Solving two-body problems with Newton s Second Law You ll get multiple equations from the x and y directions, these equations can be solved simultaneously to find unknowns 1. Draw a separate free body diagram for each object 2. Apply Newton s second law to each object for both directions (x and y) 3. Solve the resultant equations for the unknown quantities (two-body example is done in Lab: Atwood s Machine ) (This is an example of a problem giving variables instead of numbers. Treat them as knowns, just like you would if you were given numbers.) Example 4-13 In outer space, an astronaut pushes on a box of mass m 1, with force F A1. The box is in direct contact with a second box of mass m 2. a) What is the acceleration of the two boxes? b) What is the magnitude of the force each box exerts on the other? Practice: Draw Free Body Diagrams and sum forces in both directions 1. Sled on ice being pulled by a dog on a frictionless surface 2. Sled being pulled up a snowy frictionless hill by a dog 3. Sled being pulled up a snowy hill by a dog, there is friction between hill and sled 4. A block compresses a spring that sits on a horizontal surface, with friction 5. Two different masses hanging from a frictionless pulley on either side 6. A mass on a frictionless surface attached to another mass hanging off the surface (see diagram) 7. Same as #6, but with friction This is a classic conceptual physics problem which is a good indicator to tell whether or not a student fully understands Newton s Third Law: Isaac Newton s horse won t pull a cart because the horse thinks the cart will pull back on him equally and therefore the cart won t accelerate. Is he correct? Section 5.1 Friction Static and Kinetic Friction Friction is an electromagnetic phenomenon: molecular attraction between surfaces Extreme example: Gecko foot Two kinds of friction: Static Friction: a force must overcome the static friction before an object on a surface can move Kinetic Friction: the frictional force once an object is already moving Problem solving tip: use friction to link the y and x motion max Static region Kinetic region Notice: a greater force is required to start an object moving than to keep it moving. 1
Example 5.1 A 0.40kg shuffle board disc slides along a ship deck, given an initial speed 8.5m/s, stopping after 8.0m. Find the coefficient of kinetic friction. Is the normal force always equal to mg? Is the frictional force always equal to µmg? New topic: Center of Mass A ball thrown follows a parabolic path. A baton thrown seems to follow a more complicated path but really the center of mass of the baton follows the same parabolic path. To find the center of mass of an object, we need calculus! Important information However you choose your origin, that s how your equation will report the answer back to you! Example: two masses example finding the center of mass: An Arc of Mass 1. Draw a picture, designating the origin 2. Define dm (a small piece of mass), replace it with something you can integrate over 3. Define vector r, write it in terms of things that can be integrated 4. Integration limits represent where there is mass 2
Review Chapter 5 Friction Center of Mass Static Friction: you must overcome the static friction before an object on a surface can move Kinetic Friction: the frictional force once an object is already moving Tip: use friction to link the y and x motion The way you choose your origin will be how your answer is reported back to you. Use a sum for finite elements and an integral for continuous objects CHALLENGE: Practice using the integral formula to find center of mass for linear density objects! Define a new term: Work Suppose you pull on a string attached to a box that moves along the x axis. The work done on the box by you is the component of the force in the direction of motion times the magnitude of the displacement Work is a scalar Units of work are Joules (J) Principle of Superposition: If there are several forces acting on an object, the total work is the sum of the work done by each force Do we use work correctly? In physics, is this a correct use of the word work? I did a lot of work when I wrote that essay It took some work to run up that flight of stairs It takes work to do isometric exercises (like pushing against a wall to gain strength) I worked hard to lift those boxes In common language, work and effort mean the same thing, but physics only gives credit for displaced results. Example 6-1 A crane exerts an upward force of 31 kn on a 3000kg truck to lift it 2.0m. Find a. The work done on the truck by the crane b. The work done on the truck by gravity c. The total work done on the truck Work-Kinetic Energy Theorem Mathematical derivation READ the problem solving strategy on page 177! Define Kinetic Energy 3
Work done by a VARIABLE force (force that changes with time) Constant force F or variable force F(x)? We need integrals Work is the area under an F vs. x curve A more sophisticated way to express work: We can make use of a mathematical tool of vector multiplication the Dot Product! Definition and examples Back to Work Always use the dot product in the work equation. But when do we use the integral form? Check if the force depends on position. Use: Regular form of the work equation for constant forces Integral form of the work equation for variable forces Initially a body moves in one direction and has kinetic energy K. Then it moves in the opposite direction with three times its initial speed. What is the kinetic energy now? A. K B. 3K C. 3K D. 9K E. 9K The SI unit of energy can be expressed as A. kg m/s B. kg m/s 2 C. m/(kg s) D. kg m s 2 E. kg m 2 / s 2 A 5-kg object slides down a frictionless surface inclined at an angle of 30º from the horizontal. The total distance moved by the object along the plane is 10 meters. The work done on the object by the normal force of the surface is A. zero B. 0.50 kj C. 0.43 kj D. 0.58 kj E. 0.25 kj 4
Always use the dot product in the work formula, but Variable forces are to be used in integral formula for work Constant forces can use non-integral formula for work Example with dot product: A particle travels through a displacement l = (3.00ˆ i 4.00 ˆ j )m While a force acts on it F = (5.00ˆ i + 2.00 ˆ j )N a) Find the work done by this force on the particle b) Find the length of this displacement Power: rate of change of work Example 6-10 A small motor is used to operate a lift that raises a load of bricks weighing 500N to a height of 10 m in 20s at constant speed. The lift weighs 300N. What is the power output of the motor? Negative work means A. the kinetic energy of the object increases. B. the applied force is variable. C. the applied force is perpendicular to the displacement. D. the applied force is opposite to the displacement. E. nothing; there is no such thing as negative work. Negative work means loosing (kinetic) energy! 5
A body moves with decreasing speed. Which of the following statements is true? A. The net work done on the body is positive, and the kinetic energy is increasing. B. The net work done on the body is positive, and the kinetic energy is decreasing. C. The net work done on the body is zero, and the kinetic energy is decreasing. D. The net work done on the body is negative, and the kinetic energy is increasing. E. The net work done on the body is negative, and the kinetic energy is decreasing. Begin Chapter 7 Gravitational Potential Energy An object can possess energy by virtue of its position (potential energy), not just its motion (kinetic energy) You can store that energy there by doing work to change the object s position If you lift an object to a height you do positive work on it, gravity does the same amount of negative work A spring also has potential energy (more with more compression) Example: a 10 kg book initially at rest, falls vertically for a distance of 2m onto a table. 1. What work did you need to do to lift the book from the table to the 2m height? 2. And when it falls from there to the table what kinetic energy does it have just before impact? (example of mixed concept problem) 3. What is the energy when the object at 1m high (before it impacts)? Conservative Force Work done by a conservative force is independent of path taken Example: gravity does work -mgh while you go up ski lift; +mgh while you ski down. Total work is zero (you get it all back) Conservative forces: Gravity Spring Force Integral formalism: work done by a conservative force is zero when particle moves around a closed loop Define Potential Energy A force is conservative if the work done is not zero around a closed path Nonconservative Forces (you loose energy, can t get it back by going back to the starting point): Friction Air resistance, drag Gravitational potential energy Spring potential energy 6
Spring potential Example 7.3 (work in groups) A system consists of a 110-kg basketball player, the rim of a basketball hoop, and Earth. Assume that the potential energy of this system is zero when the player is standing on the floor and the rim is horizontal. Find the total potential energy of this system when the player is hanging on the front of the rim. Also, assume that the center of mass of the player is 0.80 m above the floor when he is standing and 1.30m when he is hanging. The force constant of the rim is 7.2kN/m and the front of the rim is displaced downward a distance of 15cm. 7