13.3
Składowe normalne i styczne: rozwiązywanie problemów
Rozważmy mężczyznę o masie 70 kg siedzącego na krześle połączonym ze wspornikiem sworzniowym poprzez element BC. Jeżeli mężczyzna utrzymuje pozycję wy…
Wyobraźmy sobie mężczyznę o masie 70 kg siedzącego na krześle. Krzesło jest połączone ze wspornikiem sworzniowym za pomocą pręta BC o długości 10 m.
Jeśli mężczyzna zawsze siedzi w pozycji pionowej, określ poziome i pionowe oddziaływania krzesła na mężczyznę, gdy członek tworzy kąt 45° z poziomem. W tym momencie mężczyzna porusza się z prędkością 5 m/s, która rośnie w tempie 1 m/s2.
Tutaj człowiek pokonuje tor krzywoliniowy, a przyspieszenie styczne człowieka wynosi 1 m/s2. Normalne przyspieszenie człowieka można obliczyć za pomocą prędkości stycznej i promienia krzywizny.
Następnie narysuj diagram swobodnego ciała człowieka i napisz równania ruchu dla składowych stycznych i normalnych.
Podstawiając znane wartości i zakładając, że przyspieszenie ziemskie wynosi 10 m/s2, ustala się dwa równania z wymaganymi siłami reakcji.
Ich jednoczesne rozwiązanie daje wielkości sił reakcji wzdłuż kierunku poziomego i pionowego.
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Q1: How do you calculate normal acceleration for an object moving along a curvilinear path?
Normal acceleration is calculated using the tangential speed and the radius of curvature. The formula relates the square of the velocity to the radius of the curved path. In the chair problem, with a speed of 5 m/s and a member length of 10 m acting as the radius, normal acceleration can be determined. This component is perpendicular to the direction of motion and points toward the center of the curved path.
Q2: What is the difference between tangential and normal acceleration in curvilinear motion?
Tangential acceleration changes the speed of an object along its path, while normal acceleration changes the direction of motion. In the problem, tangential acceleration is 1 m/s², indicating the speed increases at this rate. Normal acceleration acts perpendicular to velocity, directed toward the center of curvature. Both components are essential for analyzing motion on curved paths and must be considered separately in equations of motion.
Q3: Why is a free-body diagram necessary when solving curvilinear motion problems?
A free-body diagram isolates the object and shows all forces acting on it, enabling systematic analysis of motion. For the seated man, the diagram displays weight, normal force, and reaction forces from the chair. This visual representation clarifies force directions and magnitudes, making it easier to write accurate equations of motion for tangential and normal components. Without it, force relationships become unclear and errors in problem-solving increase.
Q4: How do you set up equations of motion for normal and tangential components?
Equations of motion are derived by applying Newton's second law separately to tangential and normal directions. For the tangential direction, sum forces parallel to velocity and set equal to mass times tangential acceleration. For the normal direction, sum forces perpendicular to velocity and set equal to mass times normal acceleration. Substituting known values like mass (70 kg), accelerations, and gravity (10 m/s²) yields two equations with unknown reaction forces that can be solved simultaneously.
Q5: What role does the member angle play in determining reaction forces on the chair?
The member angle of 45° determines the orientation of the support structure and affects how forces distribute between horizontal and vertical directions. At this angle, the geometry influences both the normal and tangential components of acceleration relative to the chair's support. The angle directly impacts the direction of reaction forces the chair exerts on the man, requiring careful geometric analysis when decomposing forces into horizontal and vertical components for solving the problem.
Q6: How does increasing speed affect the normal acceleration in this curvilinear motion problem?
Normal acceleration depends on the square of the tangential speed and the radius of curvature. As speed increases from 5 m/s at a rate of 1 m/s², the normal acceleration increases proportionally to the square of the new speed. With a fixed radius of 10 m, higher speeds produce greater normal acceleration, requiring larger normal forces from the chair to maintain the curved path. This relationship demonstrates why faster motion on curves demands greater support forces.
Q7: What assumptions are made when solving for reaction forces in this chair problem?
Key assumptions include the man remaining upright throughout motion, the member BC maintaining constant length of 10 m, and gravity acting at 10 m/s². The man is treated as a point mass, and the chair provides only normal and tangential reaction forces. The curvilinear path is assumed to have a constant radius equal to the member length. These simplifications allow the problem to be solved using standard equations of motion without accounting for body rotation or deformation.