perpendicularly mounted dynamic force sensors. The amplitude of the components and their phase difference will then be evaluated.

###### 23. Variable g Pendulum

**OBJECTIVE:** Measure the period of an oscillating pendulum as a function of the effective component of the gravitational acceleration.

**SUMMARY:** The period of a pendulum is lengthened by tilting its axis away from the horizontal, since the effective component of the gravitational acceleration is reduced.

###### 24. Kater’s Reversible Pendulum

**OBJECTIVE:** Work out the local acceleration due to gravity with the help of a reversible pendulum.

**SUMMARY:** A reversible pendulum is a special design of a normal physical pendulum. It is able to swing from either of two mounting points and can be set up in such a way that the period of oscillation is the same from both these points. The reduction in the length of the pendulum then matches the distance between the two mounting points. This makes it easier to determine the local acceleration due to gravity from the period of oscillation and the reduced pendulum length. Matching of the reversing pendulum is achieved by moving a weight between the mounts as appropriate while a rather larger counterweight outside that length remains fixed.

###### 25. Simple Harmonic Oscillations

**OBJECTIVE:** Measure the oscillations of a coil spring pendulum using an ultrasonic motion sensor.

**SUMMARY:** The oscillations of a coil spring pendulum are a classic example of simple harmonic oscillation. In this experiment, those oscillations are recorded by an ultrasonic motion sensor, which detects the distance to the weight suspended from the spring pendulum.

###### 26. Pohl’s Torsion Pendulum

**OBJECTIVE:** Measurement and analysis of simple harmonic rotary oscillation.

**SUMMARY:** Pohl’s wheel or rotating (torsional) pendulum allows for the investigation of simple harmonic rotary oscillation. The only forces acting on the wheel are the restoring torque provided by a spiral spring and damping torque supplied by means of an eddy current brake with an adjustable current. This experiment demonstrates how the period of oscillation is not dependent on the initial deflection or the initial velocity, and analyses the amplitudes of the oscillations.

###### 27. Forced Harmonic Rotary Oscillation

**OBJECTIVE:** Measurement and analysis of forced harmonic rotary oscillation.

**SUMMARY:** Pohl’s wheel or rotating (torsional) pendulum allows for the investigation of forced harmonic rotary oscillation. For this purpose, the oscillating system is connected to an excitation linkage which is driven by an adjustable-speed DC motor so that the restoring spring periodically extends and compresses. In this experiment the amplitude is measured as a function of the excitation frequency for various degrees of damping and the phase shift between the excitation and the actual oscillation is observed.

###### 28. Coupled Oscillations

**OBJECTIVE:** Record and evaluate oscillation of two identical coupled pendulums.

**SUMMARY:** The oscillation of two identical, coupled pendulums is distinguished by the period of oscillation and the beat period. The beat period is the interval between two points in time when one pendulum is swinging at its minimum amplitude. Both values can be calculated from the natural periods of oscillation for the coupled pendulums when the oscillations are in phase and out of phase.

###### 29. Mechanical Waves

**OBJECTIVE:** Investigate standing waves along a stretched coil spring and a taut rope.

**SUMMARY:** Some examples of where mechanical waves arise include a stretched coil spring, where the waves are longitudinal, or a taut rope where the waves are transverse. In either case, standing waves will be set up if one end of the carrier medium is fixed. This is because the incoming wave and the wave reflected at the fixed end have the same amplitude and are superimposed on one another. If the other end is also fixed, the only way that waves can propagate is if resonance conditions are met. In this experiment the coil spring and the rope are fixed at one end. The other end, a distance L from the fixed point, is fixed to a vibration generator, which uses a function generator to drive small-amplitude oscillations of variable frequency f. This end can also be regarded as a fixed point to a good approximation. The intrinsic frequency of the vibration will be measured as a function of the number of nodes in the standing wave. The speed of propagation of the wave can then be calculated from this data.

###### 30. Speed of Sound in Air

**OBJECTIVE:** Measuring the propagation time of sound pulses in Kundt’s tube.

**SUMMARY:** Sound waves propagate longitudinally in gases. The group velocity here is equal to the phase velocity. In this experiment, we will measure the propagation time of a sound pulse between two microphone probes in Kundt’s tube, and use the result to calculate the speed of sound. The temperature dependence of the speed of sound is examined between room temperature and 50°C. The measurement result matches the result of Laplace’s derivation.

###### 31. Measure Standing Sound Waves in Kundt’s Tube

**OBJECTIVE:** Generate and measure standing sound waves in Kundt’s tube.

**SUMMARY:** Sound waves propagate in gases in the form of longitudinal waves. The overall velocity is equivalent to the phase velocity. In this experiment a standing wave is generated inside Kundt’s tube with both ends closed off. The fundamental frequency is measured as a function of the length of the tube, and the frequencies of the fundamental and overtones are also measured for a fixed length of tube.

###### 32. Propagation of Sound in Rods

**OBJECTIVE:** Investigation of longitudinal sound waves in cylindrical rods and determination of propagation velocity for longitudinal sound waves.

**SUMMARY:** Sound waves can propagate through solids in the form of longitudinal, transverse, dilatational or flexural waves. An elastic longitudinal wave propagates along a rod by means of a periodic sequence of expansion and contraction along the length of the rod. The speed of propagation depends only on the modulus of elasticity and the density of the material when the diameter of the rod is small in comparison to its length. In this experiment, it will be determined from the time it takes sound pulses to travel along the rod.

###### System Requirements

- CPU: Intel/AMD, at least 2 GHz;

- RAM: at least 1 GB;

- VRAM: at least 512 MB;

- Screen Resolution: at least 1024x768x32;

- OpenGL version 2.0;

- DirectX version 9.0.c (for Windows OS);

- Standard keyboard and computer mouse with scroll wheel;

- Means of playing sound (audio speakers or headphones).