Physics I Notes: Ch 5 Work and Energy

“Work” has a variety of meanings in every day language…BUT in physics, its meaning is VERY specific .

I. Work – Work is energy transferred by a force acting through a distance. Work is a scalar quantity so it has no direction associated with it. • Work is done only if a force acts over some distance, so Work = Force x distance (W=Fd) BUT the component of the force used must be parallel to the displacement of the object. The units for work are N • m = J (joule). One joule is about the amount of work you do in lifting your calculator to a height of one meter.

Questions: 1. You lift a barbell and do some work…How much work is done if you lift a barbell that is twice as heavy the same distance? 2. How much if you lift that twice as heavy barbell twice as high? 3. How much if you hold the barbell over your head for 10.0 minutes? 4. How much work is done when you carry a 50.0 N stack of books in your arms across the 12.0 m long room?

Work can be done against another force. Ex: against gravity or friction and, Work can be done to change the motion of an object. Ex: stopping a car

Work can be done ON an object…Work can be done BY an object…BUT objects CANNOT Possess Work! Objects can – and definitely DO possess or contain Energy!

When work is done on or by an object, it changes the energy that object possesses…energy is the ability to do work! Calculating Work (W): W = Fd cos θ

Fsinθ F

θ Fcosθ

d

• θ is the angle between the direction of the force and the objects displacement; when F is parallel to the displacement, θ = 0, and cos 0 = 1, and W = Fd • Since W = Fd; if an object at rest has zero displacement, therefore W = 0 • When the Force is parallel to the motion , maximum work will be done. If the Force is exerted at some angle between θ = 0 o and θ = 90 o to the motion, then less work is done depending on how large the parallel component of the Force is. When F is perpendicular to motion, W = 0 • The sign of work is important. Work can either be positive or negative, depending on whether the parallel component of force is in the same direction (+) or opposite direction () of the displacement • Kinetic friction does negative work. It is negative because the force involved in producing the work is OPPOSITE to the direction of motion.

• Example 1: How much work is done on a car being pushed 1550.0 m by a force of 3750.0 N? How about if the force is applied by a tow truck cable at an angle of 42 o from the horizontal?

Page 1 of 6 • Example 2: How much work is done by the brakes in stopping a 12500 kg railroad car going 20 mph (9.0 m/s) if it travels 150 m and takes 45 seconds to stop?

II. Mechanical energy and the WorkEnergy Theorem • Energy is the ability to do work; and, when work is done, there is always a transfer of energy involved. The energy transferred can be calculated by multiplying the displacement times the component of force parallel to the displacement just like work is calculated; both energy and work are scalar quantities; units are N ••• m = J (joule) This is because the amount of work done on a system is exactly equal to the change in energy of the system. This statement is called the work-energy theorem .

So When you push a crate along the floor with a force of 1N for 1 meter, you do 1 Joule of work . One Joule is also the approximate energy necessary to lift a quarterpound cheeseburger from the table to over your head. It is a very small amount of work or energy…so we often use kilojoules (kJ) or megajoules (MJ).

There are two forms of mechanical energy kinetic and potential

A. Kinetic energy – energy of an object due to its motion • A moving object can do work on another object it strikes; therefore, an object moving relative to another object has energy innate “in itself”; this energy is called kinetic energy—energy of motion • Kinetic energy (in Joules) depends on speed (in m/s) and mass (in kg) • KE= ½ mv 2

1 2 1 2 • Wnet = KE or W = mv − mv 2 f 2 i • Some or all of the work done on a system can be transformed into heat energy and we usually say that the energy that becomes heat is

Questions: 1. What happens to a car’s kinetic energy if the speed is tripled? 2. How much more WORK would it take to stop it? 3. If the maximum braking force was used in both cases, what does this tripling of the speed do to the stopping distance required?

• Example 3: How much work is required to accelerate a 3.00 g bullet from rest to a speed of 40.0 m/s?

B. Potential energy – “stored energy”; energy of position or condition • Potential energy is present in an object that has the potential to move because of its position or condition relative to some other location. • Three types of potential energy we will study in physics are gravitational, elastic, and electric 1. Gravitational potential energy – energy associated with an object due to the object’s position relative to a gravitational source (ie, due to its height above the ground or a base level)

• PE g = mgh • The higher an object is, the more gravitational PE it has. • h is relative since we are concerned with the h; pick a convenient point to call ground zero

Page 2 of 6 • The amount of work required to lift a mass to a certain height against gravity only depends upon the weight of the object and h, not the path taken. The change in gravitational potential energy is the same for all three cases below.

2. Elastic potential energy – energy stored in a deformed elastic object; s tretch or compress a spring (or any elastic object) and it will try to return to its relaxed length; therefore, there is energy stored in any spring that is not at its equilibrium position 2 • Uelastic or PE elastic = ½kx • x is the displacement from the spring’s relaxed length • k is the spring constant which depends on the nature of the spring; flexible springs would have a small value of k compared to a stiff spring which would have a large value for k.

3. Electric potential energy will be covered the second semester

III. Conservation of mechanical energy and the WorkEnergy Theorem • When we say something is conserved , we mean that the total amount of it remains constant. • Mechanical energy is conserved as long as no nonconservative forces (friction, air resistance, etc.) are working on an object. • Conservative and Non-conservative Forces • A force is conservative when the work it does is independent of the path between the objects initial and final positions (gravity, electric, and elastic ). For example, work done against gravity does NOT depend on a path taken, it simply depends on h. A potential energy can be defined for a conservative force, but not for a nonconservative force. • NON-conservative forces (friction for example) do depend on the path taken. W = Fd cos θ and if d increases so does the work. Nonconservative forces include friction, air resistance, tension, motor or rocket propulsion, push or pull by person and can either add (positive work) or remove (negative work) energy from the system. • Mechanical energy is defined as the sum of the kinetic and potential energies of an object. • Conservation of mechanical energy • KE i + PE i = KE f + PE f

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• ***Energy conservation occurs even when acceleration is not constant!!! • The work-energy theorem says that the work done on an object or system is equal to its change in mechanical energy.

Bowling Ball example: Let’s say that I lift a 200N bowling ball up to a shelf that is 1.5m above the ground. Q: How much work was done on the ball?

Q: How much potential energy does it now have?

The work done gives the ball gravitational potential energy due to its position above the ground. Now let’s say that the ball falls from the shelf Q: As it falls what happens to the PE it had? (inc, dec, stay the same?)

Q: What is happening to the amount of kinetic energy it has as it falls? (inc, dec, stay the same?)

Q: How much kinetic energy will it have right before it hits whatever it will hit on the floor?

Q: What happens to all of that kinetic energy if it were to hit your toe? It will do an amount of work equal to the energy it had right before hitting!

Q: How much work will it do on the unfortunate JP’s toe that is standing under the shelf when the ball hits?

Q: If the ball hits with 30,000N of force, how far (ideally) will it compress the JP’s toe?

Q: The “Energy Epiphany” Problem: A pebble is shot from a slingshot at the top of a building at a speed of 12.0 m/s. The building is 30.0 m tall. Ignoring air resistance, find the speed with which the pebble strikes the ground when the pebble is fired in each of these scenarios: a. horizontally

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b. at an angle of 35 degrees above the top of the building

c. vertically straight up

d. vertically straight down

• Example 4. Starting from rest, a 25.0 kg child zooms down a frictionless slide from an initial height of 3.00 m. What is her speed at the bottom of the slide?

• Example 5. If it takes an average force of 80.0 N to pull back a bow string and arrow a distance of 0.50m, a. How much work is done on the bow and arrow?

This work is now stored as the potential energy in the bow and arrow! This energy will be transformed into kinetic energy of the arrow when it is shot!

b. If the arrow’s mass is 0.025 kg, how fast will the arrow go as it leaves the bowstring?

Page 5 of 6 The kinetic energy of the arrow will be transformed again as the arrow flies through the air. If the arrow was shot straight up, and we neglect air resistance, then all of the KE will become PE g by the time the arrow reaches the top of its path.

c. How high will the arrow go?

IV. Power – the rate of doing work; rate of energy transfer or conversion from one form to another Power = Work/time or Power = Force x velocity • Work can be done slowly or quickly, but the time taken to perform the work doesn’t affect the amount of work which is done, since there is no element of time in the definition for work. However, if you do the work quickly, you are operating at a higher power level than if you do the work slowly . • Power is defined as the rate at which work is done. Oftentimes we think of electricity when we think of power, but it can be applied to mechanical work and energy as easily as it is applied to electrical energy. The equation for power is Work P = time and has units of joules/second or watts (W).

A machine is producing one watt of power if it is doing one joule of work every second. A 75watt light bulb uses 75 joules of energy each second.

• Units of power are J/s or W (Watt) and 1 horsepower is about 750 Watts or 0.75 kW • If you double the power that means that you can do the same work in ½ the time or you can do double the work in the same time.

• Example 6. A student weighing 700 N climbs at constant speed to the top of an 8.0 m vertical rope in 10 s. Calculate the average power expended by the student to overcome gravity.

Regular Physics Ch 5 HW P. 193 – 196 #’s 5, 7, 9, 10, 16, 19, 28, 30, 33, 35, 45, 48

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