The Energy Principle M&I Chapter 6

“It is important to realize that in physics today, we have no knowledge of what energy is.” Richard Feynman

We may not know what energy is, but we do know how to calculate it.

System and surroundings System Surroundings

The Energy Principle System Surroundings Work 功

The total energy of the universe is conserved. System

Energy is transferred between system and surroundings. Never created, never destroyed. So for any choice of system:

Energy of a single particle Units of energy: the joule (J). 1 J = 1 kg (m/s) 2.

For a particle at rest, γ = 1. A particle not moving (v = 0) has energy proportional to its mass. Mass and energy are the same thing!

Mass turning into energy.

For a moving particle, γ > 1 E particle mc 2 K γmc 2 Total energy is the rest energy plus an extra amount, related to the particle’s motion.

For a moving particle, γ > 1 Rest energy 静能 E particle mc 2 K γmc 2 Kinetic energy 动能

Kinetic energy E particle mc 2 K γmc 2

Your turn… Consider an electron (mass 9e-31 kg) moving with speed v = 0.9c. What is its rest energy? 1) 7.3e-31 J 2) 8.1e-14 J 3) 1.05e-13 J 4) 1.86e-13 J 5) 2.7e8 m/s e-e-

Your turn… Consider an electron (mass 9e-31 kg) moving with speed v = 0.9c. What is its total particle energy? 1) 7.3e-31 J 2) 8.1e-14 J 3) 1.05e-13 J 4) 1.86e-13 J 5) 2.7e8 m/s e-e-

Your turn… Consider an electron (mass 9e-31 kg) moving with speed v = 0.9c. Its rest energy is 0.81e-13 J, and its (total) particle energy is 1.86e-13 J. What is its kinetic energy? 1) 7.3e-31 J 2) 3.28e-14 J 3) 8.1e-14 J 4) 1.05e-13 J 5) 1.86e-13 J e-e-

Your turn… A ball whose mass is 2 kg travels at a velocity of m/s. What is the rest energy of the ball? 1) 0 J 2) 25 J 3) 6e8 J 4) 9e16 J 5) 1.8e17 J

Your turn… A ball whose mass is 2 kg travels at a velocity of m/s. What is the kinetic energy of the ball? 1) 0 J 2) 2 J 3) 10 J 4) 25 J 5) J

Kinetic energy at low speeds When v << c, we can use a binomial expansion to simplify the expression for kinetic energy: 二项展开式

Kinetic energy at low speeds E particle mc 2 K γmc 2

Kinetic energy at low speeds E particle mc 2 K γmc 2

Your turn… A ball whose mass is 2 kg travels at a velocity of m/s. What is the kinetic energy of the ball? 1) 0 J 2) 2 J 3) 10 J 4) 25 J 5) J

High speed: K > rest energy Low speed: K << rest energy

Work

Change in momentum is related to impulse: force multiplied by time.

What is related to force multiplied by distance?

Change in energy. This quantity is called “work” ( 功 ).

Impulse = Force x time Work = Force x distance …changes momentum …is a vector ( 矢量 ) …changes energy …is a scalar ( 标量 )

Work done by a constant force Units of work: the joule (J). Work is energy.

The vector dot product For any two vectors A and B: θ

Work done by a constant force where

Your turn… A figure skater slides in the –x direction along the ice, toward her partner. When she gets close he pushes on her in the +x direction, to slow her down. Does he do positive, negative, or zero work? 1) positive 2) negative 3) zero

Your turn… A fan cart moves in the –x direction. The fan is on, and the force on the cart by the air is also in the –x direction. Is the work done by the air positive, negative, or zero? 1) positive 2) negative 3) zero

Δx = +2 m F x = +3 N Is the work done on the block positive (+) or negative (-) ? The block is speeding up, so the work is positive.

Δx = +2 m F x = -3 N Is the work done on the block positive (+) or negative (-) ? The block is slowing down, so the work is negative.

Δx = -2 m F x = -3 N Is the work done on the block positive (+) or negative (-) ? The block is speeding up, so the work is positive.

Δx = -2 m F x = -3 N Is the work done on the block positive (+) or negative (-) ? The block is slowing down, so the work is negative.

θ 1.0 m x y How much work is done by the force F on the red block? Example

Only the component of the force in the direction of motion does work.

Work done by a changing force (see textbook, section 6.7)

Example A baseball (0.15 kg) is moving straight up with a speed of 20 m/s. Later, it is moving up with a speed of 15 m/s. (a) How much work was done on the ball? (b) How far did it move?