Chapters 7, 8 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 = mv 2 /2 SI unit: kg*m 2 /s 2 = J (Joule) Kinetic energy describes object’s ‘state of motion’ Kinetic energy is a scalar James Prescott Joule ( )

Chapter 7 Problem 31 A 3.00-kg object has a velocity of (6.00^i – 2.00^j) m/s. (a) What is its kinetic energy at this moment? (b) What is the net work done on the object if its velocity changes to (800^i ^j) m/s?

Work–kinetic energy theorem W net – 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 F(x)

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 spring force Hooke’s law in 1D From the work–kinetic energy theorem

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 W a 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*m 2 /s 3 = W (Watt) James Watt ( )

Power of a constant force In the case of a constant force

Chapter 8 Problem 32 A 650-kg elevator starts from rest. It moves upward for 3.00 s with constant acceleration until it reaches its cruising speed of 1.75 in/s. (a) What is the average power of the elevator motor during this time interval? (b) How does this power compare with the motor power when the elevator moves at its cruising speed?

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

Chapter 7 Problem 44 A single conservative force acting on a particle varies F = (– Ax + Bx 2 ) ^i N, where A and B are constants and x is in meters. (a) Calculate the potential energy function U(x) associated with this force, taking U = 0 at x = 0. (b) Find the change in potential energy and the change in kinetic energy of the system as the particle moves from x = 2.00 m to x = 3.00 m.

Gravitational potential energy For an upward direction the y axis

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

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

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

Work done by an external force Work is transferred to or from the system by means of an external force acting on that system The total energy of a system can change only by amounts of energy that are transferred to or from the system Power of energy transfer, average and intantaneous

Conservation of mechanical energy: pendulum

Potential energy curve

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

Chapter 8 Problem 55 A 10.0-kg block is released from point A. The track is frictionless except for the portion between points B and C, which has a length of 6.00 m. The block travels down the track, hits a spring of force constant 2250 N/m, and compresses the spring m from its equilibrium position before coming to rest momentarily. Determine the coefficient of kinetic friction between block and the rough surface between B and C.

Answers to the even-numbered problems Chapter 7 Problem 2: (a) 3.28 × 10 −2 J (b) × 10 −2 J

Answers to the even-numbered problems Chapter 7 Problem 10: 16.0

Answers to the even-numbered problems Chapter 7 Problem 46: (7−9x 2 y)ˆi−3x 3 ˆj

Answers to the even-numbered problems Chapter 8 Problem 14: (a) m/s (b) m/s

Answers to the even-numbered problems Chapter 8 Problem 28: 8.01 W

Answers to the even-numbered problems Chapter 8 Problem 34: 194 m

Answers to the even-numbered problems Chapter 8 Problem 50: (a) J (b) J (c) 2.42 m/s (d) U C = J, K C = J