Impulse is the accumulation of force with time, and its function is to change the momentum of an object; Work is the accumulation of force in space, and its function is to change the energy of objects; The change of impulse and momentum, and the change of work and energy are causal. On this basis, it is easy to understand the conditions of conservation law. If conservation is to be achieved, there should be no reason for change. Energy is still a main thread running through the whole physics. It is an important and universal idea to analyze and think about problems from the perspective of energy.
When applying momentum theorem and kinetic energy theorem, the research object is generally a single object, while when applying momentum conservation law and mechanical energy conservation law, the research object must be a system; In addition, these laws apply to physical processes, not to a certain state (or moment). Therefore, when using them to solve problems, we must first choose the research object and research process. The choice of object and process is directly related to whether the problem can be solved or not. Pay attention to the following points when choosing:
1. The choice of research object and research process should be based on the analysis of physical process. The critical state should usually be regarded as the beginning or end state of the research process.
2. Be able to properly idealize the research process.
3. Some seemingly scattered and independent objects can be circled together as a system to study, sometimes this can greatly simplify the problem.
4. For some problems, you can choose this part of the object as the research object, or you can choose that part as the research object; You can choose this process as the research process, or you can choose that process as the research process; At this point, large objects and long processes are preferred.
After determining the object and process, we should choose physical laws to solve the problem on the basis of analysis. The general principle of choosing law is:
1. For a single object, momentum theorem and kinetic energy theorem should be selected, among which momentum theorem should be selected for problems involving time and kinetic energy theorem should be selected for problems involving displacement.
2. If the system consists of multiple objects, the two conservation laws take precedence.
3. If the relative displacement (distance) of objects in the system is involved, and friction is involved, the law of conservation of energy should be considered.
For example 1 Figure 1, one end of the light spring is fixed, and the other end is connected with the slider B, and B is placed on the horizontal straight guide rail, and the spring is in the original long state. Another slider A with the same mass as B slides from point P to point B on the guide rail at a certain initial speed. When A slides over a certain distance, it collides with B for a short time. After the collision, A and B are close together, but they won't stick together. It is known that the last A just returned to the starting point P and stopped. The friction coefficient between the sliders A and B and the guide rail is 0, the maximum deformation of the spring is 0 and the acceleration of gravity is 0. Find the initial velocity of a when it starts from point p.
Analysis: First of all, the whole physical process should be analyzed clearly, and the interaction objects and motion properties of different stages should be made clear, thus paving the way for the correct division into several stages for research. That is to say, the first one slides to the left from point P, and decelerates at a constant speed through friction. Let the velocity of A when it just touches B be, for A, according to the kinetic energy theorem, there is
Then a and b collide and momentum is conserved. Let the * * * velocities of A and B be the same at the moment after the collision. According to the law of conservation of momentum, for systems A and B, there are
Then A and B compress the spring to the left, assuming that the potential energy obtained by the spring is, for the system composed of A, B and the spring, according to the functional relationship, there are:
A and B are bounced back by the spring until the spring returns to its original long stage. Let the velocity of a and b be. According to the functional relationship, the system consisting of A, B and spring is as follows
Finally, A and B separate, A slides to point P and stops. The kinetic energy theorem is applied to A, and there are
It is obtained from the above solution.
Comments: The comprehensive problem of momentum and energy usually has multiple physical processes. In the analysis, the whole process should be divided into several simple sub-processes according to the stress characteristics and movement conditions at different stages, thus laying the foundation for solving problems by using the relationship between momentum and energy.
Example 2 Above the ground, a small circle A is sleeved on a uniform straight bar B, and the masses of A and B are both m, so the sliding friction between them. At the beginning, A is at the lower end of B, and B is placed vertically. There is an "interaction" area c meters below a, and the height of the area c is meters, which is fixed in the air as shown by the dotted line in Figure 2. When a enters region c, a is acted by a constant force f in the vertical direction. Area c does not apply force to the rod. A and B fall together from a standstill. It is known that the speed of A and B is the same when rod B falls to the ground. Excluding air resistance and gravity acceleration. What is the minimum length of bar B?
Analysis: Through inspection, the physical process and state are subdivided as shown in Figure 2- 1, 2-2, 2-3. Figure 2-3 shows that A and B have just reached the same speed, and the relative displacement length of A and B is the minimum length of the rod.
In the process of objects A and B changing from the state shown in Figure 2 to the state shown in Figure 2- 1, for systems A and B, mechanical energy is conserved (where is the velocity of states A and B shown in Figure 2- 1), and a solution is obtained.
In the process of changing the physical state from the state shown in Figure 2- 1 to the state shown in Figure 2-2:
For object A, it is obtained by kinetic energy theorem.
(For the speed of object A in the state shown in Figure 2-2), the solution is obtained.
For systems A and B, the resultant force is zero, which is obtained by the conservation of momentum.
(For the velocity of object B in the state shown in Figure 2-2), find the solution.
For object B, it is obtained by kinetic energy theorem (where is the falling height of object B in the process).
Solve.
In the process of changing the physical state from that shown in Figure 2-2 to that shown in Figure 2-3:
Right a is obtained by momentum theorem.
From the kinetic energy theorem (where is the falling height of object A in the process; In the state of Figure 2-3, A and B have the same speed. )
Right b is obtained by momentum theorem.
From the kinetic energy theorem (where is the falling height of process B). )
Get m, m from the above formula.
The length of the pole shall be at least 100 meter.
Comment: There are many physical problems involving critical state. When solving this kind of problem, we should examine the meaning of the problem, make clear the physical process by sketching the figure, find out the turning point, grasp the quantity connecting the past and the future, and determine the critical condition. A good schematic diagram is a silent inspiration. With the help of schematic diagram, it can help us to examine questions, enrich our imagination of physical scenes and knock on the door to solve problems correctly.
In nuclear reactors, graphite is used as a moderator, which makes the fast neutrons produced by uranium nuclear fission slow down through continuous collision with carbon nuclei. Assuming that the neutron collides with the carbon core elastically, the carbon core is stationary before the collision. It is known that the mass of carbon nucleus is about 12 times that of neutron, and the original kinetic energy of neutron is E0. Try to find:
(1) What is the neutron energy after the collision?
(2) If E0 = 1.76 MeV, how many times can the neutron energy be reduced to 0.025eV?
Analysis: According to the law of elastic frontal collision, we can find out how much neutron velocity becomes after each collision, and the corresponding kinetic energy can also be solved; According to the relationship between the ratio of kinetic energy before and after each collision and the original kinetic energy E0, the number of collisions (which must be rounded off) is obtained by logarithm.
(1) Elastic frontal collision obeys two laws of conservation of momentum and energy. Let the neutron mass m and the nuclear mass m of carbon atoms. There are:
Through the above two types of finishing.
Kinetic energy of neutrons after collision
(2) It can also be obtained.
……
Suppose that after n collisions, the kinetic energy of neutrons will be reduced to zero. 025eV, that is, en = 0.025 ev and E0 = 1.75 mev.
N≈54, used to solve the above formula.
Comments: Momentum and energy problems are generally closely combined with practical problems. Whether a practical problem can be transformed into a typical physical model and a familiar process is the key to solve this kind of problem.
Example 4 As shown in Figure 3, the widths of horizontal metal rails M and N are such that the widths of metal rails M' and N' are long enough. They are connected by a metal rod EF and are in a uniform magnetic field with vertical magnetic induction intensity. The right boundary of the magnetic field is gh, and the cd metal strip is placed vertically on the M and N rails. The ab metal strip is subjected to a horizontal external force F=5N on a smooth horizontal platform, and the force F and ab strip are withdrawn after the action time. Ab and cd rods are always parallel and will not collide. When cd and cd rod reach EF successively, ab and cd rod have reached a stable speed, which is known, regardless of all friction resistance. ask
(1) The final speed of AB and cd cards.
(2) The total heat generated by the current in the whole device.
Analysis: The initial velocity of (1)ab rod in flat throwing motion is, which is obtained according to the momentum theorem.
According to the meaning of the question, the initial velocity of ab rod horizontally to the left on M and N guide rails is, the momentum of ab rod and cd rod is conserved, and the same velocity has been reached before cd rod reaches EF, so there is a solution.
When the cd rod is on the M' and N' tracks and the ab rod is moving on the M, N tracks, the ampere force on the ab rod is twice that on the cd rod at every moment, and the relationship between the impulses of the ampere force acting on the two rods at the same time is as follows.
Before ab rod reaches EF, ab rod and cd rod have reached a stable speed, which is set for ab rod and cd rod respectively.
When the ab stick and the cd stick have a steady speed, there are
Solution,.
When ab rod reaches EF, the momentum of ab rod and cd rod system is conserved, and finally the same speed is reached, and then there is
Solve.
(2) (2) After the AB rod falls to the M and N rails, the energy of the whole system is conserved, and the total heat generated by the current is equal to the loss of the mechanical energy of the system, so there are
Comments: In the electromagnetic induction problem, the metal bar often moves at a non-uniform speed, and the force on the conductor bar changes due to the speed change of the conductor bar. Therefore, the quantitative calculation of non-uniform variable-speed motion can not be directly solved by the law of uniform change or constant force impulse, but can often be solved by the momentum theorem. In bimetallic rods, the common momentum theorem is solved by taking two rods as the research objects respectively. Of course, bimetallic rods can sometimes be regarded as a system, which can be solved directly by momentum conservation.