Work
- Work is the energy transferred to or from an object by means of a force acting on said object. In other words, it is the form of energy that is sent from one place to another, giving us a link between force and energy. Work is done on an object when the object is displaced by a force.
- Work (\( W \)) is measured in Joules. \( 1 \) Joule \( = 1 J = 1 N\cdot m = 1 \frac{kg \cdot m^{2}}{s^{2}} \)
- To calculate the work done on an object by a force during a displacement, we use only the value of the force with the object's displacement. Work can be defined as the following:
- \( W = F \cdot \Delta x \)
- \( W = F \cdot cos \theta \cdot \Delta x \)
- Where:
- \( F \) is the magnitude of the force.
- \( \Delta x \) is the magnitude of the object's displacement (could also be \( \Delta d \)).
- \( \theta \) is the angle between the force and the displacement.
- This formula gives us no information about the time it took for the displacement to occur, or the velocity or acceleration of the object. This makes sense, as work is a scalar quantity.
- A force does positive work when it is in the same direction as the displacement. This is also called a motive work.
- \( 0^{\circ} \leq \theta < 90^{\circ} \rightarrow cos\theta \leq 1 \rightarrow +W \)
- A force does zero work when it is perpendicular to the displacement (for example, horizontal displacement, vertical force).
- \( \theta = 90^{\circ} \rightarrow cos\theta = 0 \rightarrow W = 0 \)
- A force does negative work when it is in the opposite direction of the displacement. This is also called a resistive work.
- \( 90^{\circ} < \theta \leq 180^{\circ} \rightarrow cos\theta \leq -1 \rightarrow -W \)
- If there are several forces acting on the same object simultaneously, the net work done is the sum of the amount of work done by each individual force.
- \( W_{net} = W_{1} + W_{2} + W_{3} +... \)
OTHER NOTES ABOUT WORK
- The work done by solely the force of gravity can be defined as the following:
- \( W = mgh \cdot cos 0 = mgh \)
- Work done by gravity is separate from the horizontal motion of the object. Whatever the path taken by the object looks like, the displacement is always the direct distance between the start point and the end point.
- Work can be done by friction! In fact, the energy an object loses due to friction can be converted into heat, sound, or light.
- The area under any force-displacement graph is the work done.
Energy
- Energy is the ability of an object or physical system to create a change in itself or in the environment around it. In simpler terms, energy is the ability to do work!
- Energy can take shape in many different forms, including heat, kinetic or mechanical energy, light, potential energy, electrical, solar wind, and hydroelectric energy. It can be converted from one form to another, and is vital to the study of physics, chemistry, biology, geology, astronomy, and more.
- Energy is very versatile; it can be interchanged with work, and can even be used in place of Newton's Laws (allowing certain problems to be solved more simply).
- Energy (\(E \)) is measured in Joules. \( 1 \) Joule \( = 1 J = 1 N \cdot m = 1 \frac{kg \cdot m^{2}}{s^{2}} \)
- Energy can be defined in several different ways, as it has so many forms. Specific formulas can be found in each type of energy's glossary item.
- Energy can transferred using a variety of methods. We can utilize:
- Work - applying a force, causing a displacement of the system.
- Heat - the collision of molecules causes heat energy to be passed on to other systems.
- A spoon in a cup of tea heats up because some of the \(KE\) of the molecules in the tea is transferred to the molecules of the spoon.
- Mechanical waves - a disturbance transmitting through a medium.
- Examples of waves include sound, water, and seismic waves
- Electrical transmission - electrical currents transfer energy
- This is how energy enters any electrical device
- Electromagnetic radiation - this can be any form of electromagnetic waves, including light, micro, and radio waves.
- And more...
kinetic energy
- Kinetic energy is the form of energy possessed by an object due to its motion.
- Kinetic energy (\( E_{K} \)) is measured in Joules. \( 1 \) Joule \( = 1 J = 1 N \cdot m = 1 \frac{kg \cdot m^{2}}{s^{2}} \)
- Kinetic energy can be defined as \( E_{K} = \frac{1}{2} mv^{2} \), where
- \( m \) is the mass of the object,
- and \( v \) is the speed of the object.
- Kinetic energy is a scalar quantity and has the same units as work (Joules).
- Kinetic energy can also be seen as the amount of work the moving object could do in coming to rest.
- For example, a moving hammer has a particular amount of kinetic energy and can do work on a nail.
mechanical energy
- When work is done on a object, said object gains energy; this energy is known as mechanical energy.
- Mechanical energy (\( E_{M} \)) can be measured in Joules. \( 1 \) Joule \( = 1 J = 1 N \cdot m = 1 \frac{kg \cdot m^{2}}{s^{2}} \)
- Mechanical energy can be
- kinetic energy (\( E_{K} \)),
- some form of potential energy (\( E_{P} \)),
- or the sum of the two.
- \( E_{M} = E_{K} + E_{P} \)
gravitational potential energy
- Potential energy is the form of energy an object possesses due to its position with regards to a specific reference point. In the case of gravitational potential energy, this reference point is usually the surface of the Earth.
- Gravitational potential energy exists because of the gravitational pull the Earth has on an object, and it refers to the system made up of the Earth and the object (rather than solely the object).
- Gravitational potential energy (\( E_{g} \)) is measured in Joules. \( 1 \) Joule \( = 1 J = 1 N \cdot m = 1 \frac{kg \cdot m^{2}}{s^{2}} \)
- Gravitational potential energy can be defined as the following:
- \( E_{g} = F_{g} \cdot h \)
- \( \Rightarrow E_{g} = mg \cdot h \)
- Where:
- \( F_{g} \) or \( mg \) is the gravitational force on the object (its weight),
- and \( h \) is the distance from the surface of the Earth (height).
- When working with problems dealing with gravitational potential energy, a point where the gravitational potential energy is zero needs to be chosen, no matter what.
- The location of the chosen point doesn't necessarily matter, as the value important to us is the change in potential energy.
- Once the point is chosen, it cannot change for the rest of the problem.
elastic potential energy
- Elastic potential energy is the form of energy an object possesses due to its ability to change shape (compressing or stretching) and return to its original shape. This potential energy is stored in an elastic object because of the change in shape, such as the stretching of a spring.
- Elastic potential energy (\( E_{e} \)) is measured in Joules. \( 1 \) Joule \( = 1 J = 1 N \cdot m = 1 \frac{kg \cdot m^{2}}{s^{2}} \)
- Combining Hooke's law formula with the formula for work, elastic potential energy can be defined as
\( E_{e} = \frac{1}{2} kx^{2} \), where- \( x \) is the distance the spring stretched from said spring's equilibrium point,
- and \( k \) is the spring constant (which depends on how the spring was formed, the material it is made from, thickness of the wire, etc.).
Conservative forces
Non-conservative forces
- Non-conservative forces are the opposite of conservative forces; with non-conservative forces, the path an object takes does matter! The work done in moving an object between two positions is linked with the path taken to get there.
- Non-conservative forces are typically only involved with systems that deal with thermal energy, and where the mechanical energy is not conserved.
impulse
- Impulse is a force multiplied by the amount of time it acts over.
- It can also be defined as the change in momentum!
- Impulse (\( J \), not to be confused with Joules) is measured in Newton seconds. \( 1 \) Newton second \( = 1 N \cdot s = 1 \frac{kg \cdot m}{s} \).
- Impulse can be defined as \( J = F \cdot t \), where
- \( F \) is the net force acting on an object,
- and \( t \) is the amount of time the force is acting on said object.
- Remember Newton's Second Law: \( F = ma \), where
- \( F \) is the net force acting on an object,
- \( m \) is the mass of said object,
- and \( a \) is the acceleration of the object.
- Consider the following transformations:
- \( F = ma \)
- \( a = \frac{\Delta v}{t} \)
- \( \Rightarrow F = m \cdot \frac{\Delta v}{t} \)
- \( \Rightarrow F \cdot t = m \cdot \Delta v \)
- We know impulse is equal to \( F \cdot t \).
- Momentum is known to be \( m \cdot v \), so we can conclude that change in momentum is equal to \( m \cdot \Delta v \)
- Therefore, impulse = change in momentum!
momentum
- Momentum refers to the amount of motion an object possesses. In other words, momentum talks about mass in motion.
- Momentum (\( p \)) is measured in kilogram meters per second. \( 1 \) kilogram meter per second \( = 1 \frac{kg \cdot m}{s} \).
- Momentum can be defined as \( p = mv \), where
- \( m \) is the mass of an object,
- and \( v \) is the velocity of said object.
- Momentum is a vector quantity.
reference point
power
- Power is the rate of transferring energy (doing work).
- Power (\( P \)) is measured in Joules/second, or Watts. \( 1 \) Watt \( = 1 \frac{J}{s} = 1 \frac{N \cdot m}{s} = 1 \frac{kg \cdot m^{2}}{s^{3}} \)
- Both Watts and work use the capital 'W' as their notation, so be very careful!
- Power can be defined as \( P = \frac{W}{t} \), where
- \( W \) is the work done,
- and \( t \) is time.
- In the United States, things are slightly different. They use the term horsepower (hp) to represent power.
- Luckily, there's a conversion factor! \( 1 hp = 745.7 W \)
Watt
Efficiency
Isolated System
- A system is an assortment of 2+ objects, which can be affected by any external forces.
- An isolated system is a system that is not affected by any 'net external forces'; no force from outside the system causes any changes in energy inside the system.
- A "net external force" can be a few things. It could be defined as
- a force that stems from something from outside the system,
- or a force that is not equalized by other forces.
collision
elastic collision
- A perfectly elastic collision is a collision where both the kinetic energy and momentum are conserved (following the laws of conservation of energy and conservation of momentum). No energy whatsoever is lost in the collision, and none of it is transformed.
- However, for most non-microscopic (a.k.a. macroscopic) objects involved in a collision, there is always some energy lost (transformed into other types of energy), stopping the collision from being perfectly elastic.
- Collisions between hard spheres (such as steel or billiard balls) can be almost perfectly elastic, so we usually assume the various conservation laws remain true and pretend it's a perfectly elastic collision.
- Collisions of atoms in ideal conditions (gasses) also get very close to perfectly elastic collisions, as well as sub-atomic particles 'colliding' with an electromagnetic force.
- A handful of massive interactions such as the gravitational interactions between satellites and planets are perfectly elastic. Note, these aren't true 'collisions.'
inelastic collision
Work-energy theorem
- The work-energy theorem states that the work done on an object equals the change in mechanical energy of said object.
- When work is done on an object and the only change in energy is kinetic, the work done is equivalent to the change in said object's kinetic energy. This means the speed will increase if the work is positive, and decrease if the work is negative.
- \( W = \Delta E_{K} \rightarrow W = E_{Kf} - E_{Ki} \)
- The same principle can be extended to include potential energy:
- \( W = ( E_{Kf} - E_{Ki} ) + ( E_{Pf} - E_{Pi} ) \)
- Where:
- \( E_{K} \) is the total kinetic energy.
- \( E_{P} \) is the total potential energy.
- With these two equations combined, we see that work is indeed equal to the change in mechanical energy of our object.
- \( \Delta E_{M} = \Delta E_{K} + \Delta E_{P} \)
- \( \Rightarrow W = \Delta E_{M} \)
WORK-ENERGY THEOREM REGARDING A SPRING
- \( W = ( E_{Kf} - E_{Ki} ) + ( E_{gf} - E_{gi} ) + ( E_{ef} - E_{ei} ) \), where
- \( E_{K} \) is the kinetic energy,
- \( E_{g} \) is the gravitational potential energy,
- and \( E_{e} \) is the elastic potential energy.
Conservation of energy
- The law of conservation of energy states that energy cannot be created or destroyed (the energy is conserved). Energy can be transformed from one form to another, but the total energy in an isolated, perfectly insulated, and closed system remains constant.
- If we say that a number of physical objects are conserved, we mean that the quantity of these objects stays the same, no matter what's done to it.
- In an isolated system of objects only dealing with conservative forces, the total mechanical energy of this system is conserved.
- We can represent this law using a variety of forms of energy. In this example, we'll start with mechanical energy and break it down.
- \( E_{Mi} = E_{Mf} \)
- \( \Rightarrow E_{Ki} + E_{gi} = E_{Kf} + E_{gf} \)
- If we want to add another form of energy, such as elastic potential energy, we simply add it to either side of the broken down equation.
- \( E_{Mi} = E_{Mf} \)
- \( \Rightarrow E_{Ki} + E_{gi} + E_{ei} = E_{Kf} + E_{gf} + E_{ef} \)
- Where:
- \( E_{M} \) is mechanical energy,
- \( E_{K} \) is kinetic energy,
- \( E_{g} \) is gravitational potential energy,
- and \( E_{e} \) is elastic potential energy.
hooke's law
- Hooke's law states that the force required to stretch or compress the shape of the spring is proportional to the distance the spring is warped.
- Hooke's law is defined as \( F = k \cdot x \), where
- \( F \) is the force applied to the spring,
- \( x \) is the displacement from the object's equilibrium point,
- and \( k \) is the spring constant (which depends on how the spring was formed, the material it is made from, thickness of the wire, etc.).