This resource guides the user through characteristics of circular motion. The same concepts and principles used to describe the motion of an object can also be used to describe and explain the motion of objects in circular pathways. This tutorial is broken into five sections addressing: the mechanics of circular motion, centripetal force, algebraic and trigonometric problems and solutions, and a full chapter that debunks the centrifugal "force" misconception. The interactive problems use diagrams and force vectors to help students visualize how vector components affect the way circular motion is characterized.

6-8: 4F/M3b. If a force acts towards a single center, the object's path may curve into an orbit around the center.

9-12: 4F/H8. Any object maintains a constant speed and direction of motion unless an unbalanced outside force acts on it.

9. The Mathematical World

9B. Symbolic Relationships

9-12: 9B/H1b. Sometimes the rate of change of something depends on how much there is of something else (as the rate of change of speed is proportional to the amount of force acting).

9-12: 9B/H5. When a relationship is represented in symbols, numbers can be substituted for all but one of the symbols and the possible value of the remaining symbol computed. Sometimes the relationship may be satisfied by one value, sometimes by more than one, and sometimes not at all.

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)

The motion of an object is determined by the sum of the forces acting on it; if the total force on the object is not zero, its motion will change. The greater the mass of the object, the greater the force needed to achieve the same change in motion. For any given object, a larger force causes a larger change in motion. (6-8)

Newton's second law accurately predicts changes in the motion of macroscopic objects. (9-12)

Crosscutting Concepts (K-12)

Patterns (K-12)

Patterns in rates of change and other numerical relationships can provide information about natural systems. (6-8)

Systems and System Models (K-12)

When investigating or describing a system, the boundaries and initial conditions of the system need to be defined and their inputs and outputs analyzed and described using models. (9-12)

NGSS Science and Engineering Practices (K-12)

Obtaining, Evaluating, and Communicating Information (K-12)

Obtaining, evaluating, and communicating information in 9–12 builds on K–8 and progresses to evaluating the validity and reliability of the claims, methods, and designs. (9-12)

Communicate scientific information (e.g., about phenomena and/or the process of development and the design and performance of a proposed process or system) in multiple formats (including orally, graphically, textually, and mathematically). (9-12)

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.1 (+) Recognize vector quantities as having both magnitude and direction. Represent vector quantities by directed line segments, and use appropriate symbols for vectors and their magnitudes (e.g., v, |v|, ||v||, v).

High School — Algebra (9-12)

Seeing Structure in Expressions (9-12)

A-SSE.2 Use the structure of an expression to identify ways to rewrite it.

High School — Functions (9-12)

Interpreting Functions (9-12)

F-IF.6 Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.

F-IF.8.b Use the properties of exponents to interpret expressions for exponential functions.

F-IF.9 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).

Common Core State Reading Standards for Literacy in Science and Technical Subjects 6—12

Key Ideas and Details (6-12)

RST.9-10.3 Follow precisely a complex multistep procedure when carrying out experiments, taking measurements, or performing technical tasks, attending to special cases or exceptions defined in the text.

RST.11-12.2 Determine the central ideas or conclusions of a text; summarize complex concepts, processes, or information presented in a text by paraphrasing them in simpler but still accurate terms.

Integration of Knowledge and Ideas (6-12)

RST.11-12.9 Synthesize information from a range of sources (e.g., texts, experiments, simulations) into a coherent understanding of a process, phenomenon, or concept, resolving conflicting information when possible.

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.

Henderson, T. (2004, December 12). Physics Classroom: Motion Characteristics for Circular Motion. Retrieved April 26, 2015, from http://www.physicsclassroom.com/Class/circles/U6L1a.cfm

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