Energy Transformations and Conservation of Mechanical Energy 8.01 W05D2

Today’s Reading Assignment: W05D2 Young and Freedman: ,12.3 Experiment 3: Energy Transformations

Review: Potential Energy Difference Definition: Potential Energy Difference between the points A and B associated with a conservative force is the negative of the work done by the conservative force in moving the body along any path connecting the points A and B.

Review: Examples of Potential Energy with Choice of Zero Point (1) Constant Gravity: (2) Inverse Square Gravity (3) Spring Force

Review: Work-Energy Theorem: Conservative Forces The work done by the total force in moving an object from A to B is equal to the change in kinetic energy When the only forces acting on the object are conservative forces then the change in potential energy is Therefore

Forms of Energy kinetic energy gravitational potential energy elastic potential energy thermal energy electrical energy chemical energy electromagnetic energy nuclear energy mass energy

Energy Transformations  Falling water releases stored ‘gravitational potential energy’ turning into a ‘kinetic energy’ of motion.  Human beings transform the stored chemical energy of food into catabolic energy  Burning gasoline in car engines converts ‘chemical energy’ stored in the atomic bonds of the constituent atoms of gasoline into heat  Stretching or compressing a spring stores ‘elastic potential energy’ that can be released as kinetic energy

Energy Conservation Energy is always conserved It is converted from one form into another, as the system transforms from an “initial state” to a “final state”, each form of energy can undergo a change Energy can also be transferred from a system to its surroundings

Concept Question: Energy Transformations 1. The potential energy of the system increases. 2. The kinetic energy of the system decreases. 3. The earth does negative work on the system. 4. You do negative work on the system. 5. Two of the above. 6. None of the above. You lift a ball at constant velocity from a height h i to a greater height h f. Considering the ball and the earth together as the system, which of the following statements is true?

Mechanical Energy When a sum of conservative forces are acting on an object, the potential energy function is the sum of the individual potential energy functions with an appropriate choice of zero point potential energy for each function Definition: Mechanical Energy The mechanical energy function is the sum of the kinetic and potential energy function

Conservation of Mechanical Energy When the only forces acting on an object are conservative Equivalently, the mechanical energy remains constant in time

Non-Conservative Forces Definition: Non-conservative force Whenever the work done by a force in moving an object from an initial point to a final point depends on the path, then the force is called a non-conservative force and the work done is called non-conservative work

Non-Conservative Forces Work done on the object by the force depends on the path taken by the object Example: friction on an object moving on a level surface

Change in Energy for Conservative and Non-conservative Forces Force decomposition: Work done is change in kinetic energy: Mechanical energy change:

Concept Question: Energy and Choice of System 1. block 2. block + spring 3. block + spring + incline 4. block + spring + incline + Earth A block of mass m is attached to a relaxed spring on an inclined plane. The block is allowed to slide down the incline, and comes to rest. The coefficient of kinetic friction of the block on the incline is µ k. For which definition of the system is the change in energy of the system (after the block is released) zero?

Worked Example: Block Sliding off Hemisphere A small point like object of mass m rests on top of a sphere of radius R. The object is released from the top of the sphere with a negligible speed and it slowly starts to slide. Find an expression for the angle θ f with respect to the vertical at which the object just loses contact with the sphere.

Strategy: Using Multiple Ideas Energy principle: No non-conservative work For circular motion, you will also need to Newton’s Second Law in the radial direction because no work is done in that direction hence the energy law does not completely reproduce the equations you would get from Newton’s Second Law Constraint Condition : Conservation

Worked Example: Energy Changes

Worked Example: Free Body Force Diagram Newton’s Second Law Constraint condition: Radial Equation becomes

Worked Example: Combining Concepts Newton’s Second Law Radial Equation Energy Condition: Combine Concepts:

Modeling the Motion: Newton’s Second Law  Define system, choose coordinate system.  Draw free body force diagrams.  Newton’s Second Law for each direction.  Example: x -direction  Example: Circular motion

Modeling the Motion Energy Concepts Change in Mechanical Energy: Identify non-conservative forces. Calculate non-conservative work Choose initial and final states and draw energy diagrams. Choose zero point P for potential energy for each interaction in which potential energy difference is well- defined. Identify initial and final mechanical energy Apply Energy Law.

Table Problem: Loop-the-Loop An object of mass m is released from rest at a height h above the surface of a table. The object slides along the inside of the loop-the-loop track consisting of a ramp and a circular loop of radius R shown in the figure. Assume that the track is frictionless. When the object is at the top of the track (point a) it just loses contact with the track. What height was the object dropped from?

Demo slide: Loop-the-Loop B95 ?page=demo.php?letnum=B 95&show=0 A ball rolls down an inclined track and around a vertical circle. This demonstration offers opportunity for the discussion of dynamic equilibrium and the minimum speed for safe passage of the top point of the circle.

Demo slide: potential to kinetic energy B97 p?page=demo.php?letnum=B 97&show=0 This demonstration consists of dropping a ball and a pendulum released from the same height. Both balls are identical. The vertical velocity of the ball is shown to be equal to the horizontal velocity of the pendulum when they both pass through the same height.

Worked Example: Cart-Spring on an Inclined Plane An object of mass m slides down a plane that is inclined at an angle θ from the horizontal. The object starts out at rest. The center of mass of the cart is an unknown distance d from an unstretched spring with spring constant k that lies at the bottom of the plane. Assume the inclined plane to be frictionless. The spring compress a distance x when the mass first comes to rest? Find an expression for the distance d.

Table Problem: Experiment 3 Cart-Spring on an Inclined Plane An object of mass m slides down a plane that is inclined at an angle θ from the horizontal. The object starts out at rest. The center of mass of the cart is a distance d from an unstretched spring with spring constant k that lies at the bottom of the plane. Now assume that the inclined plane has a coefficient of kinetic friction μ. How far will the spring compress when the mass first comes to rest? How much energy has been transformed into heat due to friction?

Experiment 3 Energy Transformation

Potential Energy and Force In one dimension, the potential difference is Force is the derivative of the potential energy Examples: (1) Spring Potential Energy: (2) Gravitational Potential Energy:

Energy Diagram Choose zero point for potential energy: Potential energy function: Mechanical energy is represented by a horizontal line since it is a constant Kinetic energy is difference between mechanical energy and potential energy (independent of choice of zero point) Graph of Potential energy function U ( x ) vs. x

Table Problem: Energy Diagram The figure above shows a graph of potential energy U(x) verses position for a particle executing one dimensional motion along the x-axis. The total mechanical energy of the system is indicated by the dashed line. At t =0 the particle is somewhere between points A and G. For later times, answer the following questions. a)At which point will the magnitude of the force be a maximum? b)At which point will the kinetic energy be a maximum? c)At how many of the labeled points will the velocity be zero? d)At how many of the labeled points will the force be zero?

Table Problem: Potential Energy Diagram A body of mass m is moving along the x- axis. Its potential energy is given by the function U(x) = b(x 2 -a 2 ) 2 where b = 2 J/m 4 and a = 1 m. a) On the graph directly underneath a graph of U vs. x, sketch the force F vs. x. b) What is an analytic expression for F(x)?

Next Reading Assignment: W05D3 Young and Freedman , 12.3,