Chapters 6, 7 Energy

Energy What is energy? Energy - is a fundamental, basic notion in physics Energy is a scalar, describing state of an object or a system Description of a system in ‘energy language’ is equivalent to a description in ‘force language’ Energy approach is more general and more effective than the force approach Equations of motion of an object (system) can be derived from the energy equations

Scalar product of two vectors The result of the scalar (dot) multiplication of two vectors is a scalar Scalar products of unit vectors

Scalar product of two vectors The result of the scalar (dot) multiplication of two vectors is a scalar Scalar product via unit vectors

Some calculus In 1D case

Some calculus In 1D case In 3D case, similar derivations yield K – kinetic energy

Kinetic energy K = mv2/2 SI unit: kg*m2/s2 = J (Joule) James Prescott Joule (1818 - 1889) Kinetic energy K = mv2/2 SI unit: kg*m2/s2 = J (Joule) Kinetic energy describes object’s ‘state of motion’ Kinetic energy is a scalar

Work-kinetic energy theorem Wnet – work (net) Work is a scalar Work is equal to the change in kinetic energy, i.e. work is required to produce a change in kinetic energy Work is done on the object by a force

Work: graphical representation 1D case: Graphically - work is the area under the curve Fx(x)

Chapter 6 Problem 52 A force with magnitude F = a√x acts in the x-direction, where a = 9.5 N/m1/2. Calculate the work this force does as it acts on an object moving from (a) x = 0 to x = 3.0 m; (b) 3.0 m to 6.0 m; and (c) 6.0 m to 9.0 m.

Net work vs. net force We can consider a system, with several forces acting on it Each force acting on the system, considered separately, produces its own work Since

Work done by a constant force If a force is constant If the displacement and the constant force are not parallel

Work done by a constant force

Work done by a spring force Hooke’s law in 1D From the definition of work

Work done by the gravitational force Gravity force is ~ constant near the surface of the Earth If the displacement is vertically up In this case the gravity force does a negative work (against the direction of motion)

Lifting an object We apply a force F to lift an object Force F does a positive work Wa The net work done If in the initial and final states the object is at rest, then the net work done is zero, and the work done by the force F is

Power Average power Instantaneous power – the rate of doing work SI unit: J/s = kg*m2/s3 = W (Watt) James Watt (1736-1819)

Chapter 6 Problem 36 A 75-kg long-jumper takes 3.1 s to reach a prejump speed of 10 m/s. What’s his power output?

Conservative forces The net work done by a conservative force on a particle moving around any closed path is zero The net work done by a conservative force on a particle moving between two points does not depend on the path taken by the particle

Conservative forces: examples Gravity force Spring force

Potential energy For conservative forces we introduce a definition of potential energy U The change in potential energy of an object is being defined as being equal to the negative of the work done by conservative forces on the object Potential energy is associated with the arrangement of the system subject to conservative forces

Potential energy For 1D case A conservative force is associated with a potential energy There is a freedom in defining a potential energy: adding or subtracting a constant does not change the force In 3D

Gravitational potential energy For an upward direction the y axis

Gravitational potential energy

Elastic potential energy For a spring obeying the Hooke’s law

Chapter 7 Problem 37 A particle moves along the x-axis under the influence of a force F = ax2 + b, where a and b are constants. Find its potential energy as a function of position, taking U = 0 at x = 0.

Conservation of mechanical energy Mechanical energy of an object is When a conservative force does work on the object In an isolated system, where only conservative forces cause energy changes, the kinetic and potential energies can change, but the mechanical energy cannot change

Conservation of mechanical energy From the work-kinetic energy theorem When both conservative a nonconservative forces do work on the object

Internal energy The energy associated with an object’s temperature is called its internal energy, Eint In this example, the friction does work and increases the internal energy of the surface

Chapter 7 Problem 53 A spring of constant k = 340 N/m is used to launch a 1.5-kg block along a horizontal surface whose coefficient of sliding friction is 0.27. If the spring is compressed 18 cm, how far does the block slide?

Conservation of mechanical energy: pendulum

Potential energy curve

Potential energy curve: equilibrium points Neutral equilibrium Unstable equilibrium Stable equilibrium

Questions?

Answers to the even-numbered problems Chapter 6 Problem 14: 9.6 × 106 J

Answers to the even-numbered problems Chapter 6 Problem 40: The hair dryer consumes more energy.

Answers to the even-numbered problems Chapter 6 Problem 50: 360 J

Answers to the even-numbered problems Chapter 7 Problem 14: (a) 7.0 MJ (b) 1.0 MJ

Answers to the even-numbered problems Chapter 7 Problem 24: (a) ± 4.9 m/s (b) ± 7.0 m/s (c) ≈ 11 m

Answers to the even-numbered problems Chapter 7 Problem 38: 95 m