7.2 Kinetic Energy and the Work-Energy Theorem
4 min read•june 18, 2024
and work are closely linked concepts in physics. The shows how the work done on an object changes its , while energy conservation principles explain how energy transforms between different forms.
Forces play a crucial role in work-energy problems. The on an object determines its acceleration and change in kinetic energy. Understanding these relationships helps us analyze and predict the motion of objects in various scenarios.
Kinetic Energy and Work-Energy Theorem
Work-energy theorem calculations
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- states the net work done on an object equals the change in its kinetic energy
- Equation:
- represents the net work done on the object
- represents the final kinetic energy
- represents the initial kinetic energy
- Equation:
- Work is the product of the force applied to an object and the of the object in the direction of the force
- Equation:
- represents the force applied
- represents the
- represents the angle between the force and the displacement
- Example: Pushing a box 5 m with a force of 20 N at a 30° angle to the horizontal
- Equation:
- Kinetic energy is the energy an object possesses due to its motion
- Equation:
- represents the of the object
- represents the of the object
- Example: A 2 kg ball moving at 3 m/s has a kinetic energy of 9 J
- Equation:
Kinetic energy in energy transfer
- Energy transfer occurs when work is done on an object, changing its kinetic energy
- Positive work increases kinetic energy (acceleration)
- Negative work decreases kinetic energy (deceleration)
- Example: A car accelerating from rest to 60 km/h
- states that in the absence of non-conservative forces, the total energy of a system remains constant
- Equation:
- represents the change in
- Example: A roller coaster at the top of a hill has high potential energy and low kinetic energy, while at the bottom it has low potential energy and high kinetic energy
- Equation:
- is the rate at which work is done or energy is transferred
- Equation:
- represents power
- represents work
- represents the time interval
- Example: A 100 W light bulb converts 100 J of electrical energy into light and heat every second
- Equation:
Forces in work-energy problems
- Net force is the vector sum of all forces acting on an object
- If the net force is zero, the object's velocity remains constant ()
- If the net force is non-zero, the object's velocity changes ()
- Example: A book resting on a table has a net force of zero, while a falling book has a net force equal to its weight
- Acceleration is the rate of change of an object's velocity
- Equation:
- represents acceleration
- represents the net force
- Example: A 1000 kg car accelerating at 2 m/s² experiences a net force of 2000 N
- Equation:
- states the work done by the net force on an object equals the change in its kinetic energy
- Equation:
- Example: A 5 kg box pushed 3 m by a 10 N force at a 60° angle to the horizontal experiences a change in kinetic energy of 15 J
- Energy conservation states that in the absence of non-conservative forces, the total energy (kinetic + potential) remains constant
- Equation:
- Example: A pendulum swinging back and forth, constantly converting kinetic energy to potential energy and vice versa
- is a non- that can cause energy to be dissipated as heat
Energy, Momentum, and Motion
- Kinetic energy and are both related to an object's motion
- Momentum is the product of an object's mass and velocity (p = mv)
- Kinetic energy depends on the square of velocity (KE = 1/2mv²)
- The relationship between mass, velocity, and energy affects various physical phenomena
- A change in an object's velocity results in changes to both its kinetic energy and momentum
- occurs when one form of energy is transformed into another
- Example: In a hydroelectric dam, the potential energy of water is converted to kinetic energy of the turbines, which is then converted to electrical energy
Key Terms to Review (37)
A = F_{net}/m: The equation $$a = \frac{F_{net}}{m}$$ expresses the relationship between acceleration (a), net force (F_{net}), and mass (m) of an object. This foundational principle illustrates that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. Understanding this equation is crucial for grasping how forces influence the motion of objects, especially in the context of kinetic energy and energy transformations.
Atomic mass: Atomic mass is the weighted average mass of an atom's naturally occurring isotopes, measured in atomic mass units (amu). It reflects both the mass and relative abundance of each isotope.
Conservation of Energy: Conservation of energy is a fundamental principle in physics that states the total energy of an isolated system remains constant, it is said to be conserved over time. Energy can neither be created nor destroyed; rather, it can only be transformed or transferred from one form to another.
Conservative force: A conservative force is a force with the property that the work done in moving an object between two points is independent of the path taken. The mechanical energy (kinetic + potential) in a system influenced only by conservative forces remains constant.
Conservative Force: A conservative force is a type of force that does not depend on the path taken by an object, but only on its initial and final positions. The work done by a conservative force depends solely on the object's starting and ending positions, and not on the specific path it takes to get there.
Displacement: Displacement is a vector quantity that refers to the change in position of an object. It has both magnitude and direction, indicating how far and in what direction the object has moved from its initial position.
Displacement: Displacement is the change in position of an object, measured from a reference point or origin. It describes the straight-line distance and direction an object has moved, without regard to the path taken.
Elastic potential energy: Elastic potential energy is the energy stored in an object when it is deformed elastically, such as when a spring is stretched or compressed. It can be calculated using the formula $U = \frac{1}{2} k x^2$, where $k$ is the spring constant and $x$ is the displacement from equilibrium.
Electric power: Electric power is the rate at which electrical energy is transferred by an electric circuit. It is typically measured in watts (W).
Energy Conversion: Energy conversion is the process by which energy is transformed from one form to another, such as the conversion of potential energy to kinetic energy or the conversion of electrical energy to mechanical energy. This term is central to understanding the concepts of kinetic energy and the work-energy theorem in physics.
Friction: Friction is the resistive force that occurs when two surfaces interact, opposing the relative motion between them. It acts parallel to the surfaces in contact and can be either static or kinetic.
Internal kinetic energy: Internal kinetic energy is the sum of the kinetic energies of all particles within a system. It plays a crucial role in understanding how energy is distributed and conserved during elastic collisions.
Joule: A joule is the SI unit of work or energy, equivalent to one newton-meter. It measures the amount of work done when a force of one newton displaces an object by one meter in the direction of the force.
Joule: The joule (J) is the standard unit of energy in the International System of Units (SI). It represents the amount of work done or energy expended when a force of one newton acts through a distance of one meter. The joule is a fundamental unit that connects various topics in physics, from work and energy to thermodynamics and electricity.
KE = 1/2 mv^2: KE = 1/2 mv^2 is the formula for calculating the kinetic energy of an object, where 'KE' represents the kinetic energy, 'm' is the mass of the object, and 'v' is the velocity of the object. This formula is fundamental in understanding the relationship between an object's motion and the energy it possesses due to that motion.
Kinetic energy: Kinetic energy is the energy possessed by an object due to its motion. It is calculated using the formula $KE = \frac{1}{2}mv^2$, where $m$ is mass and $v$ is velocity.
Kinetic Energy: Kinetic energy is the energy of motion possessed by an object. It is the energy an object has by virtue of being in motion and is directly proportional to the mass of the object and the square of its velocity. Kinetic energy is a crucial concept in physics, as it relates to the work done on an object, the conservation of energy, and various other physical phenomena.
Law of conservation of energy: The law of conservation of energy states that energy cannot be created or destroyed, only transformed from one form to another. The total energy in an isolated system remains constant over time.
Mass: Mass is a fundamental property of matter that represents the amount of material in an object. It is a measure of the object's resistance to changes in its motion, and it is a scalar quantity, meaning it has magnitude but no direction. Mass is a crucial concept in physics, as it is a key factor in determining an object's behavior under the influence of forces.
Mechanical Energy: Mechanical energy is the sum of kinetic energy and potential energy in a system, representing the total energy available for performing work. This concept encompasses various forms of energy related to motion and position, and is crucial for understanding how objects interact under the influence of forces.
Momentum: Momentum is a vector quantity that represents the product of an object's mass and velocity. It is a measure of an object's quantity of motion and is conserved in a closed system, meaning the total momentum of a system remains constant unless acted upon by an external force.
Net Force: Net force is the vector sum of all the individual forces acting on an object. It represents the overall force that determines the object's acceleration or lack thereof, in accordance with Newton's laws of motion.
Newton-Meter: A newton-meter (N⋅m) is a unit of torque, which is a measure of the rotational force that causes an object to rotate about an axis, fulcrum, or pivot. This unit combines the units of force (newton) and distance (meter), representing the product of force and the perpendicular distance from the axis of rotation to the line of action of the force. The newton-meter is a fundamental unit in the study of rotational dynamics and equilibrium conditions.
Newton-meters: A newton-meter (N·m) is the unit of torque in the International System of Units (SI). It measures the amount of force applied over a distance, typically represented as the rotational equivalent of work.
Newton's First Law: Newton's First Law, also known as the Law of Inertia, states that an object at rest will remain at rest, and an object in motion will continue moving at a constant velocity, unless acted upon by an unbalanced force. This law establishes the fundamental principle that an object's state of motion will not change unless an external force is applied.
Newton's Second Law: Newton's second law of motion states that the acceleration of an object is directly proportional to the net force acting upon it and inversely proportional to its mass. This law describes the relationship between an object's motion and the forces acting upon it, providing a fundamental principle for understanding the dynamics of physical systems.
Potential Energy: Potential energy is the stored energy an object possesses due to its position or state, which can be converted into kinetic energy or other forms of energy. This term is central to understanding various physical phenomena and energy transformations in the context of introductory college physics.
Power: Power is the rate at which work is done or energy is transferred. It is the measure of the amount of energy expended per unit of time. Power is a fundamental concept in physics that is essential for understanding various topics, including work, energy, and simple machines.
Power (P = W/Δt): Power is the rate at which work is done or energy is transferred. It is defined as the amount of work (W) done divided by the time (Δt) taken to do that work. This relationship is expressed mathematically as P = W/Δt, where P is power, W is work, and Δt is the change in time.
Velocity: Velocity is a vector quantity that describes the rate of change in the position of an object over time. It includes both the speed of the object and the direction of its motion. Velocity is a crucial concept in understanding the motion of objects and the fundamental principles of physics.
W = F · d · cos$\theta$: The formula W = F · d · cos$\theta$ represents the work done by a force. It is the product of the force applied, the distance over which the force is applied, and the cosine of the angle between the force and the displacement. This formula is central to understanding the relationship between work, force, and motion in the context of kinetic energy and the work-energy theorem.
W = F * d: The equation W = F * d, where W represents work, F represents force, and d represents the displacement or distance over which the force is applied, is a fundamental relationship in physics. This equation is central to the concepts of kinetic energy and the work-energy theorem, which describe the transfer and transformation of energy in physical systems.
W_{net} = ΔKE = KE_f - KE_i: The net work done on an object is equal to the change in its kinetic energy, which is the difference between its final kinetic energy (KE_f) and its initial kinetic energy (KE_i). This relationship is known as the work-energy theorem and is a fundamental principle in classical mechanics.
Work-energy theorem: The work-energy theorem states that the net work done by forces on an object is equal to the change in its kinetic energy. It is a fundamental principle connecting the concepts of work and energy.
Work-Energy Theorem: The Work-Energy Theorem states that the work done by all forces acting on an object equals the change in its kinetic energy. This relationship highlights how work and energy are interchangeable; when work is done on an object, it results in a change in that object's energy state. Understanding this theorem is crucial because it connects the concept of work with energy, showing how forces impact motion and energy transformations.
Work-Kinetic Energy Relationship: The work-kinetic energy relationship is a fundamental principle in physics that describes the connection between the work done on an object and the resulting change in its kinetic energy. This relationship is a crucial concept in understanding the dynamics of moving objects and the transfer of energy.
ΔKE = -ΔPE: The principle that the change in kinetic energy (ΔKE) of an object is equal to the negative change in its potential energy (ΔPE). This relationship is a fundamental concept in the work-energy theorem and the conservation of energy.