BackChapter 7: Energy – Physics with Calculus Study Notes
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Energy
Definition and Nature of Energy
Energy is a fundamental property of a system that enables it to do work. It is both a thing (a measurable quantity) and a process (something that is transferred or transformed). Energy is always conserved in physical processes, meaning it cannot be created or destroyed, only changed from one form to another.
Energy is observed when it is transferred or transformed.
It is a conserved quantity in the universe.
Examples include electromagnetic waves from the Sun and the energy stored in matter.
Matter is the substance that occupies space and can be seen, smelled, or felt, while energy is what moves or changes matter.
Work
Definition and Calculation of Work
Work is done when a force acts on an object and causes it to move in the direction of the force. The amount of work done depends on both the magnitude of the force and the distance over which it acts.
Work involves both force and distance.
Mathematically, work is given by:
Where W is work, F is the force applied, and d is the distance moved in the direction of the force.
Work is only done if the object moves in the direction of the applied force.
Example: Pushing against a stationary wall does no work on the wall because there is no movement.
Examples of Work
Lifting two loads one story high requires twice as much work as lifting one load the same distance, because the force is doubled.
Lifting a load two stories high requires twice as much work as lifting it one story, because the distance is doubled.
Units of Work
The SI unit of work is the joule (J), which is equivalent to a newton-meter (Nm).
Example: A weightlifter raising a barbell from the floor does work on the barbell.
Power
Definition and Calculation of Power
Power is the rate at which work is done or energy is transferred. It measures how quickly work can be accomplished.
Mathematically, power is given by:
Where P is power, W is work, and t is time.
The SI unit of power is the watt (W), where 1 watt = 1 joule/second.
1 kilowatt (kW) = 1000 watts (W).
Example: A worker running up stairs uses more power than climbing slowly, even if the total work is the same.
Mechanical Energy
Forms of Mechanical Energy
Mechanical energy is the energy due to the position or motion of an object, or both. It is divided into two main forms:
Potential Energy (PE): Energy stored due to position or configuration.
Kinetic Energy (KE): Energy of motion.
Potential Energy
Potential energy is stored energy that has the potential to do work due to an object's position or state.
Gravitational Potential Energy: Energy stored due to an object's elevation above the ground.
Calculated as:
Where m is mass, g is acceleration due to gravity, and h is height above a reference point.
Examples: Water in an elevated reservoir, a stretched bow, or a raised ram of a pile driver.
Example: A car hoisted for repairs has increased potential energy relative to the floor.
Example: The potential energy of a 10-N ball is the same in all three cases shown, because the work done in elevating it is the same.
Kinetic Energy
Kinetic energy is the energy an object possesses due to its motion. It depends on both the mass of the object and the square of its speed.
Calculated as:
Where m is mass and v is velocity.
If the speed of an object is doubled, its kinetic energy increases by a factor of four.
Work-Energy Theorem
Relationship Between Work and Energy
The work-energy theorem states that the net work done on an object is equal to its change in kinetic energy.
Doubling the speed of an object requires four times the work.
This theorem also applies to decreasing speed, such as when brakes are applied to a moving car.
Example: The work done by brakes (friction force x stopping distance) brings a moving car to a stop. If the initial speed is doubled, the stopping distance is four times greater.
Conservation of Energy
Law of Conservation of Energy
The law of conservation of energy states that energy cannot be created or destroyed; it can only be transformed from one form to another. The total amount of energy in a closed system remains constant.
Example: In a pile driver, potential energy is converted to kinetic energy as the weight falls, and then to other forms (such as heat and sound) upon impact.
Example: Drawing a bow stores potential energy; releasing the bowstring transfers most of this energy to the arrow as kinetic energy, with some lost as heat.
Kinetic Energy and Momentum Compared
Similarities and Differences
Both momentum and kinetic energy are properties of moving objects.
Momentum is a vector quantity (depends on direction), while kinetic energy is a scalar (depends only on magnitude).
Momentum can be canceled by equal and opposite momentum; kinetic energy cannot be canceled.
Momentum depends linearly on velocity (), while kinetic energy depends on the square of velocity ().
Example: An object moving with twice the velocity of another (with the same mass) has twice the momentum but four times the kinetic energy.
