December 8, 2017 - Page 3 of 6 - Shawn Zhong - 钟万祥

3.4 – Retarding & Drag Forces

Retarding Forces • Sometimes the frictional force is a function of an objects velocity (such as air resistance) • These forces are called drag, or retarding, forces. The Skydiver • Assume we drop Alex from an airplane • Typically the drag forces on a free-falling object take the form F_drag=bv or F_drag=cv^2, where b and c are constants • For this problem, lets assume F_drag=bv Velocity as a Function of Time Acceleration as a Function of Time Graph of Acceleration, Velocity, and Displacement 2005 Free Response Question 1 2013 Free Response Question 2

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3.5 – Ramps & Inclines

Drawing FBDs for Ramps • Choose the object of interest and draw it as a dot or box • Draw and label all the external forces acting on the object • Sketch a coordinate system, choosing the direction of the object's motion as one of the positive axes Pseudo-FBDs • When forces don't line up with axes, you can draw a pseudo-FBD • Break forces that don't line up with axes into components that do • Redraw your diagram with all forces parallel to axes

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3.6 – Atwood Machines

Atwood Machine • Atwood Machine - two objects are connected by a light string over a massless pulley • Ideal pulleys are frictionless and massless (they add no inertia to the system) • Tension is constant in a light string passing over an ideal pulley Solving Atwood Machine Problems 1. Adopt a sign convention for positive and negative motion 2. Draw a free body diagram for each mass 3. Write Newton's Second Law equations for each mass 4. Solve for unknowns Alternate Solution • Analyze the system as a whole Mass and Pulley on a Ramp

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4.1 – Work

What is Work? • Work is the process of moving an object by applying a force • The object must move for work to be done • The force must cause the movement • Work is a scalar quantity • Units are Joules Work in One Dimension • Only the force in the direction of the displacement contributes to the work done Non-Constant Forces • Work done is the area under the force vs. displacement graph Hooke's Law • The more you stretch or compress a string, the greater the force of the spring • The spring's force is opposite the direction of its displacement from equilibrium • Model this as a linear relationship, where the force applied by the spring is equal to a constant (the spring constant) multiplied by the spring's displacement from its equilibrium (rest) position • F_s=−kx • Slope of the graph gives you the spring constant, k (in N/m) Work Done in Compressing a Spring • A spring obeys Hooke's Law. How much work is done in compressing the spring from equilibrium to some point x? Work Done in Compressing a Non-Linear Spring Work in Multiple Dimensions Work-Energy Theorem Example: Velocity from an F-d Graph • Determine the object's final speed.

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4.2 – Energy & Conservative Forces

What is Energy? • Energy is the ability or capacity to do work ○ Work is the process of moving an object • Energy is the ability or capacity to move an object Energy Transformations • Energy can be transformed from one type to another • You can transfer energy from one object to another by doing work • Work-Energy Theorem ○ Work done on a system by an external force changes the energy of the system Kinetic Energy • Kinetic Energy is energy of motion ○ The ability or capacity of a moving object to move another object • K=1/2 〖mv〗^2 Potential Energy • Potential Energy (U) is energy an object possesses due to its position or state of being ○ Gravitational Potential Energy (Ug) is the energy an object possesses because of its position in a gravitational field ○ Elastic Potential Energy (Us) ○ Chemical Potential Energy ○ Electric Potential Energy ○ Nuclear Potential Energy • A single object can have only kinetic energy, as potential energy requires an interaction between objects Internal Energy • The internal energy of a system include the kinetic energy of the objects that make up the system and the potential energy of the configuration of the objects that make up the system • Changes in a system's structure can result in changes in internal energy Gravitational Potential Energy (Ug) Conservative Forces • A force in which the work done on an object is independent of the path taken is known as conservative force • A force in which the work done moving along a closed path is zero • A force in which the work done is directly related to a negative change in potential energy (W=-ΔU) Conservative Forces Non-Conservative Forces Gravity Elastic Forces Friction Drag Air Resistance Coulombic Forces Newton's Law of Universal Gravitation • The gravitational force of attraction between any two objects with mass • F_g=−(〖Gm〗_1 m_2)/r^2 r ̂ Force from Potential Energy Summary

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