Forces in Two Dimensions - Complete Toolkit

Objectives

1. To use vector concepts to add force vectors and determine the resultant or net force.
2. To use vector concepts to determine the components of a force that is directed at an angle.
3. To use Newton’s laws and vector concepts to analyze objects that are at equilibrium (e.g., an object hung by two cables stretched diagonally to support points).
4. To use vector resolution and Newton’s second law to determine the acceleration of an object along a level surface when a force is applied to it at an angle to the surface.
5. To use free-body diagrams and Newton’s second law to analyze the motion of an object moving along an inclined plane.
6. To conduct a Newton’s second law analysis of a two-body system that includes two objects connected by a rope that is stretched over a pulley (e.g., an Atwood’s or modified Atwood’s machine).

Readings from The Physics Classroom Tutorial

1. The Physics Classroom Tutorial, Motion and Forces in Two-Dim. Chapter, Lesson 3
   http://www.physicsclassroom.com/class/vectors

Interactive Simulations

1. Vector Addition
   
   The Vector Addition Interactive provides learners with a tool for visualizing the addition of vectors using either the head-to-tail method or the component method. Up to three vectors can be added and the resultant is drawn. The components of each vector can be toggled on and off; the magnitudes of the components are displayed as a numerical value. The Interactive can serve as a great tool for learning how vectors add together. This HTML5 interactive works on all devices.
   
   
   
   
   
   
    
2. Name That Vector
    
   This skill-building activity presents 12 randomly-selected challenges to students. Each challenge involves the task of using components to add vectors. Vectors are presented graphically on a background grid so that the horizontal and vertical components can be easily determined. The Interactive reinforces the concept that the components of the resultant vector is the sum of the components of the individual vectors. This HTML5 Interactive works on tablets and Chromebooks; perfect for the 1:1 classroom.
   
   
   
   
   
    
3. Sliding Down an Incline Plane
   
   This Java simulation shows a stone block lying at rest on an inclined plane. Initially, the component of gravity along the plane surface is exceeded by the force of static friction. The slope of the ramp can be increased or decreased, allowing students to see the exact point when the component of gravity equals the force of static friction. If the ramp is raised any further, the block will slide down. The coefficient of static friction is set at 0.2. As the slope is increased or decreased, students will see changing numerical values for the gravitational component, the force of static friction, and the force of kinetic friction.
   
   
   
   
    
4. PhET:  The Ramp
   
   For teachers who want to take a deeper dive into mechanical advantage and real-life connections, "The Ramp" simulation is a good start. You can push a variety of objects up a virtual ramp: a file cabinet, refrigerator, piano, crate, or a sleepy dog! Set the coefficient of friction, set the ramp angle from 1-90 degrees, and change the applied force from 0-3000 Newtons. This gives the simulation powerful flexibility. And if that's not enough ... students can view related free-body diagrams and graphs of energy and forces.
    
   For a turn-key activity to accompany this simulation, try The Ramp Student Guide  Developed by a high school teacher, this lesson is simple enough to be completed without extensive prior knowledge. It will keep beginners on task as they tackle the multiple elements of this simulation. It asks probing questions about the nature of the forces at play when objects move on an incline plane.
   For a more in-depth lesson that introduces calculation of the ideal and actual mechanical advantage of the inclined plane, click here: "The Ramp" - Mechanical Advantage    Allow 60-90 minutes
    

Video and Animations

Interactive Homework Problems

Labs and Investigations

Minds On Physics Internet Modules:

1. Forces in Two Dimensions module, Ass’t 2D1 -  Vector Analysis
    
2. Forces in Two Dimensions module, Ass’t 2D2 -  F_net = m•a and Forces at Angles
    
3. Forces in Two Dimensions module, Ass’t 2D3 -  Equilibrium Concepts
    
4. Forces in Two Dimensions module, Ass’t 2D5 -  Static Equilibrium and Analysis
    
5. Forces in Two Dimensions module, Ass’t 2D5 -  Inclined Plane Concepts
    
6. Forces in Two Dimensions module, Ass’t 2D6 -  Incline Plane Analysis

Concept Building Exercises:

1. The Curriculum Corner, Forces in Two Dimensions, Another Angle on F-m-a
    
2. The Curriculum Corner, Forces in Two Dimensions, Adding and Resolving Forces
    
3. The Curriculum Corner, Forces in Two Dimensions, Analyzing Equilibrium Situations
    
4. The Curriculum Corner, Forces in Two Dimensions, Analyzing Accelerations on Level Surfaces
    
5. The Curriculum Corner, Forces in Two Dimensions, Inclined Plane Analysis
    

Problem-Solving Exercises:

1. The Calculator Pad, Vectors and Forces in Two Dimensions, Problems #1 - #27

Real Life Connections:

1. CIESE – Navigational Vectors
   
   This high school instructional unit features nine lessons relating to vectors.  Students build understanding of vector properties as they learn airplane navigation.  Project-based activities include reading real-time weather maps, tracking airplanes flying in U.S. skies, calculating vector components, analyzing effects of wind velocity, and completing training similar to a private pilot certification program.  The unit culminates with an online pilot flight test. Comprehensive teacher guides, student guides , reference materials, and assessments are included. 
    

Elsewhere on the Web:

1. Tilt Maze Game Model Java-Based game
   
   The Tilt Maze Game Model is a recent addition to Open Source Physics. The model is set on a plane that can be tilted to make the ball roll due to gravity. The plane can be tilted up, down, to the left and to the right. By tilting the planes, the user must cause the ball to move through the maze to get to the hole at the top left side of the maze...but be careful, there are some traps laid out that would stop you from accomplishing your goal.
   
   
   
    

Standards:

• HS-PS2.1.i  Newton’s second law accurately predicts changes in the motion of macroscopic objects

• Algebraic thinking is used to examine scientific data and predict the effect of a change in one variable on another

• When investigating or describing a system, the boundaries and initial conditions of the system need to be defined.

• Analyze data using computational models in order to make valid and reliable scientific claims.

• Develop and use a model based on evidence to illustrate the relationships between systems or between components of a system.
• Use a model to provide mechanistic accounts of phenomena.

• Plan and conduct an investigation individually and collaboratively to produce data to serve as the basis for evidence … and consider limitations on the precision of the data
• Select appropriate tools to collect, record, analyze, and evaluate data.
• Collect data about a complex model or system to identify failure points or improve performance relative to criteria for success or other variables.

• Construct and revise an explanation based on valid and reliable evidence obtained from a variety of sources (including students’ own investigations, models, theories, simulations) and the assumption that theories and laws that describe the natural world operate today as they did in the past and will continue to do so in the future.

• Use mathematical representations of phenomena to support claims.
• Use mathematical representations of phenomena to describe explanations.
  Create or revise a computational model or simulation of a phenomenon, designed device, process, or system.

• Reason abstractly and quantitatively
• Model with mathematics
• Look for and express regularity in repeated reasoning

• 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.
• N-VM.2   Find the components of a vector.
• N-VM.4.a  Add vectors end-to-end, component-wise, and by the parallelogram rule. Understand that the magnitude of a sum of two vectors is typically not the sum of its magnitudes.
• N-VM.4.b  Given two vectors in magnitude and direction form, determine the magnitude and direction of their sum.
• N-VM.4.c  Understand vector subtraction v – w as v + (-w) where –w is the additive inverse of w, with the same magnitude as w and pointing in the opposite direction. Represent vector subtraction graphically by connecting the tips in the appropriate order, and perform vector subtraction component-wise.

• A-REI.4.b   Solve quadratic equations by inspection (e.g., for x squared = 49), taking square roots, completing the square, the quadratic formula, and factoring, as appropriate to the initial form of the equation.
• A-REI.10   Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve.

• F-IF.4   For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship.
• 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.

• Use special triangles to determine geometrically the values of sine, cosine, tangent.

• RST.9-10.2  Determine the central ideas or conclusions of a text; trace the text’s explanation or depiction of a complex process, phenomenon, or concept; provide an accurate summary of the text.
• 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.11-12.4   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.

• RST.11-12.7    Integrate and evaluate multiple sources of information presented in diverse formats and media in order to address a question or solve a problem.
• RST.11-12.8    Evaluate the hypotheses, data, analysis, and conclusions in a science or technical text, verifying the data when possible and corroborating or challenging conclusions with other sources of information.
• 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.)

• RST-9-10.10    By the end of Grade 10, read and comprehend science/technical texts in the grades 9-10 text complexity band independently and proficiently.

• The linear acceleration of an object is directly proportional to the vector sum of all the forces acting on the object and inversely proportional to the object’s mass (a = "F/m). The vector sum of all the forces (net force) is not a real force caused by an interaction with another object. The single force that could replace the original multiple forces and cause the same acceleration of the object is the vector sum of forces.
• A special case of Newton’s second law occurs when the vector sum of all the forces (net force) on an object is zero. In this case, there is no acceleration and the object remains at rest or maintains a constant speed and a constant direction of motion.

• Analyze force diagrams to determine if they accurately represent different situations involving multiple contact, gravitational and/or electrical interactions. Interpret the meaning of the vector sum of all the forces (net force).
• Analyze different problems involving at least two different types of interactions (contact, gravitational, magnetic and/or electrical) and an identified object of interest.
• Represent the forces acting on the object of interest by drawing a force diagram showing both the vertical and horizontal forces. When appropriate, use vector addition to determine the relative size and direction of the sum of all the forces (net force), and interpret the meaning of the net force.
• Explain the observed motion of the object of interest. Justification is based on Newton’s second law.

• The force of kinetic friction always acts in the opposite direction of the relative velocity of the object with respect to the surface it is sliding over. The magnitude of the kinetic friction depends on the types of materials that make up the two surfaces sliding past each other and the magnitude of the compression force acting on the object. This can be mathematically represented by Fk = μkN.

• A contact interaction occurs when the surfaces of two solid objects are pressed together because of other interactions on one or both objects (e.g., a solid sitting on or sliding along a table; a magnet attached to a refrigerator). This is called a compression interaction. A compression (normal) force applied to an object is always a push directed at right angles from the surface of the other interacting object.
• A contact interaction occurs when a cord (e.g., rope, wire, rod) pulls on another object or system and the cord is not slack. A tension force on an object always points in the direction the cord is pulling.

• Explain why all objects near Earth’s surface fall with approximately the same acceleration, despite having different masses and weights. Justify by using the universal law of gravitation and Newton’s second law.
• Analyze different problems to determine whether a constant gravitational acceleration can be assumed. Justify the analysis by using the universal law of gravitation. When appropriate, calculate a gravitational force.