# Conveyor Belt Equations

- Jan 14, 2019-

1 Chasing the encounter problem

The movement of two objects on the same line often involves problems such as chasing, encountering or avoiding collisions. The essential condition of such problems is to see whether the two objects can reach the same position in space at the same time.

1.1 Tracing the problem: chasing and chasing the two objects at the same speed (same direction motion) is the critical condition for catching up and the extreme distance between the two. There are two common situations: the first type - the speed is large Deceleration (such as uniform deceleration linear motion) chasing speed is small (such as uniform motion): 1 When the two speeds are equal, if the chaser displacement is still less than the chased displacement, it will never catch up, at this time there is shortest distance. 2 If the displacements of the two are equal, and the speeds of the two are equal, then it can catch up, and it is also the critical condition for avoiding collision. 3 If the two players are equal in displacement, the chasing speed is still greater than the speed of the chased person. Then the chased person has another chance to catch up with the chaser. When the speed is equal, the distance between the two has a larger value. The second type - the speed is small (such as the uniform acceleration of the initial speed of zero linear motion) chasing the speed of the greater (such as uniform motion): 1 when the two speeds are equal, there is a maximum distance. 2 If the two displacements are equal, then catch up. 3 When the object being chased is used for uniform deceleration, be sure to pay attention to whether the object has stopped moving before it catches up.  1.2 Encounter problems: Encounter problems are divided into two situations: chasing and encountering and facing movement. The main condition is that the coordinates of the two objects at the same position are the same. When analyzing such problems, we must pay attention to grasping one condition and two relations: one condition is the critical condition that is satisfied when the speeds of the two objects are equal, such as whether the distance between the two objects is the largest or the smallest, whether it just catches up. The two relationships are the time relationship and the displacement relationship. The basic idea of solving is: 1 separately study two objects; 2 draw a schematic diagram of the motion process; 3 find the time relationship, velocity relationship and displacement relationship of the two objects; 4 establish the equation, solve the result, and discuss if necessary.

Example: Example 1. Both objects of A and B move in the same direction on the same straight line at the same time. A makes a uniform linear motion at a speed of 6m/s. The initial velocity is zero and the acceleration is 2m/s2. motion. When is the distance between the two? What is the maximum distance?

Analysis: At the beginning of a period of time, A fast and slow, A is in front, the distance between the two becomes larger, the speed of A is v = 6m / s, the acceleration of B is a = 2m / s2, when the speed of B reaches 6m / s When the distance between the two is the largest, the velocity formula v=at is obtained as t=vA/a=6/2s=3s. Within this 3s, the displacement of A is s = v A = t = 6 × 3m = 18m, the displacement of B is s = = 2 / 2 = 2 × 32 / 2m = 9m, the maximum distance between the two △ s = s - s B = 18m-9m = 9m.

2 conveyor problem

Regarding the conveyor belt problem, the interaction between the slider and the conveyor belt is the sliding friction force, which is one of the important reasons for participating in changing the movement state of the slider. The size is in accordance with the calculation formula of the sliding friction force, which is independent of the speed of the slider relative to the conveyor belt. The direction depends on the relative movement direction of the conveyor belt, and the direction of the sliding friction force changes, which will cause the transition of the slider movement state, so that the same physical environment Multiple physical processes may occur at the same time. Therefore, such propositions are often quite difficult. The moment when the slider and the conveyor belt are at the same speed is the moment of the relative movement direction and the direction of the sliding friction force, and is also the critical point of the transition of the slider motion state.

Example: As shown in Figure 2-1, the conveyor belt is at an angle of θ=37° to the ground. It rotates counterclockwise at a speed of 10 m/s. Gently place an object of mass m=0.5 kg on the upper end of the conveyor belt. The dynamic friction factor μ=0.5, the length of the conveyor belt from A→B is L=16m, what is the time required for the object from A to B?

Analysis: After the object is placed on the conveyor belt, it starts for a period of time, and its motion acceleration is a=■=10m/s2. Such acceleration can only be maintained until the speed of the object reaches 10m/s, and the corresponding time and displacement are: t1=■=■s=1s, s1=■=5m<16m, and the frictional force of the object becomes The acceleration along the conveyor belt is (because mgsin θ > μmgcos θ). A2=■=10m/s2.  Let the time taken by the object to complete the remaining displacement s2 is t2, then s2=u0t2+■a2t2, 11m=10t2+t22, the solution is: t211s, or t22=-11s (rounded), ∴t total = 1s+1s=2s. The difficulty of this type of problem is to determine the movement of the object relative to the ground and the relative conveyor belt, and to judge the error. The breakthrough method is to use the theoretical basis that "force is the reason for changing the motion state of the object", to make a correct analysis of the motion properties of the object, to determine the acceleration and velocity relationship between the object and the conveyor belt, and to be clear about the block as it is When the speed reaches the same speed as the conveyor belt, it is the turning point of the friction direction and the size change. Draw a sketch analysis to find the displacement of the object and the belt and the relationship between the two.

The method for solving such problems is as follows: selecting a research object and isolating the selected research object is a good way to make it difficult. The object gently placed on the moving conveyor belt, due to the backward sliding of the opposite conveyor belt, is subjected to the sliding frictional force in the direction of the movement of the conveyor belt, which determines that the object will be uniformly accelerated under the sliding friction force given by the conveyor belt. Until the object reaches the same speed as the belt, it is no longer subject to friction, and a uniform linear motion is carried out with the conveyor belt. The conveyor belt always makes a uniform linear motion. If you want to combine the two together, you need to draw a displacement diagram of the motion process to make it easy for students to grasp. In short, as long as the object is on the conveyor belt, it wants to reach the same speed as the conveyor belt. As for whether it can be achieved depends on the actual conditions. Simplification is: 1 study the acceleration of the block; 2 draw a schematic diagram of the motion process; 3 find out the time relationship, velocity relationship, displacement relationship and displacement relationship of the object; 4 establish the equation, solve the result, discuss if necessary .

3 plate model

This type of problem is usually a small slider moving on a wooden board, and small pieces and long boards are linked by a pair of sliding friction or static friction. Separate the selected research objects, select the ground as the reference system, apply Newton's second law and kinematics knowledge, and find the displacement of the board to the ground. The key to solving such problems lies in the in-depth analysis and establish a picture in the mind. A clear and dynamic physical picture, for which you should carefully draw a sketch. During the relative movement of the wooden board and the wooden block, the sliding friction force f acting on the wooden block is the power, and the sliding friction force f' acting on the wooden board is the resistance, and the displacement of the wooden board is exactly equal to the block in the relative movement. The sum of the displacement Sm of the left end of the board leaving the board and the length L of the board, and their respective uniform acceleration movements are all completed within the same time t.

Example: As shown in the figure, the car with mass M=8kg is parked on a smooth horizontal surface, and a horizontal constant force F=8N is applied to the right end of the car. When the moving speed of the car to the right reaches 3m/s, a small mass of m=2kg is lightly placed at the right end of the trolley, and the dynamic friction factor of the block and the small workshop is μ=0.2, assuming that the trolley is long enough, ask: 1 How long does the block stop moving relative to the small workshop? What is the displacement of 2 small blocks from t0=3.0s after being placed on the car? (g takes 10m/s2)

Analysis: 1 After the small block is placed, the uniform acceleration motion is performed under the action of friction. The acceleration a=μg=2m/s2, and the car accelerates with acceleration a', a'=■=■m/s2=0.5m/ S2, set the elapsed time t, the block accelerates to the same speed as the trolley, and stops the relative movement with the small workshop. Then there is at=v+a't (2 points) substituting the data solution t=2s, 2 small blocks are put up and 2s before the acceleration motion, after 1 second uniform acceleration motion a''=■=0.8m/s2 then s= ■ at2+at(3-t)+■a''(3-t)2=8.4m.

The methods to solve such problems are: 1 to study the acceleration of the block and the board; 2 to draw a schematic diagram of the respective motion processes; 3 to find the time relationship, velocity relationship, relative displacement relationship of the object motion; 4 to establish the equation, solve the result, necessary Discuss when.

In these three models, especially the most complex model of the plate requires students to analyze the acceleration of the wood and wood blocks, write the displacement and velocity expressions, and find the time to reach the common speed. The second is the conveyor belt model. In general, only the acceleration and motion of the object need to be analyzed, and the conveyor belt is generally uniform motion without additional analysis. Finally, it is to catch up with the encounter problem. It is only a kinematic problem and does not involve the problem of force analysis. Relatively the simplest, as long as the displacement relationship speed formula can be a problem. For the above three models, we can easily find that their commonalities are: 1 to write displacement and velocity expressions respectively; 2 to obtain unknowns based on the relationship between displacement and velocity. I think that in the three models, as long as you are proficient in analyzing the plate model, the other two models can be solved based on the known conditions. This can reduce the students' memory of the number of models and achieve twice the result with half the effort.