What winter holidays would be complete without a mountain of tangerines and a couple of juicy...
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1 2 Uniform motion in a circle is a motion in which a material point passes circles of equal length in equal intervals of time. Uniform motion in a circle Solution of problems 10 3 4 5 6 7 8 9 Lyakhovich E.Yu., MBVSOU “VSOSH No. 3”, Nizhnekamsk
Period of revolution 2 1 10 3 4 5 6 7 8 9 Lyakhovich E.Yu., MBVSOU "VSOSH No. 3", Nizhnekamsk The time of one revolution around the circle is called the period of rotation T N - the number of revolutions made during time t. The unit of circulation frequency is 1 revolution per second (1 s -1)
3 2 10 1 4 5 6 7 8 9 Lyakhovich E.Yu., MBVSOU “VSOSH No. 3”, Nizhnekamsk Angular velocity
4 2 10 3 1 5 6 7 8 9 Lyakhovich E.Yu., MBVSOU "VSOSH No. 3", Nizhnekamsk The modulus of the linear velocity vector is equal to:
5 2 10 3 4 1 6 7 8 9 Lyakhovich E.Yu., MBVSOU "VSOSH No. 3", Nizhnekamsk The module of the centripetal acceleration vector is equal to:
6 2 10 3 4 5 1 7 8 9 Lyakhovich E.Yu., MBVSOU “VSOSH No. 3”, Nizhnekamsk Problem. What is the linear speed of the points on the wheel rim of a steam turbine with a wheel diameter of 1 m and a rotation speed of 300 rpm? Show solution
7 2 10 3 4 5 6 1 8 9 Lyakhovich E.Yu., MBVSOU “VSOSH No. 3”, Nizhnekamsk Problem. How many times will the centripetal acceleration of a body change if it moves uniformly around a circle of twice the radius with the same angular velocity? Show solution
8 2 10 3 4 5 6 7 1 9 Lyakhovich E.Yu., MBVSOU “VSOSH No. 3”, Nizhnekamsk Problem. The angular speed of the fan blades is 20π rad/s. Find the number of revolutions in 30 minutes. Show solution
1 Option 2 Option 1. The angular speed of the fan blades is 20π rad/s. Find the number of revolutions in 30 minutes. 2. The rotation speed of the aircraft propeller is 1500 rpm. How many revolutions will the propeller make on a path of 90 km at a flight speed of 180 km/h 2? A diesel locomotive moves at a speed of 60 km/h. How many revolutions per second do its wheels make if their radius is 50 cm? 1 . When turning, a tram car moves at a constant absolute speed of 5 m/s. What is its centripetal acceleration equal to if the radius of curvature of the path is 50 m. 9 2 10 3 4 5 6 7 8 1 Lyakhovich E.Yu., MBVSOU "VSOSH No. 3", Nizhnekamsk
ANSWERS 1 Option 2 Option 1. 18000. 2. 45000 2. 5.31 1 . 0.5 m/s 2. 1 2 10 3 4 5 6 7 8 9 Lyakhovich E.Yu., MBVSOU “VSOSH No. 3”, Nizhnekamsk
1 2 10 3 4 5 6 7 8 9 Lyakhovich E.Yu., MBVSOU “VSOSH No. 3”, Nizhnekamsk Show solution
On the topic: methodological developments, presentations and notes
A lesson in solving problems on the topic "Dynamics of motion in a circle." In the process of solving problems in groups, students learn from each other....
A lesson in learning a new topic using presentations, videos....
The work is intended for 10th grade students, presented in two versions. Definition knowledge tasks, graphic tasks and matching tasks....
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Movement in a circle (closed track) Elena Mikhailovna Savchenko, mathematics teacher of the highest qualification category. Municipal educational institution gymnasium No. 1, Polyarnye Zori, Murmansk region. State (final) certification Training modules for distance self-training X IV All-Russian competition methodological developments"One Hundred Friends"
If two cyclists simultaneously begin to move around a circle in one direction with speeds v 1 and v 2, respectively (v 1 > v 2, respectively), then the 1st cyclist approaches 2 with speed v 1 – v 2. At the moment when the 1st cyclist catches up with the 2nd for the first time, he covers one lap more distance. Continue Show At the moment when the 1st cyclist catches up with the 2nd cyclist for the second time, he covers a distance of two laps and more, etc.
1 2 1. From one point on a circular track, the length of which is 15 km, two cars started simultaneously in the same direction. The speed of the first car is 60 km/h, the speed of the second is 80 km/h. How many minutes will pass from the start before the first car is exactly 1 lap ahead of the second? 1 red 2 green 60 80 v, km/h 15 km less (1 lap) Equation: Answer: 45 x we get in hours. Don't forget to convert to minutes. t , h x x S, km 60х 80х Show
2 1 2. From one point on a circular track, the length of which is 10 km, two cars started simultaneously in the same direction. The speed of the first car is 90 km/h, and 40 minutes after the start it was one lap ahead of the second car. Find the speed of the second car. Give your answer in km/h. 1 car 2 cars 90 x v, km/h 10 km more (1 lap) Answer: 75 t, h 2 3 2 3 S, km 2 3 90 2 3 x Equation: Show
3. Two motorcyclists start simultaneously in the same direction from two diametrically opposite points on a circular track, the length of which is 14 km. How many minutes will it take for the motorcyclists to meet each other for the first time if the speed of one of them is 21 km/h greater than the speed of the other? 1 red 2 blue x x+21 v, km/h 7 km less (half a circle) Equation: Answer: 20 t is obtained in hours. Don't forget to convert to minutes. t, h t t S, km t x t(x +21) How many laps each motorcyclist drove is not important to us. It is important that blue traveled half a circle more to the meeting point, i.e. at 7 km. Another way is in the comments. Show
start finish 2 1 2 1 1 2 2 1 1 2 Let the full circle be 1 part. 4. Ski competitions take place on a circular track. The first skier completes one lap 2 minutes faster than the second and an hour later is exactly one lap ahead of the second. How many minutes does it take the second skier to complete one lap? Show
4. Ski competitions take place on a circular track. The first skier completes one lap 2 minutes faster than the second and an hour later is exactly one lap ahead of the second. How many minutes does it take the second skier to complete one lap? 1 lap more Answer: 10 1 skier 2 skier v, lap/min t, min 60 60 S, km x x+2 1 1 t, min 1 skier 2 skier S, part v, part/min 1 x+2 1 x 1 x+2 1 x 60 x 60 x+2 First, let's express the speed of each skier. Let the first skier complete a circle in x minutes. The second one is 2 minutes longer, i.e. x+2. 60 x 60 x+2 – = 1 This condition will help you enter x...
5. From one point on a circular track, the length of which is 14 km, two cars started simultaneously in the same direction. The speed of the first car is 80 km/h, and 40 minutes after the start it was one lap ahead of the second car. Find the speed of the second car. Give your answer in km/h. 1 yellow 2 blue S, km 80 x v, km/h t, h 2 3 2 3 2 3 80 2 3 x 14 km more (1 lap) Equation: You could first find the speed in pursuit: 80 – x Then the equation will be look like this: v S t Answer: 59 You can press the button several times. How many laps each car drove is not important to us. It is important that the yellow car drove 1 lap more, i.e. at 14 km. Show 1 2
6. A cyclist left point A of the circular route, and 30 minutes later a motorcyclist followed him. 10 minutes after departure, he caught up with the cyclist for the first time, and another 30 minutes after that he caught up with him for the second time. Find the speed of the motorcyclist if the length of the route is 30 km. Give your answer in km/h. 1 motorcycle. 2 bicycles S, km x y v, km/h t, h 1 6 2 3 2 3 y 1 equation: 1 6 x = Show 1 meeting. The cyclist was 40 minutes (2/3 hours) before the first meeting, the motorcyclist was 10 minutes (1/6 hours). And during this time they traveled the same distance.
6. A cyclist left point A of the circular route, and 30 minutes later a motorcyclist followed him. 10 minutes after departure, he caught up with the cyclist for the first time, and another 30 minutes after that he caught up with him for the second time. Find the speed of the motorcyclist if the length of the route is 30 km. Give your answer in km/h. 1 motorcycle. 2 bicycles S, km x y v, km/h t, h 1 2 1 2 1 2 y 30 km more (1 lap) 2nd equation: Answer 80 1 2 x Required value – x Show (2) 2nd meeting. The cyclist and motorcyclist were on the road for 30 minutes (1/2 hour) before the 2nd meeting. And during this time the motorcyclist traveled 1 lap more.
7. The clock with hands shows 8 hours 00 minutes. In how many minutes will the minute hand line up with the hour hand for the fourth time? minute hour x S, circle v, circle/h t, h 1 1 12 x 1x 1 12 x on a circle more than 2 3 3 1x – = 1 12 x 2 3 3 Answer: 240 min 2 3 1 3 For the first time the minute hand you have to go one more lap to catch up with the minute hand. The 2nd time – 1 more lap. The 3rd time – 1 more lap. The 4th time – 1 more lap. Total 2 3 for more circles 2 3 3
6 12 1 2 9 11 10 8 7 4 5 3 Show (4) The first time the minute hand has to go one more circle to catch up with the minute hand. The 2nd time – 1 more lap. The 3rd time – 1 more lap. The 4th time – 1 more lap. Total 2 3 more circles 2 3 3 Check Another way is in the comments.
Unified State Exam 2010. Mathematics. Problem B12. Edited by A. L. Semenov and I. V. Yashchenko http://www.2x2abc.com/forum/users/2010/B12.pdf Open bank of tasks in mathematics. Unified State Exam 2011 http://mathege.ru/or/ege/Main.html Drawings by the author http://le-savchen.ucoz.ru/index/0-67 Skier http://officeimg.vo.msecnd.net/en -us/images/MH900282779.gif Materials published on the author’s website “Mathematics teacher’s website” Section “Preparation for the Unified State Exam”. Task B12. http://le-savchen.ucoz.ru/publ/17
