3 laws of conservation of momentum and energy. Moscow State University of Printing Arts
Energy and momentum are the most important concepts in physics. It turns out that in general, conservation laws play an important role in nature. The search for conserved quantities and the laws from which they can be derived is the subject of research in many branches of physics. Let us derive these laws in the simplest way from Newton's second law.
Impulse conservation law.Pulse, or amount of movementp defined as the product of mass m material point at speed V: p= mV... Newton's second law using the definition of momentum is written as
= dp= F, (1.3.1)
here F- resultant forces applied to the body.
Closed system is called a system in which the sum of external forces acting on the body is equal to zero:
F= å Fi= 0 . (1.3.2)
Then the change in the momentum of the body in a closed system according to Newton's second law (1.3.1), (1.3.2) is
dp= 0 . (1.3.3)
In this case, the momentum of the particle system remains constant:
p= å pi= const. (1.3.4)
This expression is momentum conservation law, which is formulated as follows: when the sum of external forces acting on a body or a system of bodies is equal to zero, the momentum of a body or a system of bodies is constant.
Law of energy conservation. In everyday life, by the concept of "work" we mean any useful work of a person. In physics, however, is studied mechanical work, which occurs only when the body moves under the action of force. Mechanical work ∆A is defined as the dot product of the force F applied to the body and displacement of the body Δ r as a result of the action of this force:
A A= (F, Δ r) = F A r cosα. (1.3.5)
In formula (1.3.5), the sign of the work is determined by the sign of cos α.
Wanting to move the cabinet, we press on it with force, but if it does not move at the same time, then we do not perform mechanical work. You can imagine a case when the body moves without the participation of forces (by inertia),
in this case, no mechanical work is performed either. If a system of bodies can do work, then it has energy.
Energy is one of the most important concepts not only in mechanics, but also in other areas of physics: thermodynamics and molecular physics, electricity, optics, atomic, nuclear and particle physics.
In any system belonging to the physical world, energy is conserved in any process. Only the form into which it passes can change. For example, when a bullet hits a brick, part kinetic energy(moreover, the larger one) turns into heat. The reason for this is the presence of frictional force between the bullet and the brick, in which it moves with great friction. When the turbine rotor rotates, mechanical energy is converted into electrical energy, and a current arises in a closed circuit. The energy released during the combustion of chemical fuels, i.e. the energy of molecular bonds is converted into thermal energy. The nature of chemical energy is the energy of intermolecular and interatomic bonds, which is essentially molecular or atomic energy.
Energy is a scalar quantity that characterizes the body's ability to do work:
E2- E1 = ∆A. (1.3.6)
When performing mechanical work, the energy of the body passes from one form to another. The energy of the body can be in the form of kinetic or potential energy.
Energy mechanical movement
W kin =.
are called kinetic energy translational movement of the body. Work and energy in SI units are measured in joules (J).
Energy can be caused not only by the movement of bodies, but also by their mutual arrangement and shape. This energy is called potential.
Potential energy is possessed relative to each other by two weights connected by a spring, or a body located at a certain height above the Earth. This last example refers to the gravitational potential energy when a body moves from one height above the Earth to another. It is calculated by the formula
The figure shows the graphs of the dependence of the impulse on the speed of movement of two bodies. What body mass is more and how many times?
1) The masses of bodies are the same
2) Body weight 1 is 3.5 times more
3) Body weight 2 more
4) According to the charts, it is impossible
compare body masses
Plasticine ball mass T, moving at speed V , hits a resting plasticine ball with a mass 2t. After hitting, the balls stick together and move together. What is the speed of their movement?
1) v /3
3) v /2
4) There is not enough data to answer
Wagons weight m = 30 t and m= 20 t are moving along a rectilinear railway track at speeds, the dependence of the projections on an axis parallel to the tracks on time is shown in the figure. In 20 s, an automatic coupling took place between the cars. At what speed and in which direction will the coupled cars go?
1) 1.4 m / s, towards the initial movement 1.
2) 0.2 m / s, towards the initial movement 1.
3) 1.4 m / s, towards the initial movement 2 .
4) 0.2 m / s, towards the initial movement 2 .
Energy (E) is a physical quantity that shows what kind of work the body can do
Perfect work equals a change in body energy
The coordinate of the body changes in accordance with the equation x : = 2 + 30 t - 2 t 2 written in SI. Body weight 5 kg. What is the kinetic energy of the body 3 s after the start of movement?
1) 810 J
2) 1440 J
3) 3240 J
4) 4410 J
The spring is stretched by 2cm . At the same time, work is being done 2 J. What work should be done to stretch the spring another 4 cm.
1) 16 J
2) 4 J
3) 8 J
4) 2 J
Which of the formulas can be used to determine the kinetic energy E k, which the body has at the top point of the trajectory (see figure)?
2) E K = m (V 0) 2/2 + mgh-mgH
4) E K = m (V 0) 2/2 + mgH
The ball was thrown from the balcony 3 times with the same initial speed. The first time the velocity vector of the ball was directed vertically downward, the second time - vertically upward, and the third time - horizontally. Neglect air resistance. The ball speed module when approaching the ground will be:
1) more in the first case
2) more in the second case
3) more in the third case
4) the same in all cases
The skydiver descends evenly from point 1 to point 3 (Fig.). At which point of the trajectory is its kinetic energy most important?
1) At point 1.
2) At point 2 .
3) At point 3.
4) At all points, the values
the energies are the same.
Having driven off the slope of the ravine, the sledge rises along the opposite slope to a height of 2 m (to the point 2 in the figure) and stop. The weight of the sled is 5 kg. Their speed at the bottom of the ravine was 10 m / s. How the total mechanical energy of the sled changed when moving from point 1 to point 2?
1) Has not changed.
2) Increased by 100 J.
3) Decreased by 100 J.
4) Decreased by 150 J.
8.3. The law of conservation of angular momentum. Equation of moments
It is known that angular momentum(angular momentum) of a material point is a vector physical quantity that is numerically equal to the product of its impulse (momentum) by the shoulder, i.e. for the shortest distance from the direction of the pulse to the axis (or center) of rotation:
L i = m i v i r i = m i ω i r i r i = m i r i 2 ω i = I i ω, (8.22)
where I i is the moment of inertia of a material point relative to the selected axis (selected center) of rotation;
ω - angular velocity of a material point.
In vector form
L i= I i ω or L = [r p]. (8.23)
Moment of momentum of a rigid body(system) relative to the selected axis (or center) of rotation is equal to the sum of the angular momentum of the individual material points of the body (bodies of the system) relative to the same axis (the same center) of rotation. Wherein
L= I ω , (8.24)
where is the moment of inertia of the body (system);
ω - angular velocity.
The basic equation of the dynamics of the rotational motion of a material point has the form
,
(8.25)
where L i - angular momentum of a material point relative to the origin of coordinates;
- total torque acting on the i-th material point;
- the resultant moment of all internal forces acting on a material point;
- the resultant moment of all external forces acting on a material point.
For a body consisting of n material points (a system of n bodies):
.
(8.26)
Because 
-the moment of all internal forces is zero, then
or 
,
(8.27)
where L 0 - angular momentum of the body (system) relative to the origin;
M hn is the total torque of external forces acting on the body (system).
From (8.27) it follows that the angular momentum of the body (system) can change under the influence of the moment of external forces, and the rate of its change is equal to the total torque of external forces acting on the body (system).
If M ext = 0, then
, a L 0 = const. (8.28)
Thus, if the external torque does not act on the body (closed system), then its angular momentum remains constant. This statement is called angular momentum conservation law.
For real systems, the law of conservation of angular momentum can be written as
and L 0 x = const. (8.29)
From the law of conservation of angular momentum follows: if the body did not rotate
(ω = 0), then at M = 0 it will not come into rotation; if the body performed rotational motion, then at M = 0, it will perform uniform rotary motion.
Equations 
,
are called moment equations, respectively, for a body (system) or material point.
The equation of moments indicates how the angular momentum changes under the action of forces. Since d L 0 = M∙ dt, then the moment of forces coinciding in direction with the moment of impulse increases it. If the moment of forces is directed towards the moment of impulse, then the latter decreases.
The equation of moments is valid for any arbitrarily chosen fixed axis of rotation.
Here are some examples:
a ) when a cat unexpectedly falls from a great height, it vigorously rotates its tail in one direction or another, achieving the optimal turn of its body for a favorable landing;
b )
a person moves along the edge of a round, freely rotating platform: let the moments of momentum of the platform and the person, respectively, be equal 
and 
, then, taking the system closed, we obtain
,
,
.
Those. angular velocities of rotation of these bodies around them common axis will be opposite in sign, and in magnitude - inversely proportional to their moments of inertia;
v ) experience with the Zhukovsky bench. The person in the middle of the bench and rotating with the platform attracts weights. Neglecting friction in the support bearings, we consider the moment of force to be zero:
,
,
.
,
.
At 
,
, if 
, then 
;
d) in figure skating, the athlete, performing rotation, folds and at the same time accelerates his rotation;
d )
gyroscopes
- devices, the principle of operation of which is based on the law of conservation of the angular momentum of a body: 
... Designed for fixing the initially specified direction in space on an object that moves in an arbitrary direction and unevenly (space rockets, tanks, etc.).
The motion of a body with a constant speed, as follows from Newton's laws, can be carried out in two ways: either without the action of forces on a given body, or under the action of forces, the geometric sum of which is equal to zero. Between them there is fundamental difference... In the first case, no work is done, in the second, the work is done by forces.
The term "work" is used in two meanings: to denote a process and to denote a scalar physical quantity, which is expressed by the product of the projection of the force by the direction of displacement by the length of the displacement vector, the formula "src =" http://hi-edu.ru/e-books/ xbook787 / files / f150.gif "border =" 0 "align =" absmiddle "alt =" (! LANG:
In mathematics, the dot product of two vectors by the cosine of the angle between them is called dot product vectors, therefore the work is equal to the scalar product of the force vector F and the displacement vector formula "src =" http://hi-edu.ru/e-books/xbook787/files/f152.gif "border =" 0 "align =" absmiddle "alt = "(! LANG:
If the angle between the direction of the force and the direction of displacement is sharp, then the force does positive work, if it is dull, then the work of the force is negative.
In the general case, when the force changes in an arbitrary way and the trajectory of the body is arbitrary, calculating the work is not so easy. The entire path of the body is divided into such small sections that on each of them the force can be considered constant. At each of these sites, they find basic work formula "src =" http://hi-edu.ru/e-books/xbook787/files/f154.gif "border =" 0 "align =" absmiddle "alt =" (! LANG:
The total work when moving the body from point 1 to point 2 is equal to the area of the figure under the graph F (r), Fig. eighteen 
.
In practice, it is important to know the speed of the work. The quantity that characterizes the speed at which work is done is called power.
Power is numerically equal to the ratio of work formula "src =" http://hi-edu.ru/e-books/xbook787/files/f156.gif "border =" 0 "align =" absmiddle "alt =" (! LANG:, for which it is performed:
defined "> average power, and the limit of this ratio at defined"> instantaneous power:
example "> dA = defined"> power is determined by the scalar product of the vectors of the acting force and the speed of the body:
example "> v is different with respect to two frames of reference moving relative to each other.
The ability of a particular body to do work is characterized by energy.
In general, energy appears in physics as uniform and universal measure different forms motion of matter and their corresponding interactions.
Since motion is an inalienable property of matter, then any body, system of bodies or fields has energy. Therefore, the energy of a system quantitatively characterizes this system in relation to possible transformations of motion in it. It is clear that these transformations occur due to interactions between parts of the system, as well as between the system and external environment... For various forms of motion and the corresponding interactions, introduce different types of energy- mechanical, internal, electromagnetic, nuclear, etc.
We'll consider mechanical energy... A change in the mechanical movement of a body is caused by forces acting on it from other bodies. To quantitatively characterize the process of energy exchange between interacting bodies in mechanics, the concept of work of force is used. In mechanics, kinetic and potential energies are distinguished.
Kinetic energy A moving material point is called a value, defined as half of the product of the mass of a point by the square of its speed:
example "> m moving forward with speed v is also equal to example"> F acts on a body at rest and causes it to move with speed v, then it does work, and the energy of the moving body increases by the amount of work expended. The increment in the kinetic energy of the body under consideration is equal to the total work of all forces acting on the body:
formula "src =" http://hi-edu.ru/e-books/xbook787/files/f165.gif "border =" 0 "align =" absmiddle "alt =" (! LANG:- the difference between the final and initial values of the kinetic energy.
Statement (3.1) is called kinetic energy change theorem.
The forces acting on the body can differ in their nature and properties. In mechanics, a division of forces has developed into conservative and non-conservative.
Conservative (potential) forces are called, whose work does not depend on the trajectory of the body, but is determined only by its initial and final position, therefore, the work on a closed trajectory is always zero. Such forces are, for example, the force of gravity and the force of elasticity.
Non-conservative (dissipative) forces are called, the work of which depends on the shape of the trajectory and the distance traveled. Non-conservative are, for example, sliding friction force, air or liquid resistance forces.
In the general case, the work of any conservative forces can be represented as a decrease in some quantity P, which is called potential energy body:
defin-e "> The decrease in the value differs from the increment in the sign of the defin-e"> Potential energy is a part of the mechanical energy of the system, determined by the mutual arrangement of bodies and the nature of the interaction between them.
Potential energy is determined by the work that would be performed by the acting conservative forces, moving the body from the initial state, where it is possible to consider by the appropriate choice of coordinates that the potential energy P1 is equal to zero, to a given position.
Expression (3.2) can be written as:
formula "src =" http://hi-edu.ru/e-books/xbook787/files/f169.gif "border =" 0 "align =" absmiddle "alt =" (! LANG:
Therefore, if the function is known, then (3.3) completely determines the force F modulo and direction:
formula "src =" http://hi-edu.ru/e-books/xbook787/files/f171.gif "border =" 0 "align =" absmiddle "alt =" (! LANG:
The vector on the right in (3.4) in square brackets and constructed using the scalar function is called gradient functionП and is denoted by gradП. Designation example "> P in the x direction, respectively example"> y, and example "> z.
Then we can say that the force acting on a material point in a potential field is equal to the gradient of the potential energy of this point taken with the opposite sign:
example "> x from initial state 1 to final state 2:
defin-e "> The potential energy can have a different physical nature and the specific form of the function P depends on the nature of the force field. For example, the potential energy of a body of mass m, located at a height h above the earth's surface, is equal to P = mgh, if the zero level is conventionally taken surface of the earth Since the origin is chosen arbitrarily, the potential energy can have a negative value.
The potential energy of a body under the action of the elastic force of a deformed spring is equal to example "> x is the amount of deformation of the spring, k is the stiffness of the spring.
You can find work against elastic forces. We apply the force F = -kх to the elastic body, then the work at lengthening from the formula "src =" http://hi-edu.ru/e-books/xbook787/files/f179.gif "border =" 0 "align =" absmiddle "alt =" (! LANG::
is determined by the "> function of the state of the system. It depends only on the configuration of the system and its position in relation to external bodies.
The work of the friction force depends on the path, and therefore on the shape of the trajectory. Consequently, the friction force is non-conservative.
A physical quantity equal to the sum of the kinetic and potential energies of the body is called mechanical energy E = example "> P.
It can be shown that the increment of mechanical energy is equal to the total work formula "src =" http://hi-edu.ru/e-books/xbook787/files/f183.gif "border =" 0 "align =" absmiddle "alt =" (! LANG:
Hence, if non-conservative forces are absent or such that they do not perform work on the body during the time of interest to us, then the mechanical energy of the body remains constant during this time: E = const... This statement is known as mechanical energy conservation law.
Consider a system of N particles, between which only conservative forces operate the formula "src =" http://hi-edu.ru/e-books/xbook787/files/f185.gif "border =" 0 "align =" absmiddle "alt = "(! LANG:.
Let's write Newton's second law for all N particles of the system:
the formula "src =" http://hi-edu.ru/e-books/xbook787/files/f187.gif "border =" 0 "align =" absmiddle "alt =" (! LANG:), their sum is equal to zero..gif "border =" 0 "align =" absmiddle "alt =" (! LANG:- the impulse of the entire system.
As a result of the addition of the equations, we obtain
determin-e "> the law of change in the impulse of the system.
For a system of particles, one or another averaging is often used. This is much more convenient than keeping track of every single particle. Such averaging is the center of mass - a point, the radius vector of which is determined by the expression:
formula "src =" http://hi-edu.ru/e-books/xbook787/files/f192.gif "border =" 0 "align =" absmiddle "alt =" (! LANG:is the mass of a particle with a radius vector example "> m is the mass of the system, equal to the sum of the masses of all its particles.
Since mass is a measure of inertia, the center of mass is called center of inertia of the system... Sometimes it is also called the center of gravity, meaning that at this point the resultant of the gravity forces of all particles of the system is applied.
When the system moves, the center of mass changes with the speed
formula "src =" http://hi-edu.ru/e-books/xbook787/files/f195.gif "border =" 0 "align =" absmiddle "alt =" (! LANG:- the momentum of the system, equal to the vector sum of the momenta of all its particles.
Based on (3.8), expression (3.6) can be represented as:
formula "src =" http://hi-edu.ru/e-books/xbook787/files/f197.gif "border =" 0 "align =" absmiddle "alt =" (! LANG:- acceleration of the center of inertia of the system.
Thus, the center of inertia of the system moves under the action of external forces, as material point with a mass equal to the mass of the entire system.
The right side of (3.6) can be zero in two cases: if the system is closed or if external forces cancel each other out. In these cases, we get:
def-e "> If the sum of external forces is zero (the system is closed), the momentum of the system of bodies remains constant for any processes occurring in it (the law of conservation of momentum).
Equation (3.9) - the law of conservation of momentum of a closed system - is one of the most important laws of nature. Like the law of conservation of energy, it is fulfilled always and everywhere - in the macrocosm, microcosm and on the scale of space objects.
Special role physical quantities- energy and momentum is explained by the fact that energy characterizes the properties of time, and momentum characterizes the properties of space: their homogeneity and symmetry.
Time uniformity means that any phenomena at different points in time proceed in exactly the same way.
Uniformity of space means that it has no landmarks, no features. Therefore, it is impossible to determine the position of a particle "relative to space", it can only be determined relative to another particle. Any physical phenomena in all points of space proceed in exactly the same way.
Define "> absolutely elastic (or simply elastic). So, for example, the central collision of two steel balls can be considered absolutely elastic.
the change in mechanical energy during such collisions, as a rule, is characterized by a decrease and is accompanied, for example, by the release of heat. If the bodies after the collision move as a whole, then such a collision is called absolutely inelastic.
Inelastic blow. Let the balls considered above, after impact, move as a whole with a velocity u. We use the law of conservation of momentum:
the formula "src =" http://hi-edu.ru/e-books/xbook787/files/f222.gif "border =" 0 "align =" absmiddle "alt =" (! LANG:
The mechanical energy of the system in the case of an inelastic impact is not conserved since non-conservative forces are at work. Let us find the decrease in the kinetic energy of the balls. Before impact, their energy is equal to the sum of the energies of both balls:
formula "src =" http://hi-edu.ru/e-books/xbook787/files/f224.gif "border =" 0 "align =" absmiddle "alt =" (! LANG:
Energy change
definition "> An example of using the laws of conservation of momentum and mechanical energy
TASK. A bullet of mass m, flying horizontally at a speed v, hits a ball of mass M, suspended by a thread, and gets stuck in it. Determine the height h, to which the ball will rise along with the bullet.
defined "> SOLUTION
The collision of a bullet and a ball is inelastic. According to the law of conservation of momentum for a closed-loop bullet-ball system, we can write:
example "> u is the speed of the ball and bullet.
According to the law of conservation of mechanical energy:
the formula "src =" http://hi-edu.ru/e-books/xbook787/files/f229.gif "border =" 0 "align =" absmiddle "alt =" (! LANG:
Test questions and tasks
1. What is work of force? How to graphically define the work of force?
2. Give the definition of the kinetic energy of the body.
3. What is the theorem about the change in the kinetic energy of a body?
4. What characterizes potential energy?
5. How to determine the specific type of potential energy of the body in a particular force field?
6. What is the change in the potential energy of a spring with stiffness k when it is stretched by?
7. What is total mechanical energy?
8. Formulate the law of conservation of mechanical energy of the body.
9. What is power? What does it depend on?
10. How is the law of conservation of momentum written mathematically?
11. What particular cases of fulfillment of the law of conservation of momentum do you know?
12. What equations can describe an absolutely elastic and absolutely inelastic collision of two bodies?
E full = E kin + U
E kin = mv 2/2 + Jw 2/2 - kinetic energy of translational and rotational motion,
U = mgh - potential energy of a body of mass m at a height h above the Earth's surface.
F tr = kN - sliding friction force, N - normal pressure force, k - friction coefficient.
In the case of off-center impact, the law of conservation of momentum
S p i= const is written in projections on the coordinate axes.
The law of conservation of angular momentum and the law of the dynamics of rotational motion
S L i= const is the law of conservation of angular momentum,
L os = Jw - axial angular momentum,
L orb = [ rp] –Orbital angular momentum,
dL / dt = SM ext - the law of the dynamics of rotational motion,
M= [rF] = rFsina - moment of force, F - force, a - angle between radius - vector and force.
А = òМdj - work during rotational movement.
Mechanics section
Kinematics
Task
Task. The time dependence of the path traveled by the body is given by the equation s = A – Bt + Ct 2. Find the speed and acceleration of the body at time t.
Solution example
v = ds / dt = -B + 2Ct, a = dv / dt = ds 2 / dt 2 = 2C.
Variants
1.1. The dependence of the path traveled by the body on time is given
equation s = A + Bt + Ct 2, where A = 3m, B = 2 m / s, C = 1 m / s 2.
Find the speed in a third second.
2.1. The dependence of the path traveled by the body on time is given
by the equation s = A + Bt + Ct 2 + Dt 3, where C = 0.14 m / s 2 and D = 0.01 v / s 3.
How long after the start of movement is the acceleration of the body
will be equal to 1 m / s 2.
3.1 The wheel, rotating uniformly accelerated, has reached the angular velocity
20 rad / s in N = 10 revolutions after the start of the movement. Find
angular acceleration of the wheel.
4.1 A wheel with a radius of 0.1 m rotates so that the dependence of the angle
j = A + Bt + Ct 3, where B = 2 rad / s and C = 1 rad / s 3. For points lying
on the wheel rim, find 2 s after the start of the movement:
1) angular velocity, 2) linear velocity, 3) angular
acceleration, 4) tangential acceleration.
5.1 A wheel with a radius of 5 cm rotates so that the dependence of the angle
the rotation of the wheel radius versus time is given by the equation
j = A + Bt + Ct 2 + Dt 3, where D = 1 rad / s 3. Find for points lying
on the wheel rim, the change in tangential acceleration for
every second of movement.
6.1 A wheel with a radius of 10 cm rotates so that the dependence
linear velocity of points lying on the rim of the wheel, from
time is given by the equation v = At + Bt 2, where A = 3 cm / s 2 and
B = 1 cm / s 3. Find the angle made up by the vector of the total
acceleration with wheel radius at time t = 5s after
start of movement.
7.1 The wheel rotates so that the dependence of the angle of rotation of the radius
wheel versus time is given by the equation j = A + Bt + Ct 2 + Dt 3, where
B = 1 rad / s, C = 1 rad / s 2, D = 1 rad / s 3. Find the radius of the wheel,
if it is known that by the end of the second second of movement
normal acceleration of points lying on the wheel rim is
and n = 346 m / s 2.
8.1 The radius vector of a material point changes with time in
the law R= t 3 I+ t 2 j. Determine for the moment of time t = 1 s:
speed module and acceleration module.
9.1 The radius vector of a material point changes with time in
the law R= 4t 2 I+ 3t j+2To. Write an expression for a vector
speed and acceleration. Determine for the moment of time t = 2 s
speed module.
10.1 A point moves in the xy plane from a position with coordinates
x 1 = y 1 = 0 with speed v= A i+ Bx j... Define Equation
the trajectory of the point y (x) and the shape of the trajectory.
Moment of inertia
distance L / 3 from the beginning of the rod.
An example of a solution.
M - the mass of the rod J = J st + J gr
L - rod length J st1 = mL 2/12 - rod moment of inertia
2m is the mass of the weight relative to its center. By theorem
Steiner find the moment of inertia
J =? of the rod relative to the axis o, spaced from the center at a distance a = L / 2 - L / 3 = L / 6.
J st = mL 2/12 + m (L / 6) 2 = mL 2/9.
According to the principle of superposition
J = mL 2/9 + 2m (2L / 3) 2 = mL 2.
Variants
1.2. Determine the moment of inertia of a rod with a mass of 2m relative to the axis spaced from the beginning of the rod at a distance L / 4. At the end of the rod, the concentrated mass m.
2.2. Determine the moment of inertia of a rod of mass m relative to
axis spaced from the beginning of the bar at a distance L / 5. At the end
the lumped mass of the rod is 2m.
3.2. Determine the moment of inertia of a rod with a mass of 2m relative to the axis spaced from the beginning of the rod at a distance L / 6. At the end of the rod, the concentrated mass m.
4.2. Determine the moment of inertia of a rod with a mass of 3m about an axis spaced from the beginning of the rod at a distance L / 8. At the end of the rod, the concentrated mass is 2m.
5.2. Determine the moment of inertia of a rod with a mass of 2m about the axis passing through the beginning of the rod. Lumped masses m are attached to the end and middle of the rod.
6.2. Determine the moment of inertia of a rod with a mass of 2m about the axis passing through the beginning of the rod. A lumped mass 2m is attached to the end of the rod, and a lumped mass 2m is attached to the middle.
7.2. Determine the moment of inertia of a rod with mass m relative to the axis spaced from the beginning of the rod by L / 4. Lumped masses m are attached to the end and middle of the rod.
8.2. Find the moment of inertia of a thin homogeneous ring of mass m and radius r relative to the axis lying in the plane of the ring and spaced from its center by r / 2.
9.2. Find the moment of inertia of a thin homogeneous disk of mass m and radius r relative to the axis lying in the plane of the disk and spaced from its center by r / 2.
10.2. Find the moment of inertia of a homogeneous ball of mass m and radius
r relative to the axis spaced from its center by r / 2.
