Chapter 3, Energy and Conservation Laws

Physics 2001, Professor Ray Merry

Conservation Laws:

What does it mean for a physical quantity to be conserved?

It means:

The amount of that thing does not change.

E.G. Conservation of mass, means the amount of mass in a system stays constant.

Take the example of a burning cigarette.

Cigarette before burning weighs 10 g.

After burning the butt plus the ash weighs 5 g.

What happened? Is mass conserved?

What is the Closed System

Note for all the conservation laws we must include all the parts of the system!

For example, not considering the air in the cigarette example led to an error.

This is the secret to good scientific analysis. What is the total environment or system?

Linear momentum, p:

The mass of an object times its velocity.

p = mv Note since v is a vector so is p!

Example, m = 5 kg, v = 10 m/s what is p?

p = 5 x 10 kg m/s = 50 kg m/s

Note units are kg m/s or mass x vel. Units.

Newton’s Second Law w Impulse

F= D(mv)/Dt is another way of stating Newton’s second law, in terms of

D(mv), change in momentum and

Dt time for the change to occur.

We can also state this as

F Dt = D(mv) Where F Dt is the impulse of a force.

Impulse Diagram-Ball Hitting Floor

Impulse Problem

Suppose a ball of .1 kg is in contact with a bat for .2 s, and the force the bat exerts is 5 n. What would the speed of the ball be?

F Dt = D(mv) since original velocity of the ball is 0, D(mv) = m x final vel.(vf)

5n x .2s = .1kg x v final

Vf = 1 ns/.1kg = 10 m/s

(n=kg m/s/s nxs = kg m/s n/kg = m/s²)

Law of Conservation of Linear Momentum:

The total linear momentum of an isolated system is constant.

Remember this is a vector relationship

If all velocities are in the same direction it is simple, if not we must use vectors to solve

total mv before = total mv after

1 D Momentum Problem

A 200lb boy moving at 10 ft/sec hits 2 50 kg midgets at rest. Both go off together, but how fast ?

Momentum before = momentum after

Mom.before =200 kg x 10m/s = 2000 kgm/sec

Mass after = 50 +50 + 200 = 300 kg

After mom = 300 x V2; 2000 = 300 V2

V2 = 2000/300 = 6.6 m/s

Work

The force that acts times the distance moved in the direction of the force.

Work = F x d (Joules =Newtons x meters)

Work units are Joules, same as energy.

E.G. Force of 10 N moves 5 m

W = 10 N x 5 m = 50 Nm or 50 J

Note Force is a vector but work is a scalar!c

Force vs Work

Note work = force x distance in direction of force.

If Force is perpendicular to motion or displacement no work is done!

Examples circular motion, moving parallel to floor

Gravitational Work

When we lift a body we do work against gravity, but when a body falls, gravity does work on it. If the distances are equal so are the works.

Work = F x d

F = weight = W=mg

W=Wd = mgd

Example 10kg falls 5m, what is the work?

W = 10kg x 9.8m/s/s x 5m =490J

Energy (E)

The measure of a system's capacity to do work.

That which is transferred when work is done.

Energy is measured in Joules, which are Newtons x meters.

Define kinetic energy:

Kinetic energy (KE) Energy due to motion. Energy that an object has when it is moving

KE = ½ mv²

Example, 20 kg mass moving at 15 m/s

KE = ½ x 20 kg x (15)²=10 x 225 = 2250 J

Potential Energy

Potential energy (PE) is Energy due to an object's position or configuration.

A stretched spring has potential energy.

A raised object has gravitational energy.

Gravitational Potential Energy

Energy due to position, PE

Gravitational PE = mgd

Other PE’s

elastic potential energy

stretched out rubber band or spring.

internal energy

unburned fuel, battery

PE Problem

We can calculate the potential energy of a given situation if we know the force.

Since work against gravity is weight x height lifted. mgd, PE is the same, mgd.

W = m g d. Lift 10 kg, 2 meters

W = 10kg x 9.8 m/s/s x 2 m

W = 196 J; PE stored = same, 196 J

Work and Potential Energy

Law of Conservation of Energy:

Energy cannot be created or destroyed, only converted from one form to another.

The total energy in an isolated system is constant.

E_o = E_f (original energy = final energy)

In terms of PE and KE:

PE + KE = constant

Conservation of Energy Problem

A man does a work against gravity of 98 Joules lifting an object.

What is its KE before it strikes if it falls back down?

Energy Conversion and Conservation

Many common devices convert energy from one form to another.

For example your radio converts chemical energy from a battery to electrical to sound energy.

Although the law of conservation of energy holds, some of the energy ends up in forms we do not need or want, for example as heat.

Energy Conservation in Society

When we talk about energy conservation in society we mean

Don’t waste energy

Renewable energy is energy that we can recover soon, like wood, or solar energy

Fossil fuels like coal and oil take millions of years to replenish

Elastic collision

Collision in which the total kinetic energy of the colliding objects is the same before and after the collision.

I.E. KE is conserved.

½ mv² before = ½ mv² after

Inelastic collision

Collision in which the total kinetic energy of the colliding bodies after the collision is not equal to the total kinetic energy before.

Torque (levers)

Torque is the rotational analog of force.

Torque = force times lever arm

Example teeter totter

Inclined Plane

Inclined plane increases distance moved but decreases force needed.

Power

Power (P): The rate of doing work.

The rate at which energy is transferred or transformed.

Work done divided by the time. Energy transferred divided by the time.

Unit of power is the watt (joule/s)

Power Problem

20 Joules of work is done in 5 s. What is the power?

P = W/t = 20J/5s = 4 watts

Angular momentum

The mass of an object times its speed and times the radius of its path.

mvr

Law of conservation of angular momentum

The total angular momentum of an isolated system is constant. mvr= constant

Ice Skater Spinning

When the skater brings their arms in, since angular momentum is constant and r is decreasing, v must increase.

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	What does it mean for a physical quantity to be conserved?
	It means:
		The amount of that thing does not change.
	E.G. Conservation of mass, means the amount of mass in a system stays constant.
	Take the example of a burning cigarette.
	Cigarette before burning weighs 10 g.
	After burning the butt plus the ash weighs 5 g.
	What happened? Is mass conserved?


	Note for all the conservation laws we must include all the parts of the system!
	For example, not considering the air in the cigarette example led to an error.
	This is the secret to good scientific analysis. What is the total environment or system?


	The mass of an object times its velocity.
	p = mv Note since v is a vector so is p!
	Example, m = 5 kg, v = 10 m/s what is p?
	p = 5 x 10 kg m/s = 50 kg m/s
	Note units are kg m/s or mass x vel. Units.


	F= D(mv)/Dt is another way of stating Newton’s second law, in terms of
	D(mv), change in momentum and
	Dt time for the change to occur.
	We can also state this as
	F Dt = D(mv) Where F Dt is the impulse of a force.


	Suppose a ball of .1 kg is in contact with a bat for .2 s, and the force the bat exerts is 5 n. What would the speed of the ball be?
	F Dt = D(mv) since original velocity of the ball is 0, D(mv) = m x final vel.(vf)
	5n x .2s = .1kg x v final
	Vf = 1 ns/.1kg = 10 m/s
	(n=kg m/s/s nxs = kg m/s n/kg = m/s²)


	The total linear momentum of an isolated system is constant.
	Remember this is a vector relationship
	If all velocities are in the same direction it is simple, if not we must use vectors to solve
	total mv before = total mv after


	A 200lb boy moving at 10 ft/sec hits 2 50 kg midgets at rest. Both go off together, but how fast ?
	Momentum before = momentum after
	Mom.before =200 kg x 10m/s = 2000 kgm/sec
	Mass after = 50 +50 + 200 = 300 kg
	After mom = 300 x V2; 2000 = 300 V2
	V2 = 2000/300 = 6.6 m/s


	The force that acts times the distance moved in the direction of the force.
	Work = F x d (Joules =Newtons x meters)
	Work units are Joules, same as energy.
	E.G. Force of 10 N moves 5 m
	W = 10 N x 5 m = 50 Nm or 50 J
	Note Force is a vector but work is a scalar!c


	Note work = force x distance in direction of force.
	If Force is perpendicular to motion or displacement no work is done!
	Examples circular motion, moving parallel to floor


	When we lift a body we do work against gravity, but when a body falls, gravity does work on it. If the distances are equal so are the works.
	Work = F x d
	F = weight = W=mg
	W=Wd = mgd
	Example 10kg falls 5m, what is the work?
	W = 10kg x 9.8m/s/s x 5m =490J


	The measure of a system's capacity to do work.
	That which is transferred when work is done.
	Energy is measured in Joules, which are Newtons x meters.


	Kinetic energy (KE) Energy due to motion. Energy that an object has when it is moving
	KE = ½ mv²
	Example, 20 kg mass moving at 15 m/s
	KE = ½ x 20 kg x (15)²=10 x 225 = 2250 J


	Potential energy (PE) is Energy due to an object's position or configuration.
	A stretched spring has potential energy.
	A raised object has gravitational energy.


	elastic potential energy
		stretched out rubber band or spring.
	internal energy
		unburned fuel, battery


	We can calculate the potential energy of a given situation if we know the force.
	Since work against gravity is weight x height lifted. mgd, PE is the same, mgd.
	W = m g d. Lift 10 kg, 2 meters
	W = 10kg x 9.8 m/s/s x 2 m
	W = 196 J; PE stored = same, 196 J


	Energy cannot be created or destroyed, only converted from one form to another.
	The total energy in an isolated system is constant.
	E_o = E_f (original energy = final energy)
	In terms of PE and KE:
		PE + KE = constant


	A man does a work against gravity of 98 Joules lifting an object.
	What is its KE before it strikes if it falls back down?


	Many common devices convert energy from one form to another.
	For example your radio converts chemical energy from a battery to electrical to sound energy.
	Although the law of conservation of energy holds, some of the energy ends up in forms we do not need or want, for example as heat.


	When we talk about energy conservation in society we mean
	Don’t waste energy
	Renewable energy is energy that we can recover soon, like wood, or solar energy
	Fossil fuels like coal and oil take millions of years to replenish


	Collision in which the total kinetic energy of the colliding objects is the same before and after the collision.
	I.E. KE is conserved.
	½ mv² before = ½ mv² after


	Torque is the rotational analog of force.
	Torque = force times lever arm
	Example teeter totter


	Inclined plane increases distance moved but decreases force needed.


	Power (P): The rate of doing work.
	The rate at which energy is transferred or transformed.
	Work done divided by the time. Energy transferred divided by the time.
	Unit of power is the watt (joule/s)


	20 Joules of work is done in 5 s. What is the power?
	P = W/t = 20J/5s = 4 watts


	The mass of an object times its speed and times the radius of its path.
	mvr


	The total angular momentum of an isolated system is constant. mvr= constant


	When the skater brings their arms in, since angular momentum is constant and r is decreasing, v must increase.