• Energy is a property of an object, like age or height or mass.
• Every object that is moving has some Kinetic Energy K.
• An object in a gravitational field also has some Gravitational Potential Energy U_g.
• Energy has units, and can be measured.
• Energy is relative; kinetic energy of car is different for an observer in the car than it is for an observer standing on the side of the road.

What is “energy”?

Kinetic Energy: The energy of motion

Potential Energy: The energy of position

The most basic form of energy: Work

• involves force and distance.
• is force × distance.
• in equation form: W = F d cos θ
• Here θ is the angle between the force and displacement

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Two things occur whenever work is done:

• application of force
• movement of something by that force

SI Unit of work:

newton-meter (N·m)

or joule (J)

The Vector Dot Product

Fun Fact: Work is the dot-product (a.k.a. scalar product) of the Force vector and the Displacement vector.

Work can be positive, zero or negative

• Your hand (H) pulls a briefcase (B) to the right and it moves to the right.
• When the force and the distance are in the same direction, you are helping the motion with the force, so the work done on the object is positive.
• The force is adding energy to the object + environment.

F _{H on B}

d

• Maybe this force is speeding the object up.

Work can be positive, zero or negative

• Your hand (H) supports a briefcase (B) with an upward force, as the briefcase moves to the right.
• When the force and the distance are at right angles, you are not helping the motion with the force, so the work is zero.
• This force is not changing the energy of the object.

• This force won’t speed the object up or slow it down.

d

F _{H on B}

Work can be positive, zero or negative

• Your hand (H) pulls a briefcase (B) to the left, while, for some reason, the briefcase moves to the right.
• When the force and distance are in opposite directions, you are hindering the motion with the force, so the work done on the object is negative.
• This force is reducing the energy of the object.

• Maybe this force is slowing the object down.

d

F _{H on B}

Gravitational Potential Energy, U_g

(U_g) change

The U_g

m_bgh.

U_g change on

Consider moving a book of mass m_b.

Gravitational Potential Energy

• Gravitational potential energy stores the work done against gravity:

​

​

• Gravitational potential energy increases linearly with height h.
• This reflects the constant gravitational force near Earth’s surface.

Dominoes

• A domino is a rectangular solid which can be balanced on its edge
• When standing upright, its gravitational potential energy is a maximum

• This is a state of unstable equilibrium: a small perturbation can cause the domino to fall, transforming its gravitational potential energy into kinetic energy
• As it is falling, it can perturb its neighbor, which then releases its potential energy: a chain reaction can ensue!

U of T Physics has its own meme!

NOTE: The Zero of Potential Energy

• You can place the origin of your coordinate system, and thus the “zero of potential energy,” wherever you choose and be assured of getting the correct answer to a problem.
• The reason is that only ΔPE_g has physical significance, not PE_g itself.

Another way of looking at freefall:

Conservation of Energy

• The total energy in an isolated system is constant over time:

E_i = E_f

From the Preclass Survey Today

• why is thermal energy the type of energy usually referred to as the non conservative force? (I've seen energy lost due to friction always being referred to as thermal energy) I know friction generates heat but it also generates sound as well as slightly damaging the surfaces in question, would the energy to make sound and the energy to cause scratches on a surface also count as energy lost? I might be a bit disappointed if the answer is just ""those losses are negligible"".
• Harlow Answer: You are right, deformation of a surface (scratching, bending of metal, etc) and sound are also forms of energy that the kinetic energy gets converted to during a collision, for example. I wouldn’t say the losses are negligible, but they are complicated, and we often lump them together with thermal energy, as it all is “lost” energy that we can’t get back.

From the Preclass Survey today

Ch.7 Example. I hold a ball at a distance of 5.0 m above the ground and release it from rest. How fast is it going just before it hits the ground?

Hooke’s Law

• If you stretch a rubber band, bend a ruler or other solid object, a force appears that tries to pull or push the object back to its equilibrium, or unstretched, state.
• A force that restores a system to an equilibrium position is called a restoring force.

• k is called the “spring force constant”, which you should measure for each spring. The units of k are N/m.
• This equation only gives the magnitude of the spring force: remember it is always toward equilibrium (restoring force)
• Note that Hooke’s Law is approximate, and only works if |x| is small.

Elastic Potential Energy

• What is the work done when a Finger stretches a Spring, originally at equilibrium, out to a distance x?

Elastic Potential Energy

• Consider a before-and-after situation in which a spring launches a ball
• The compressed spring has “stored energy,” which is then transferred to the kinetic energy of the ball

• We define the elastic potential energy PE_s of a spring to be:

Ch.7 Example. A moving car has 40,000 J of kinetic energy while moving at a speed of 7.0 m/s. A spring-loaded automobile bumper compresses 0.30 m when the car hits a wall and stops. What can you learn about the bumper’s spring using this information?

• You can set the total energy E to any height you wish simply by stretching the spring to the proper length at the beginning of the motion.

• Shown is the energy diagram of a mass on a horizontal spring.
• The potential energy is the parabola:

PE_s = ½k(x – x_e)²

• The PE_s curve is determined by the spring constant; you can’t change it.

E_total line

U_s curve is

Potential-Energy Curve for an H₂ Molecule

• The potential-energy curve for a pair of hydrogen atoms shows potential energy of the covalent bond as a function of atomic separation.

Example

• The energy conservation equation includes potential energies from the conservative forces of the spring and gravity:

A spring-loaded toy gun is used to shoot a ball of mass m straight up in the air. The spring has spring constant k. The ball has speed v_B at point B.

• Here W is the work done on the ball by non-conservative forces, like my hand or sliding friction. In our case there are no significant non-conservative forces acting on the ball so W = 0.

Conservation of Energy

A spring-loaded toy gun is used to shoot a ball of mass m straight up in the air.

The spring has spring constant k.

The ball has speed v_B at point B.

Consider time A to time B.

Conservation of Energy

A spring-loaded toy gun is used to shoot a ball of mass m straight up in the air.

The spring has spring constant k.

The ball has speed v_B at point B.

Consider time B to time C.

Conservation of Energy

A spring-loaded toy gun is used to shoot a ball of mass m straight up in the air.

The spring has spring constant k.

The ball has speed v_B at point B.

Or, if you want, you can even skip B and consider time A to time C!

Friction and Energy Conversion

Can a friction force do work?

Friction and Energy Conversion

The effect of friction as a change in internal energy

• If we choose the box or car as our system object, our model does not account for the change in internal energy.
• If we pick the surface and the box or car as our system, then the friction force between the surface and the box or car is internal and does no work.
• This choice of systems allows us to construct an expression for the change in internal energy of a system:

Friction and Energy Conversion

Positive work done by rope on box

The surfaces are warmer and scratched, gaining positive internal energy

W is the work done in between the initial and final states, and we choose to place it on the left-side of the equals sign.

Wait: what is W again?

Generalized work-energy principle:

• The sum of the initial energies of a system plus the work done on the system by external forces equals the sum of the final energies of the system:

E_i + W = E_f

• This is similar to E_i = E_f, except now you can have Work, W: positive or negative energy added by outside nonconservative forces.

7.8 Power

• Measure of how fast work is done
• In equation form:

​

​

​

Unit of power

• joule per second, called the watt after James Watt, developer of the steam engine
• 1 joule/second = 1 watt
• 1 kilowatt = 1000 watts

Electric Power

• 1 kWh is the amount of energy used by a power of 1kW over 1 hour
• 1 kWh = 1000 J/s * 60 min/hour * 60 s/min
• 1 kWh = 3.6 million Joules

• The unit of power is the watt, which is defined as 1 watt = 1 W = 1 J/s
• Energy is measured by Hydro One (the company that sells electricity in this province) in kWh “kiloWatt hours”.

From today’s Preclass Quiz

Energy Sources in Ontario 2024

https://www.ieso.ca//en/Learn/Ontario-Electricity-Grid/Supply-Mix-and-Generation

My actual garage:

• Jason Harlow has 12 solar panels on the roof of his garage.
• Over the 12 months of 2023, they produced 4.76 MWh of energy (months 1=January up to 12=December are shown in the histogram).
• If the cost of electricity is $0.151 per kWh, how much money did these solar panels save Jason in 2023?

1 MWh = 1000 kWh

Electric Power

• 1 kWh is the amount of energy used by a power of 1kW over 1 hour
• 1 kWh = 1000 J/s * 60 min/hour * 60 s/min
• 1 kWh = 3.6 million Joules

• The unit of power is the watt, which is defined as 1 watt = 1 W = 1 J/s
• Energy is measured by Hydro One (the company that sells electricity in this province) in kWh “kiloWatt hours”.

• The Price of electricity purchased from Hydro One is $0.151 per kWh.

Your clothes dryer uses 5000 Watts and you need to run it for 1 hour to dry your clothes. How much does this cost?

Units of Energy

• This textbook sticks pretty close to “Joules” for energy units, which is good.
• There are two conflicting definitions of the unit “calorie”.
• The one nobody uses: The small calorie, sometimes used in physics and chemistry, is 1 calorie = 4.184 Joules. A muffin has about 400,000 small calories!
• The one everyone uses: The large calorie, most often used in nutrition and food science, is 1 calorie = 4184 Joules, or one-thousand small calories. Sometimes this is called a kilocalorie, or kcal, and sometimes it is simply called a cal, or Cal. A muffin has about 400 large callories. (your textbook calls this 400 kcal, but it’s the same thing..)

Example.

Running cross-country causes you to consume 740 Watts.

How much time would you have to spend running in order to burn off the energy in one 400 kcal muffin?

The Greenhouse Effect

• Example:
• The Sun’s power output is P = 3.8 × 10²⁶ Watts.
• The sun is d = 1.5 × 10¹¹ m away from us.
• How much energy hits the Earth every second due to the Sun?

​

• The power from the Sun, going out in all directions, is spread over a sphere of radius d by the time it gets to Earth.
• The area of this huge sphere is 4πd ².
• The Earth has a radius of R = 6.4 × 10⁶ m, and a cross-sectional area of πR².

Physics of the Greenhouse Effect

• Example:
• The Sun’s power output is P = 3.8 × 10²⁶ Watts.
• The sun is d = 1.5 × 10¹¹ m away from us.
• How much energy hits the Earth every second due to the Sun?

​

• The Earth has a radius of R = 6.4 × 10⁶ m, and a cross-sectional area of πR².
• The fraction of the Sun’s total power that hits the Earth is the ratio of these two areas: P_Earth = P × A_Earth / A_spread = P × πR ² / 4πd ² = 3.8 × 10²⁶ W × (6.4 × 10⁶ m)² / (4 × (1.5 × 10¹¹ m)²)
• So P_Earth = 1.4 × 10¹⁷ Watts. That’s how many Joules per second impact the Earth, mostly in the form of visible sunlight.

Physics of the Greenhouse Effect

• P_Earth = 1.4 × 10¹⁷ Watts. That’s how many Joules per second impact the Earth, mostly in the form of visible sunlight.
• The average albedo, or reflectivity, of Earth is 0.3. So about 70% of the sunlight that hits the Earth is absorbed.
• That means that the Earth gains about 10¹⁷ Watts energy due to sunlight.
• If the thermal energy of the surface is to remain constant, all of this energy must be re-radiated out into space in the form of infrared radiation.
• The power output of the Earth in infrared radiation depends on temperature as well as the opacity of the atmosphere to infrared radiation.
• The higher the concentration of infrared-absorbing molecules in the air, the higher the temperature must be to radiate the necessary 10¹⁷ Watts.

Physics of the Greenhouse Effect

Physics of Climate Change

• Water vapor is the primary greenhouse gas, contributing to 50% of the greenhouse effect, but its global concentrations are not directly affected by human activity.
• The main drivers of climate change is Carbon Dioxide (CO₂).

• During the 10,000 years prior to the mid-18th century, CO₂ concentration was 290 ppm.
• The current global average concentration of CO₂ in the atmosphere is 426 ppm (June 2024).

Humans are responsible for global warming

• Climate scientists have showed that humans are responsible for virtually all global heating over the last 200 years.
• Human activities are causing greenhouse gases that are warming the world faster than at any time in at least the last two thousand years.
• The average temperature of the Earth’s surface is now about 1.2°C warmer than it was in the late 1800s (before the industrial revolution) and warmer than at any time in the last 100,000 years.
• The last decade (2014-2023) was the warmest on record, and each of the last four decades has been warmer than any previous decade since 1850.
• The consequences of climate change now include: intense droughts, water scarcity, severe fires, rising sea levels, flooding, melting polar ice, catastrophic storms and declining biodiversity.

We face a huge challenge but already know many solutions

https://www.un.org/en/climatechange/what-is-climate-change

• Many climate change solutions can deliver economic benefits while improving our lives and protecting the environment.
• Three broad categories of action are: cutting emissions, adapting to climate impacts and financing required adjustments.
• Switching energy systems from fossil fuels to renewables like solar or wind will reduce the emissions driving climate change.
• Emissions must be cut in half by 2030 to keep warming below 1.5°C.
• Achieving this means huge declines in the use of coal, oil and gas: over two-thirds of today’s proven reserves of fossil fuels need to be kept in the ground by 2050 in order to prevent catastrophic levels of climate change.