"Special Reinforcement: Comprehensive Analysis of Circular Motion" Excellent Courseware of Circular Motion PPT

"Special Reinforcement: Comprehensive Analysis of Circular Motion" Excellent Courseware of Circular Motion PPT

Part One: Learning Objectives

1. Be able to analyze circular motion in the vertical plane, and master the analysis methods of circular motion under the action of light ropes and light rods.

2. Master the analysis methods of critical problems of circular motion.

Special topic to strengthen the comprehensive analysis of circular motion PPT, the second part of the content: 01 Exploring key points to improve literacy

circular motion in vertical plane

1. Light rope (roller coaster) model with circular motion in the vertical plane

As shown in Figure 1, the small ball in Figure A is affected by the tension of the rope and gravity, and the small ball in Figure B is affected by the elastic force and gravity of the track. The motion rules of the two are the same. Let's take Figure A as an example.

(1) The lowest point dynamic equation:

FT1－mg＝mv 21L

So FT1=mg＋mv 21L

(2)Highest point dynamic equation:

FT2＋mg＝mv 22L

So FT2=mv 22L-mg

Example 1 As shown in Figure 2, a light rope with a length of L=0.4 m is tied to a small ball that performs circular motion in the vertical plane. The mass of the small ball is m=0.5 kg. The radius of the small ball is not included. G is 10 m. /s2, please:

(1) The speed of the ball when it just passes the highest point;

Answer: 2 m/s

(2) When the speed of the ball passing through the highest point is 4 m/s, the tension of the rope will be greater;

Answer 15 N

(3) If the maximum tension that the light rope can withstand is 45 N, the maximum speed of the ball during its motion.

2. Light rod (tube) model with vertical in-plane circular motion

As shown in Figure 3, the small ball fixed on the thin rod and the small ball moving in the smooth tubular track make circular motions under the action of gravity and the elastic force of the rod (pipe).

(1) Minimum speed at the highest point

Since the rod and tube can produce upward supporting force for the ball at the highest point, the minimum speed at which the ball can reach the highest point is v=0. At this time, the supporting force on the ball is FN=mg.

(2) When the ball passes the highest point, the elastic force of the track on the ball

Example 2 As shown in Figure 4, a light rod OA with a length of 0.5 m makes a circular motion in the vertical plane around point O. End A is connected to a small ball with a mass m = 2 kg (radius is not included). Find the following two In this case, the magnitude and direction of the force exerted by the ball on the rod when passing through the highest point (g is 10 m/s2, π2=10):

(1) The rotation speed of the rod in uniform circular motion is 2 r/s;

Critical Problems in Circular Motion

When an object moves in a circular motion, if the linear velocity and angular velocity of the object change, it will cause certain forces (such as pulling force, supporting force, friction force) to change, and then there will be sudden changes in certain physical quantities or motion states, that is, a critical state will occur. , the method to analyze the critical problem of circular motion is to gradually increase the angular velocity or linear velocity from a small value, analyze the changes of each quantity, and find out the critical state.

Usually, the most encountered ones involve the following three kinds of forces:

(1) Critical conditions related to the elasticity of the rope: the elasticity of the rope is exactly 0.

(2) Critical conditions related to the elasticity of the support surface: the support force is exactly 0.

(3) Critical problems caused by static friction: static friction reaches the maximum value.

Example 3 As shown in Figure 5, a thin wire with a length of l=1 m is used. One end is tied to a small ball with a mass of m=1 kg (can be regarded as a particle), and the other end is fixed to the top of a smooth cone. The angle between the cone surface and the vertical direction is θ = 37°. When the angular velocity of the ball in a uniform circular motion around the axis of the cone in the horizontal plane is ω, the tension of the thin wire is FT. (g is 10 m/s2, sin 37°=0.6, cos 37°=0.8, the result can be expressed by the radical formula)

(1) If the ball is to leave the cone, what is the minimum angular velocity ω0 of the ball?

(2) If the angle between the thin line and the vertical direction is 60°, what is the angular velocity ω′ of the ball?

Example 4 (End of the first semester of high school in Zhengzhou City, 2019) As shown in Figure 6A, a block with mass m is placed on the horizontal turntable. The distance from the block to the rotating axis is r, and the kinetic friction factor between the block and the turntable is μ. , assuming that the maximum static friction force experienced by the object is equal to the sliding friction force, and the acceleration due to gravity is known to be g.

(1) When the horizontal turntable rotates at a constant speed with an angular velocity ω1, the block and the turntable are just relatively stationary. Find the value of ω1;

(2) As shown in Figure B, connect the block and the rotating shaft with a string. When the angular velocity of the turntable ω2 = μg3r, find the tension FT2 of the string;

Special topic to strengthen the comprehensive analysis of circular motion PPT, the third part of the content: 02 in-class drills, point-by-point implementation

1. (Light rope model) The acrobat performs "Water Star". At one end of a string with a length of 1.6 m, a water container with a total mass of water of m = 0.5 kg is tied to the other end of the string. One end is the center of the circle, and it performs circular motion in the vertical plane, as shown in Figure 7. If the speed of the "water star" when passing through the highest point is 4 m/s, then the following statement is correct (g is 10 m/s2)

A. When the "water star" passes through the highest point, water flows out of the container

B. When the "water star" passes the highest point, the tension of the rope and the pressure on the bottom of the container are both zero.

C. When the "water star" passes through the highest point, it is in a state of complete weightlessness and is not affected by force.

D. When the "water star" passes the highest point, the pulling force of the rope is 5 N

2. (Roller Coaster Model) (Multiple Choice) As shown in Figure 8, a small ball with mass m makes a circular motion inside a smooth ring in the vertical plane. The radius of the ring is R, and the radius of the ball is not included. The ball passes through When the highest point on the inside of the ring is just within the circle, then when it passes the highest point, which of the following statements is correct?

A. The pressure exerted by the ball on the ring is equal to mg

B. Gravity mg acts as the centripetal force required for the ball to move in a circular motion

C. The linear velocity of the ball is equal to

D. The centripetal acceleration of the ball is equal to g

3. (Light rod model) As shown in Figure 9, a small ball with mass m is fixed at one end of the rod and makes a circular motion around the other end O of the rod in the vertical plane. When the ball reaches the highest point, instantaneously The speed is v=12Lg, L is the distance from the center of the ball to point O, then the force exerted by the ball on the rod is

A. Tension force of 12mg B. Pressure of 12mg

C. Zero D. Pressure of 32mg

4. (Critical Problems of Circular Motion) (End of the first semester of Shuangfeng No. 1 Middle School and Senior High School, 2019) As shown in Figure 10, three objects A, B, and C are placed on a rotating horizontal disk. The maximum distance between the objects and the disk is The static friction force is k times its gravity. The masses of the three objects are 2m, m, and m respectively. Their distances from the axis of rotation are R, R, and 2R respectively. When the disk rotates, if the three objects A, B, and C are equal If the relative disk is at rest, then which of the following statements is correct?

A.A has the largest centripetal acceleration

B. The friction forces on B and C are equal.

C. When the disk speed increases slowly, C slides before A

D. When the disk speed increases slowly, B slides before A

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