This student tutorial illustrates how circular motion principles can be combined with Newton's Second Law to analyze physical situations. The author uses free-body diagrams to analyze various forces acting upon a car moving in a circle. Two algebraic problems and detailed solutions are provided, plus a five-step model for solving circular motion problems.
9-12: 2B/H3. Mathematics provides a precise language to describe objects and events and the relationships among them. In addition, mathematics provides tools for solving problems, analyzing data, and making logical arguments.
3. The Nature of Technology
3A. Technology and Science
9-12: 3A/H2. Mathematics, creativity, logic, and originality are all needed to improve technology.
4. The Physical Setting
9-12: 4F/H1. The change in motion (direction or speed) of an object is proportional to the applied force and inversely proportional to the mass.
9-12: 4F/H8. Any object maintains a constant speed and direction of motion unless an unbalanced outside force acts on it.
10. Historical Perspectives
10B. Uniting the Heavens and Earth
9-12: 10B/H1. Isaac Newton, building on earlier descriptions of motion by Galileo, Kepler, and others, created a unified view of force and motion in which motion everywhere in the universe can be explained by the same few rules. Newton's system was based on the concepts of mass, force, and acceleration; his three laws of motion relating them; and a physical law stating that the force of gravity between any two objects in the universe depends only upon their masses and the distance between them.
Next Generation Science Standards
Motion and Stability: Forces and Interactions (HS-PS2)
Students who demonstrate understanding can: (9-12)
Analyze data to support the claim that Newton's second law of motion describes the mathematical relationship among the net force on a macroscopic object, its mass, and its acceleration. (HS-PS2-1)
Disciplinary Core Ideas (K-12)
Forces and Motion (PS2.A)
All positions of objects and the directions of forces and motions must be described in an arbitrarily chosen reference frame and arbitrarily chosen units of size. In order to share information with other people, these choices must also be shared. (6-8)
Newton's second law accurately predicts changes in the motion of macroscopic objects. (9-12)
Crosscutting Concepts (K-12)
Scale, Proportion, and Quantity (3-12)
Algebraic thinking is used to examine scientific data and predict the effect of a change in one variable on another (e.g., linear growth vs. exponential growth). (9-12)
Scientific Knowledge Assumes an Order and Consistency in Natural Systems (1-12)
Science assumes that objects and events in natural systems occur in consistent patterns that are understandable through measurement and observation. (6-8)
NGSS Science and Engineering Practices (K-12)
Using Mathematics and Computational Thinking (5-12)
Mathematical and computational thinking at the 9–12 level builds on K–8 and progresses to using algebraic thinking and analysis, a range of linear and nonlinear functions including trigonometric functions, exponentials and logarithms, and computational tools for statistical analysis to analyze, represent, and model data. Simple computational simulations are created and used based on mathematical models of basic assumptions. (9-12)
Use mathematical representations of phenomena to describe explanations. (9-12)
Common Core State Standards for Mathematics Alignments
High School — Number and Quantity (9-12)
Vector and Matrix Quantities (9-12)
N-VM.2 (+) Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point.
N-VM.3 (+) Solve problems involving velocity and other quantities that can be represented by vectors.
High School — Algebra (9-12)
Seeing Structure in Expressions (9-12)
A-SSE.1.a Interpret parts of an expression, such as terms, factors, and coefficients.
A-SSE.2 Use the structure of an expression to identify ways to rewrite it.
Reasoning with Equations and Inequalities (9-12)
A-REI.1 Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.
High School — Functions (9-12)
Linear, Quadratic, and Exponential Models? (9-12)
F-LE.5 Interpret the parameters in a linear or exponential function in terms of a context.
Common Core State Reading Standards for Literacy in Science and Technical Subjects 6—12
Key Ideas and Details (6-12)
RST.11-12.3 Follow precisely a complex multistep procedure when carrying out experiments, taking measurements, or performing technical tasks; analyze the specific results based on explanations in the text.
Craft and Structure (6-12)
RST.11-12.4 Determine the meaning of symbols, key terms, and other domain-specific words and phrases as they are used in a specific scientific or technical context relevant to grades 11—12 texts and topics.
Range of Reading and Level of Text Complexity (6-12)
RST.11-12.10 By the end of grade 12, read and comprehend science/technical texts in the grades 11—CCR text complexity band independently and proficiently.
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