where G=6.67×10 -11 N×m 2 /kg 2 is the universal gravitational constant.
This law is called the law of universal gravitation.
The force with which bodies are attracted to the Earth is called gravity. The main feature of gravity is the experimental fact that this force to all bodies, regardless of their mass, reports the same acceleration directed towards the center of the Earth.
It follows from this that the ancient Greek philosopher Aristotle was wrong when he argued that heavy bodies fall to the Earth faster than light ones. He did not take into account that, in addition to gravity, the body is subject to a resistance force against the air, which depends on the shape of the body.
A musket bullet and a heavy cannonball, thrown by the Italian physicist Galileo Galilei from the famous 54.5 m high tower located in the city of Pisa, reached the surface of the Earth almost simultaneously, i.e. fell with the same acceleration (Fig. 4.27).
Calculations carried out by G. Galilei showed that the acceleration acquired by bodies under the influence of the Earth's gravity is equal to 9.8 m/s 2 .
Further more accurate experiments were carried out by I. Newton. He took a long glass tube into which he placed a lead ball, a stopper and a feather (Fig. 4.28).
This tube is now called a "Newton tube". Turning the tube over, he saw that the ball fell first, then the cork, and only then the feather. If the air is first pumped out of the tube using a pump, then after turning the tube over, all the bodies will fall to the bottom of the tube simultaneously. And this means that in the second case all bodies increased their speed equally, i.e. received the same acceleration. And this acceleration was imparted to them by a single force - the force of attraction of bodies to the Earth, i.e. gravity. The calculations made by Newton confirmed the correctness of the calculations of G. Galileo, since he also obtained the value of the acceleration acquired by freely falling bodies in the “Newton tube”, equal to 9.8 m/s 2. This constant acceleration is called acceleration of free fall on Earth and is designated by the letter g(from the Latin word “gravitas” - heaviness), i.e. g = 9.8 m/s 2.
Free fall is understood as the movement of a body that occurs under the influence of one single force - gravity (air resistance forces are not taken into account).
On other planets or stars, the value of this acceleration is different, as it depends on the masses and radii of the planets and stars.
Here are the values of the acceleration due to gravity on some planets of the Solar System and on the Moon:
1. Sun g = 274 N/kg
2. Venus g = 8.69N/kg
3. Mars g = 3.86 N/kg
4. Jupiter g = 23 N/kg
5. Saturn g = 9.44 N/kg
6. Moon (Earth satellite) g = 1.623 N/kg
How can we explain the fact that the acceleration of all bodies freely falling to the Earth is the same? After all, the greater the mass of a body, the greater the force of gravity acting on it. You and I know that 1 N is the force that imparts an acceleration of 1 m/s 2 to a body weighing 1 kg. At the same time, the experiments of G. Galileo and I. Newton showed that the force of gravity changes the speed of any body 9.8 times more. Consequently, a force of 9.8 N acts on a body weighing 1 kg, and a force of gravity equal to 19.6 N will act on a body weighing 2 kg, etc. That is, the greater the body mass, the greater the force of gravity will act on it, and the proportionality coefficient will be equal to 9.8 N/kg. Then the formula for calculating gravity will look like or in general:
Accurate measurements have shown that the acceleration of gravity decreases with height and changes slightly with changes in latitude due to the fact that the Earth is not a strictly spherical body (it is slightly flattened at the poles). In addition, it may depend on the geographical location on the planet, since the density of the rocks that make up the surface layer of the Earth is different. The latter fact makes it possible to detect mineral deposits.
Here are some values of the acceleration of gravity on Earth:
1. At the North Pole g = 9.832 N/kg
2. At the equator g = 9.780 N/kg
3. At latitude 45 o g = 9.806 N/kg
4. At sea level g = 9.8066 N/kg
5. At the peak of Khan Tengri, 7 km high, g = 9.78 N/kg
6. At a depth of 12 km g = 9.82 N/kg
7. At a depth of 3000 km g = 10.20 N/kg
8. At a depth of 4500 km g = 6.9 N/kg
9. At the center of the Earth g = 0 N/kg
The attraction of the Moon leads to the formation of ebbs and flows in the seas and oceans on Earth. The tide in the open ocean is about 1 m, and off the coast of the Bay of Fundy in the Atlantic Ocean it reaches 18 meters.
The distance from the Earth to the Moon is enormous: about 384,000 km. But the gravitational force between the Earth and the Moon is large and amounts to 2 × 10 20 N. this is due to the fact that the masses of the Earth and the Moon are large.
When solving problems, unless there are special reservations, the value of 9.8 N/kg can be rounded to 10 N/kg.
The lag of the pendulums of clocks synchronized on the first floor of a high-rise building is associated with a change in the quantity g. Since the value g decreases as the height increases, then the clock on the top floor will begin to lag.
Example. Determine the force with which a steel bucket weighing 500 g, volume 12 liters, completely filled with water, presses on the support.
The force of gravity is equal to the sum of the force of gravity of the bucket itself, equal to F gravity1 = m 1 g, and the force of gravity of water poured into a bucket, equal to F heavy1 = m 2 g= ρ 2 V 2 g, i.e.
F strand = m 1 g+ρ 2 V 2 g
Substituting numerical values, we get:
F strand = 0.5 kg 10 N/kg + 10 3 kg/m 3 12 10 -3 m 3 10 N/kg = = 125 N.
Answer: F strand = 125 N
Questions for self-control:
1. What force is called gravitational? What is the reason for this power?
2. What does the law of universal gravitation say?
3. What force is called gravity? What is its main feature?
4. Does gravity exist on other planets? Justify your answer.
5. For what purpose did G. Galileo conduct experiments on the Leaning Tower of Pisa?
6. What do the experiments that Newton conducted with the “Newton tube” prove to us?
7. What acceleration is called the acceleration of gravity?
8. You have two identical sheets of paper. Why does a crumpled leaf fall to the ground faster, even though each leaf has the same force of gravity?
9. What is the fundamental difference in the explanation of free fall by Aristotle and Newton?
10. Give a report on how Aristotle, Galileo and Newton studied free fall.
All bodies attract each other. For material points (or balls), the law of universal gravitation has the form
where, - masses of bodies, - distance between material points or centers of balls, - gravitational constant. The masses included in this law are a measure of the gravitational interaction of bodies. Experience shows that gravitational and inertial masses are equal.
Physical meaning: the gravitational constant is numerically equal to the force of attraction acting between two material points or balls of mass 1 kg located at a distance of 1 m from each other, . If a body of mass is located above the surface of the earth at a height, then it is acted upon by a gravitational force equal to
where is the mass of the Earth, is the radius of the Earth. Near the earth's surface, all bodies are affected by a force caused by attraction - the force of gravity.
Gravity is determined by the force of gravity of the Earth and the fact that the Earth rotates around its own axis.
Due to the small angular velocity of the Earth's rotation (), the force of gravity differs little from the force of gravity. When the acceleration created by gravity is the acceleration of gravity:
It is obvious that the acceleration of gravity is the same for all bodies.
The weight of a body is the force with which the body acts on a horizontal support or stretches a vertical suspension, and this force is applied either to the support or to the suspension.
Newton's second law. The acceleration with which a body moves is directly proportional to the force acting on the body, and inversely proportional to its mass and coincides in direction with the acting force:
If several forces act on a body, then F is understood as the resultant of all acting forces. Equation (2.7) expresses the basic law of the dynamics of a material point. The motion of a rigid body depends not only on the applied forces, but also on the point of their application. It can be shown that the acceleration of the center of gravity (center of mass) does not depend on the point of application of forces and the equation is valid
where is the mass of the body, is the acceleration of its center of gravity. If the body moves translationally, then this equation completely describes the movement of the body.
The momentum of a body is the product of the body's mass and its speed:
Impulse is a vector quantity and depends simultaneously on both the state of motion (speed) and its inertial properties (mass).
Let the momentum of the body have a value at some initial moment of time, and at a subsequent moment of time acquire a new value (the mass does not change over time). Then, during the time interval, the impulse changed by an amount. Then
From kinematics it is known that it is equal to the acceleration of the body, which means. Taking into account (2.7):
Newton's third law. For every action there is always an equal and opposite reaction.
So, if two bodies A and B interact with forces F1 and F2, then these forces are equal in magnitude, opposite in direction, directed along the same straight line and applied to different bodies (Fig. 2.4).
The nature of these forces is always the same. Let's take the following example. A mass body lies on a table. The force with which the body acts on the table, P (body weight), is applied to the table, the force with which the table acts on the body, N (ground reaction force), is applied to the body (Fig. 2.5). According to Newton's 3rd law, . The force FT with which the Earth acts on a body of mass is equal to, applied to the body and directed towards the center of the Earth; the force with which the body acts on the Earth, F, is applied to the center of the Earth and directed to the center of mass of the body (Fig. 2.6).
Newton's first law is necessary in order to determine those frames of reference in which Newton's second law is valid. The reference systems in which Newton's 1st law is satisfied are called inertial, those reference systems in which Newton's 1st law is not satisfied are called non-inertial.
Consider the following example. A load is suspended from the ceiling of the stationary surge, which is visible to observer 1 sitting in the car and observer 2 located on the platform (Fig. 2.7). The pendulum thread is vertical, which is natural from the point of view of observers 1 and 2, since two vertical forces act on the load: the tension force of the thread T and the force of gravity FT, equal in magnitude and opposite in direction. If the car moves with acceleration a, then from the point of view of observer 2 the thread should deviate from the vertical, since the same forces continue to act on the load, but the resultant of these forces will no longer be equal to 0, in order to ensure the movement of the pendulum with acceleration a.
From the point of view of observer 1, the pendulum remains at rest relative to the walls of the car, and the resulting force acting on the pendulum must be equal to zero. But since the thread is deflected, the observer must assume the presence of a force, which, in sum with the tension force of the thread and the force of gravity, gives 0. This is the force of inertia. But this force is no longer the result of the interaction of bodies, but is the result of the fact that we consider the movement of a body relative to a reference frame moving with acceleration.
The system associated with observer 1 is non-inertial, the system associated with observer 2 is inertial. We will consider the motion of bodies only relative to inertial frames of reference. Let us emphasize that force is the result of the interaction of real bodies.
In connection with the importance of what has been stated, let us once again formulate Newton’s first law: there are such reference systems, called inertial ones, in which the body maintains a state of rest or uniform rectilinear motion if no forces act on it or the action of the forces is compensated. It is obvious that if there is one inertial reference system, then any other moving uniformly and rectilinearly relative to it is also an inertial reference system. To a first approximation, the reference frame associated with the Earth is inertial, although strictly speaking it is non-inertial, since the Earth rotates around its own axis and revolves around the Sun. However, the accelerations of these movements are small.
In connection with the difficulties that arise when solving problems of dynamics, especially in cases where a system of bodies is considered, we will propose a scheme according to which problems of dynamics should be solved.
1. We make a drawing and depict the forces acting on bodies from other bodies.
2. Select a reference body with respect to which we will consider the movement.
3. We associate the coordinate system with the reference body.
4. We write down the basic law of dynamics for each body separately.
5. We write the equations in projections on the coordinate axes.
6. From the obtained equations we compose a system of algebraic equations, and the number of equations must be equal to the number of unknowns.
7. We solve the system of equations and find unknown physical quantities; We check the name of the obtained values.
Rotational movement
Rotational motion is the movement of a body in which all its points move in circles, the centers of which lie on one straight line, called the axis of rotation, and the planes of the circles are perpendicular to the axis of rotation.
Complex movements can be thought of as combinations of translational and rotational movements.
In the previous chapter, the concept of angular velocity was introduced for uniform motion of a body in a circle. Angular velocity is usually considered as a vector directed along the axis of rotation according to the rule of the right screw: if the screw is rotated in the same direction as the body rotates, then the direction of movement of the screw coincides with the direction of the angular velocity.
If a body rotates through equal angles at any equal intervals of time, then such motion is called uniform rotational motion.
Using the concept of angular velocity, we can give another definition of uniform rotational motion. Uniform rotational motion is called motion with constant angular velocity ().
To describe uneven rotational motion, a value is introduced that characterizes the change in angular velocity. This quantity is the ratio of the change in angular velocity to the small time interval during which this change occurred. This quantity is called the average angular acceleration:
During accelerated rotation, the vectors and coincide in direction; during slow rotation the vector is directed opposite to the vector.
The SI unit of angular acceleration is 1.
The moment of force is a vector directed along the axis of rotation and oriented according to the rule of the right screw relative to the force vector. The modulus of the moment of force is equal to
where is the leverage of force. It is equal to the shortest distance between the axis of rotation and the direction of force.
Basic equation for the dynamics of rotational motion of a rigid body
To obtain the required equation, let us first consider the simplest case, when a material point with mass rotates on a weightless solid rod with a length around an axis (Fig. 2.9). Newton's second law for this point will be written as:
But tangential acceleration
Substituting into formula (2.10), we get:
Multiplying both sides of this equality by to reduce the action of a force to its moment, we will have:
The product of the mass of a point and the square of its distance to the axis is called the moment of inertia of the material point relative to the axis:
The SI unit of moment of inertia is .
Then expression (2.11) will take the form:
Since the vectors and are directed in the same direction along the axis of rotation, expression (2.13) can be written in vector form:
This is the basic equation for the dynamics of rotational motion.
The moment of inertia of a body is the sum of the moments of inertia of its constituent particles:
For different axes of rotation, the moment of inertia of the same body is different.
If the moment of inertia about any axis passing through the center of mass of a body is known, then to calculate the moment of inertia of this body about another axis parallel to the first and spaced apart from it, a relation known as Steiner’s theorem is used:
The table shows formulas for calculating the moments of inertia of some bodies about an axis passing through the center of mass of these bodies.
3. Body impulse. Law of conservation of momentum
The momentum of a body (amount of motion) p is a physical quantity equal to the product of the body’s mass and its speed:
The impulse of a force is a physical quantity equal to the product of a force and the period of time during which this force acts. Newton's 2nd law can be stated as follows:
The change in the momentum of the body is equal to the impulse of the force acting on it, i.e.
Obviously, law (3.2) goes into (3.1) if the mass remains constant.
If several forces act on a body, then in this case the resulting impulse of all the forces acting on the body is taken. In projections onto the coordinate axes, equation (3.2) can be written as
From (3.3) it follows that if, for example, and, then the projection of the impulse changes in only one direction, and vice versa, if the projection of the impulse changes only in one of the axes, then, consequently, the impulse of the force acting on the body has only one projection other than zero. For example, suppose a ball flying at an angle to the horizon elastically hits a smooth wall. Then, upon reflection, only the x-component of the pulse changes (Fig. 3.1). Projections of momentum onto the x axis:
Impulse change:
With an elastic impact on a wall, the velocities before and after the impact are equal: , therefore
Consequently, the ball was affected by a force impulse, the projection of which on the x-axis is, the projection on the y-axis
Impulse change:
Therefore, the projection of the force impulse on the y-axis is equal.
The concept of momentum is widely used when solving problems about the motion of several interacting bodies. A set of interacting bodies is called a system of bodies. Let us introduce the concept of external and internal forces. External forces are forces acting on the bodies of a system from bodies outside of it. Internal forces are the forces that arise as a result of the interaction of bodies included in the system. For example, a boy throws a ball. Let's consider the boy-ball system of bodies. The forces of gravity acting on the boy and the ball, the force of normal reaction acting on the boy from the floor, are external forces. The force with which the ball presses on the boy's hand, the force with which the boy acts on the ball until it comes off his hand, are internal forces.
Let's consider a system of two interacting bodies 1 and 2. Body 1 is acted upon by an external force and an internal force (from the second body). The forces and act on the second body. According to (3.2), the change in momentum of the first body over a period of time is equal to
change in momentum of the second body:
The total impulse of the system is equal to
Adding the left and right sides of equations (3.4a) and (3.4b), we obtain the change in the total momentum of the system:
According to Newton's 3rd law
where is the resulting impulse of external forces acting on the bodies of the system. So, equation (3.5) shows that the momentum of the system can only change under the influence of external forces. The law of conservation of momentum can be formulated as follows:
The momentum of the system is conserved if the resulting momentum of external forces acting on the bodies included in the system is zero.
Systems in which only internal bodies act on bodies (i.e. the bodies of the system interact only with each other) are called closed (isolated). It is obvious that in closed systems the momentum of the system is conserved. However, in open systems, in some cases, the law of conservation of momentum can be used. Let us list these cases.
1. External forces act, but their resultant is 0.
2. The projection of external forces to some direction is equal to 0, therefore, the projection of the impulse to this direction is preserved, although the impulse vector itself does not remain constant.
3. External forces are much smaller than internal forces (). The change in momentum of each body is almost equal.
4. Mechanical work and energy. Law of energy conservation
Let a constant force F act on the body, and the body moves by. Mechanical work is equal to the product of the moduli of force and displacement of the point of application of force by the cosine of the angle between the force vector and the displacement vector (Fig. 4.1):
The projection of the force onto the displacement vector is equal to
hence,
From formula (4.1) it follows that at, the work of force is positive, at, at.
In Fig. 4.2 shows the dependence on s. From formula (4.2) it is obvious that the work done by the force F is numerically equal to the area of the shaded rectangle.
If it depends on s according to an arbitrary law (Fig. 4.3), then, dividing the total displacement into small segments, within each of which the value can be considered constant, we obtain that the work of force F on displacement s is equal to the area of the curved trapezoid:
Work of elastic force. The elastic force is equal. The dependence of the elastic force on x is shown in Fig. 4.4. When the spring is stretched from x1 to x2, the work done by the elastic force is equal to the area of the shaded trapezoid, up to the sign:
The work of the elastic force during tension is negative, since the elastic force is directed in the direction opposite to the movement. When restoring the dimensions of the spring, the work of the elastic force is positive, since the elastic force coincides in direction with the displacement.
Work of gravity. The force of gravity depends on the distance from the center of the Earth r. Let us determine the work done by the gravitational force when moving a body of mass point A to point B (Fig. 4.5). At small displacements, the work done by the gravitational force
where is the mass of the Earth. If it’s not enough, then
Thus, the work when moving from point A to point B will be determined as the sum of the work on small movements:
If, a, then
is the work done by the gravitational force when moving a body from the surface of the Earth to an infinitely distant point on the trajectory.
Mechanical energy characterizes the ability of a body to perform mechanical work. The total mechanical energy of a body consists of kinetic and potential energy.
Kinetic energy is the energy possessed by a moving body. Let a force F act on the body, the displacement of the body. The work done by force F is equal to (Fig. 4.6)
According to Newton's 2nd law,
If at points 1 and 2 the speed of the body is u, then
Substituting expressions (4.7) and (4.8) into (4.6), we obtain
So, if a force F acts on a body, the work of which is different from zero, then this leads to a change in a quantity called kinetic energy:
From (4.9a) it follows that the change in kinetic energy is equal to the work of the force acting on the body. If several forces act on a body, then the change in kinetic energy is equal to the algebraic sum of the work performed for a given movement of each of the forces.
A system of bodies interacting with each other has potential energy if the interaction forces are conservative. A conservative (potential) force is a force whose work does not depend on the shape of the trajectory, but is determined only by the position of the starting and ending points of the trajectory.
Let's consider the movement of mass m from point 1 to point 2 along various trajectories (Fig. 4.7). The work of gravity of a body in a straight line is determined by the expression
Because the,
The work of gravity when a body moves along a trajectory:
Let's calculate the work done by gravity when a body moves along trajectory III. Let us imagine the trajectory with any degree of accuracy in the form of a broken line consisting of vertical and horizontal segments. Then the work done by gravity when moving horizontally is equal to zero, along vertical segments, . Total work is
As shown, the work done by gravity does not depend on the trajectory. Gravity is a conservative force. It is obvious that the work done by a conservative force along a closed loop is zero. The force of gravity and the force of elasticity are also conservative forces. When a body falls, the potential energy decreases. From (4.9) it follows
The change in potential energy is equal to the work done by the conservative force, taken with the opposite sign:
Potential energy is calculated accurate to a constant value, so it is always necessary to indicate the zero level of potential energy reference. So, the potential energy of a body raised to a height h() is equal to
The potential energy due to the force of gravity is
; at. (4.12)
The potential energy of a compressed or stretched spring is equal to
At. (4.13)
As can be seen from the examples, potential energy depends on the relative position of bodies or parts of the body. Non-conservative forces in mechanics are the friction force and the drag force.
Let's consider a two-body system. Bodies can be acted upon by external and internal forces, which can be conservative or non-conservative. The change in the kinetic energy of each body is equal to the sum of the work of all forces acting on this body, namely, for the first body:
Let's look at these forces in detail. The frictional force can be either an internal or external force; Let us denote the work of all friction forces. The body is acted upon by conservative internal forces, the work of which. The body can also be in the field of external conservative forces, the work of which will lead to a change in potential energy. An external force can also act on the body, to which we will not associate a change in potential energy. Her job is there.
Then the change in the kinetic energy of bodies is determined by the formula
Similarly, for the second body we have
Because the
adding the left and right sides of the equations and moving them to the left side, to change the total mechanical energy of the system, equal to
According to Newton's 3rd law, the sum of the work done by internal forces is 0, which means that
those. the change in mechanical energy is equal to the work of external forces and friction forces.
Law of conservation of mechanical energy
The mechanical energy of the system is conserved if the work of external forces acting on the bodies included in the system is zero and there are no friction forces, i.e. there is no transition of mechanical energy into other types of energy, for example, into heat:
Note that conservation laws make it possible to determine the final state from the initial state of the system (from the initial velocities) without clarifying all the details of the interaction of bodies and without specifying the magnitude of the interaction forces.
In practice, it is often useful to know how quickly a particular job can be completed. To characterize the speed at which work is performed, a quantity called power is introduced.
The power developed by a constant traction force is equal to the ratio of the work done by this force on a certain movement to the period of time during which this movement occurred. Power is determined by the formula
Since, then, substituting this expression into formula (4.15), we obtain
where is the speed of the body, is the angle between the vectors F and v. If the motion of the body is uniform, then in (4.16) we mean the speed of uniform motion. If the movement is not uniform, but it is necessary to determine the average power developed by the traction force on the movement s, then in (4.16) we mean the average speed of movement. If it is required to find the power at a certain given moment in time (instantaneous power), then, taking small time intervals and passing to the limit at, we obtain
those. - instantaneous speed of the body. The concept of power is introduced to estimate the work per unit of time that can be performed by some mechanism (pump, crane, machine motor, etc.). Therefore, in formulas (4.14)-(4.17), F always means only the traction force.
The SI unit of power is Watt (W)
This law, called the law of universal gravitation, is written in mathematical form as follows:
where m 1 and m 2 are the masses of the bodies, R is the distance between them (see Fig. 11a), and G is the gravitational constant equal to 6.67.10-11 N.m 2 /kg2.
The law of universal gravitation was first formulated by I. Newton when he tried to explain one of I. Kepler’s laws, which states that for all planets the ratio of the cube of their distance R to the Sun to the square of the period T of revolution around it is the same, i.e.
Let us derive the law of universal gravitation as Newton did, assuming that the planets move in circles. Then, according to Newton’s second law, a planet of mass mPl moving in a circle of radius R with speed v and centripetal acceleration v2/R must be acted upon by a force F directed towards the Sun (see Fig. 11b) and equal to:
The speed v of the planet can be expressed in terms of the orbital radius R and the orbital period T:
Substituting (11.4) into (11.3) we obtain the following expression for F:
From Kepler's law (11.2) it follows that T2 = const.R3. Therefore, (11.5) can be transformed into:
Thus, the Sun attracts a planet with a force directly proportional to the mass of the planet and inversely proportional to the square of the distance between them. Formula (11.6) is very similar to (11.1), the only thing missing is the mass of the Sun in the numerator of the fraction on the right. However, if the force of attraction between the Sun and the planet depends on the mass of the planet, then this force must also depend on the mass of the Sun, which means that the constant on the right side of (11.6) contains the mass of the Sun as one of the factors. Therefore, Newton put forward his famous assumption that the gravitational force should depend on the product of the masses of bodies and the law became the way we wrote it in (11.1).
The law of universal gravitation and Newton's third law do not contradict each other. According to formula (11.1), the force with which body 1 attracts body 2 is equal to the force with which body 2 attracts body 1.
For bodies of ordinary sizes, gravitational forces are very small. So, two cars standing next to each other are attracted to each other with a force equal to the weight of a raindrop. Since G. Cavendish determined the value of the gravitational constant in 1798, formula (11.1) has helped to make many discoveries in the “world of enormous masses and distances.” For example, knowing the magnitude of the acceleration due to gravity (g=9.8 m/s2) and the radius of the Earth (R=6.4.106 m), we can calculate its mass m3 as follows. Each body of mass m1 near the Earth’s surface (i.e., at a distance R from its center) is acted upon by a gravitational force of its attraction equal to m1g, the substitution of which in (11.1) instead of F gives:
from where we find that m W = 6.1024 kg.
Review questions:
· Formulate the law of universal gravitation?
· What is the gravitational constant?
Rice. 11. (a) – to the formulation of the law of universal gravitation; (b) – to the derivation of the law of universal gravitation from Kepler’s law.
§ 12. GRAVITY. WEIGHT. WEIGHTLESSNESS. FIRST SPACE SPEED.
Gravitational forces or otherwise gravitational forces acting between two bodies:
- long-range;
- there are no barriers for them;
- directed along a straight line connecting the bodies;
- equal in size;
- opposite in direction.
Gravitational interaction
Proportionality factor G called gravitational constant.
Physical meaning of the gravitational constant:
the gravitational constant is numerically equal to the modulus of the gravitational force acting between two point bodies weighing 1 kg each, located at a distance of 1 m from each other
Condition for the applicability of the law of universal gravitation
1. The sizes of bodies are much smaller than the distances between them;
2. Both bodies are spheres and they are homogeneous;
; 
3. One body is a large ball, and the other is located near it
(planet Earth and bodies near its surface).
Not applicable.
The difficulty is that the gravitational forces between bodies of small masses are extremely small. It is for this reason that we do not notice the attraction of our body to surrounding objects and the mutual attraction of objects to each other, although gravitational forces are the most universal of all forces in nature. Two people with masses of 60 kg at a distance of 1 m from each other are attracted with a force of only about 10 -9 N. Therefore, to measure the gravitational constant, fairly subtle experiments are needed.
Gravitational interaction is noticeably manifested when bodies of large mass interact.
Since, for example, the Earth acts on the Moon with a force proportional to the mass of the Moon, then the Moon, according to Newton’s third law, must act on the Earth with the same force. Moreover, this force must be proportional to the mass of the Earth. If the force of gravity is truly universal, then from the side of a given body a force must act on any other body proportional to the mass of this other body. Consequently, the force of universal gravity must be proportional to the product of the masses of interacting bodies.
Examples of gravitational interactions
The attraction from the Moon causes the ebb and flow of water on Earth, huge masses of which rise in the oceans and seas twice a day to a height of several meters. Every 24 hours and 50 minutes, the Moon causes tides not only in the oceans, but also in the Earth's crust and atmosphere. Under the influence of tidal forces, the lithosphere is stretched by about half a meter.
Conclusion
- In astronomy, the law of universal gravitation is fundamental, on the basis of which the parameters of the movement of space objects are calculated and their masses are determined.
- The onset of the ebb and flow of the seas and oceans is predicted.
- The flight trajectories of projectiles and missiles are determined, heavy ore deposits are explored
- One of the manifestations of universal gravitation is the action of gravity
Homework.
1. E.V. Korshak, A.I. Lyashenko, V.F. Savchenko. Physics. Grade 10, “Genesis”, 2010. Read §19 (p.63-66).
2. Solve problems No. 1, 2 exercises 10 (p. 66).
3. Complete the test task:
1.What force makes the Earth and other planets move around the Sun? Choose the correct statement.
A. Inertial force. B. Centripetal force. B. Gravitational force.
In the 7th grade physics course, you studied the phenomenon of universal gravitation. It lies in the fact that there are gravitational forces between all bodies in the Universe.
Newton came to the conclusion about the existence of universal gravitational forces (they are also called gravitational forces) as a result of studying the movement of the Moon around the Earth and the planets around the Sun.
Newton's merit lies not only in his brilliant guess about the mutual attraction of bodies, but also in the fact that he was able to find the law of their interaction, that is, a formula for calculating the gravitational force between two bodies.
The law of universal gravitation says:
- any two bodies attract each other with a force directly proportional to the mass of each of them and inversely proportional to the square of the distance between them
where F is the magnitude of the vector of gravitational attraction between bodies of masses m 1 and m 2, g is the distance between the bodies (their centers); G is the coefficient, which is called gravitational constant.
If m 1 = m 2 = 1 kg and g = 1 m, then, as can be seen from the formula, the gravitational constant G is numerically equal to the force F. In other words, the gravitational constant is numerically equal to the force F of attraction of two bodies weighing 1 kg each, located at a distance 1 m apart. Measurements show that
G = 6.67 10 -11 Nm 2 /kg 2.
The formula gives an accurate result when calculating the force of universal gravity in three cases: 1) if the sizes of the bodies are negligible compared to the distance between them (Fig. 32, a); 2) if both bodies are homogeneous and have a spherical shape (Fig. 32, b); 3) if one of the interacting bodies is a ball, the dimensions and mass of which are significantly greater than that of the second body (of any shape) located on the surface of this ball or near it (Fig. 32, c).
Rice. 32. Conditions defining the limits of applicability of the law of universal gravitation
The third of the cases considered is the basis for calculating, using the given formula, the force of attraction to the Earth of any of the bodies located on it. In this case, the radius of the Earth should be taken as the distance between bodies, since the sizes of all bodies located on its surface or near it are negligible compared to the Earth’s radius.
According to Newton's third law, an apple hanging on a branch or falling from it with the acceleration of free fall attracts the Earth to itself with the same magnitude of force with which the Earth attracts it. But the acceleration of the Earth, caused by the force of its attraction to the apple, is close to zero, since the mass of the Earth is incommensurably greater than the mass of the apple.
Questions
- What was called universal gravity?
- What is another name for the forces of universal gravity?
- Who discovered the law of universal gravitation and in what century?
- Formulate the law of universal gravitation. Write down a formula expressing this law.
- In what cases should the law of universal gravitation be applied to calculate gravitational forces?
- Is the Earth attracted to an apple hanging on a branch?
Exercise 15
- Give examples of the manifestation of gravity.
- The space station flies from the Earth to the Moon. How does the modulus of the vector of its force of attraction to the Earth change in this case; to the moon? Is the station attracted to the Earth and the Moon with equal or different magnitude forces when it is in the middle between them? If the forces are different, which one is greater and by how many times? Justify all answers. (It is known that the mass of the Earth is about 81 times the mass of the Moon.)
- It is known that the mass of the Sun is 330,000 times greater than the mass of the Earth. Is it true that the Sun attracts the Earth 330,000 times stronger than the Earth attracts the Sun? Explain your answer.
- The ball thrown by the boy moved upward for some time. At the same time, its speed decreased all the time until it became equal to zero. Then the ball began to fall down with increasing speed. Explain: a) whether the force of gravity towards the Earth acted on the ball during its upward movement; down; b) what caused the decrease in the speed of the ball as it moved up; increasing its speed when moving down; c) why, when the ball moved up, its speed decreased, and when it moved down, it increased.
- Is a person standing on Earth attracted to the Moon? If so, what is it more attracted to - the Moon or the Earth? Is the Moon attracted to this person? Justify your answers.