As we all know, the development of physics is inseparable from mathematics, which is the basis of the development of physics. The solution of many physical problems is the product of the ingenious combination of mathematical methods and physical ideas. To lay a good foundation of mathematics, we should start from high school, cultivate students' mathematical thinking and innovative ability, better integrate with university courses, and let high school students embark on the hall of physics earlier.
Let's take a topic as an example to illustrate the application of mathematical thought in physics:
As shown in figure 1, a closed glass tube with a length of L=96 cm has a section of mercury column of h 1=20 cm. When the temperature is 27 degrees Celsius and the open end is vertically upward, the closed gas column h2=60 cm, and the temperature is at least a few degrees, all mercury can overflow from the tube.
Solution: First, raise the temperature to T0, so that the mercury column rises 16 cm, and the mercury is flush with the nozzle. This process is linear. The temperature continues to rise and mercury overflows, which is no longer linear. When the temperature is t, the residual mercury is h, and the equilibrium equation at any position:
(76+H 1)×60/300 =(76+H)×(96-H)/t:
T=(-h2+20h+7296)/ 19.2
The range of H is 0-20. It can be seen that the temperature T is a quadratic function of H, and this problem is transformed into finding the value range of T in the definition domain. If Tmin
Only through the quadratic function extreme value method can the ontology be fundamentally solved. Strengthening the infiltration of mathematical ideas is a new embodiment of new teaching materials, such as "exploring the relationship between the period of spring vibrator and those factors" and "exploring the relationship between spring elasticity and elongation" should be highly valued and carried out in the actual teaching process.
College physics course and senior high school physics course span a long time, and the difficulty lies in the method of exploring physics problems by mathematical means. In addition, calculus is hard to understand. In order to better integrate with college physics courses, it is also necessary to infiltrate calculus into students in senior high school. Therefore, in the process of physics teaching in senior high school, we should seize the favorable opportunity to infiltrate infinitesimal thought and lay a good foundation for learning calculus well. The content of infiltration should have two aspects: one is the rate of change, and the other is the infinitesimal amount of change, such as:
When talking about speed, average speed v=△s/t, instantaneous speed? △s/t is the rate of change. When △s is infinitely small, V can be understood as the speed at a certain moment-even the speed. The acceleration a = △v/t, and △v/t is the speed change rate. When △ v is infinite, the acceleration A can be understood as the acceleration at a certain moment. Examples like this are w/t, I/t, △φ/t and so on. In a word, physics teachers in senior high schools should properly infiltrate the idea of calculus and practice according to students' specific conditions to consolidate their understanding. Let's discuss a related topic.
Example 2 1 A vertically placed U-shaped tube with equal cross-section is filled with a mercury column with a total length of L. When its left and right liquid surfaces freely vibrate up and down, it is proved that the vibration time of the mercury column is resonant.
Solution: Let the average velocity of two liquid levels be V0, and establish the coordinates as shown in the figure.
When the liquid level rises x and the liquid velocity is v, then according to the conservation of energy,
mv02/2 =△mgx 1+mv 12/2⑴
△m=mgx 1/L ⑵
(2) Bring in (1)
mv02/2 = mgx 12/L+mv 12/2⑶
When the liquid level rises by △x, x2=x 1+△x, then
mv02/2=mgx22/L +mv22/2 ⑷
(4) Subtract (3)
0 = (x22-x 12) mg/l+m (V22-V 12)/2 Simplified:
0 =(x 1+x2)mg△x/L+m(v 12-v 22)/2⑸
△x is very small, then the acceleration a is considered to be constant. According to the kinematic formula:
V 12-v22=2ax brought in [5].
0=2x△xmg/L+2ma△x/2 ⑹
That is, F=-2mgx/L 2mg/L is a constant k, which proves that the vibration of mercury column is simple harmonic vibration.