Glossary

The ###-energy_theorem_0### connects the work done on a to its change in . It's a powerful tool for analyzing motion, allowing us to calculate velocities and displacements without needing to know the entire path of an object.

This theorem bridges the concepts of , work, and energy. By understanding how work relates to changes in kinetic energy, we can solve complex problems involving particle motion and in various physical scenarios.

Work-Energy Theorem

Work-energy theorem for particle motion

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  • States done on a particle equals change in its kinetic energy Wnet=ΔKEW_{net} = \Delta KE
    • WnetW_{net} represents done on particle by all forces
    • ΔKE\Delta KE represents change in particle's kinetic energy
  • Calculate net work by summing work done by each force acting on particle Wnet=W1+W2+...+WnW_{net} = W_1 + W_2 + ... + W_n
    • Calculate work done by a force using product of force and in direction of force W=F⃗⋅d⃗W = \vec{F} \cdot \vec{d}
      • For constant force W=Fdcos⁡θW = Fd \cos \theta, θ\theta represents angle between force and vectors
      • For force varying with position W=∫x1x2F(x)dxW = \int_{x_1}^{x_2} F(x) dx
  • Calculate change in kinetic energy by subtracting initial from final kinetic energy ΔKE=KEf−KEi=12mvf2−12mvi2\Delta KE = KE_f - KE_i = \frac{1}{2}mv_f^2 - \frac{1}{2}mv_i^2
    • mm represents particle's
    • viv_i and vfv_f represent particle's initial and final velocities
  • Determine particle's final or displacement by applying given initial conditions and acting forces (sliding block, )
  • Work-energy theorem relates to energy transfer between different forms (e.g., kinetic to )

Forces from motion using work-energy

  • Determine net work done on particle using work-energy theorem if initial and final velocities (or kinetic energies) and displacement are known
  • Rearrange work-energy theorem to solve for net work Wnet=ΔKE=12mvf2−12mvi2W_{net} = \Delta KE = \frac{1}{2}mv_f^2 - \frac{1}{2}mv_i^2
  • Calculate work done by unknown force using net work and known forces Wunknown=Wnet−(W1+W2+...+Wn)W_{unknown} = W_{net} - (W_1 + W_2 + ... + W_n)
  • Determine average force exerted on particle using work-displacement relationship Favg=Wunknowndcos⁡θF_{avg} = \frac{W_{unknown}}{d \cos \theta} once work done by unknown force is calculated (pulling a , pushing a )

Kinetic energy changes from net work

  • Calculate change in particle's kinetic energy directly from net work done on it ΔKE=Wnet\Delta KE = W_{net}
  • To find change in kinetic energy:
    1. Calculate net work done by all forces acting on particle
    2. Equate net work to change in kinetic energy
  • Alternatively, calculate change in kinetic energy using particle's initial and final velocities ΔKE=12mvf2−12mvi2\Delta KE = \frac{1}{2}mv_f^2 - \frac{1}{2}mv_i^2
    • Equation derived from work-energy theorem and definition of kinetic energy KE=12mv2KE = \frac{1}{2}mv^2
  • Understanding relationship between work and changes in kinetic energy is crucial for analyzing motion of particles under influence of forces (, )

Energy Conservation and Mechanical Energy

  • principle states that total energy in an isolated system remains constant
  • is the sum of kinetic and potential energy in a system
  • In conservative systems, is conserved when no non-conservative forces do work
  • Power is the rate at which work is done or energy is transferred, measured in watts (W)

Key Terms to Review (32)

Action-at-a-distance force: An action-at-a-distance force is a force exerted by an object on another object that is not in physical contact with it, acting over a distance through space. Examples include gravitational, electromagnetic, and nuclear forces.
Cart: A cart is a small, wheeled vehicle used to transport objects or materials from one location to another. It is a simple machine that utilizes the principles of the wheel and axle to overcome friction and ease the movement of loads.
Collision: A collision is an event in which two or more objects interact for a short period of time, during which the objects' states (such as their velocities) change due to their interaction. Collisions are a fundamental concept in the study of mechanics, particularly in the context of the work-energy theorem.
Conservation of Energy: The conservation of energy principle states that energy cannot be created or destroyed, only transformed from one form to another. This fundamental concept links various phenomena, illustrating how mechanical, kinetic, and potential energies interconvert while keeping the total energy constant in a closed system.
Conservative force: A conservative force is a force where the work done in moving an object between two points is independent of the path taken. Examples include gravitational and electrostatic forces.
Conservative Force: A conservative force is a type of force that does not depend on the path taken by an object between two points, but only on the initial and final positions of the object. The work done by a conservative force depends solely on the start and end points, and not the specific path taken between them.
Displacement: Displacement is a vector quantity that refers to the change in position of an object. It is measured as the straight-line distance from the initial to the final position, along with the direction.
Displacement: Displacement is the change in position of an object relative to a reference point. It is a vector quantity, meaning it has both magnitude and direction, and is used to describe the movement of an object in physics.
Energy Transfer: Energy transfer is the process by which energy is moved from one system or object to another. This fundamental concept is central to understanding the principles of work, power, and energy conservation in physics.
Force: Force is a vector quantity that represents the interaction between two objects, causing a change in the motion or shape of the objects. It is the fundamental concept that underlies many of the physical principles studied in college physics, including Newton's laws of motion, work, energy, and more.
Friction: Friction is a force that opposes the relative motion between two surfaces in contact. It arises due to the microscopic irregularities on the surfaces, which create resistance to sliding or rolling. Friction is a fundamental concept in physics that plays a crucial role in various topics, including solving problems, understanding forces, and analyzing energy transformations.
Joule: A joule is the SI unit of work or energy, equivalent to one newton-meter. It represents 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.
Kinetic energy: Kinetic energy is the energy possessed by an object due to its motion. It depends on the mass and velocity of the object.
Loop-the-loop: A loop-the-loop is a circular path that an object follows, often seen in roller coasters or toy tracks. It involves the conversion of kinetic energy to potential energy and vice versa as the object travels through the loop.
Mass: Mass is a fundamental physical quantity that represents the amount of matter in an object. It is a measure of an object's resistance to changes in its state of motion, and it is a key concept in the study of mechanics and the behavior of objects under the influence of forces.
Mechanical energy: Mechanical energy is the sum of kinetic energy and potential energy in a system. It is the energy associated with the motion and position of an object.
Mechanical Energy: Mechanical energy is the sum of the kinetic energy and potential energy possessed by an object due to its motion and position within a physical system. It represents the total energy available to do work or cause change in the system.
Net work: Net work is the total work done on an object, accounting for all forces acting on it. It determines the change in the object's kinetic energy.
Net Work: Net work is the total amount of work done on an object, taking into account both the positive and negative work contributions. It represents the net change in the object's energy, which can be used to determine the object's final velocity, kinetic energy, or potential energy.
Newton-Meter: The newton-meter (N⋅m) is the unit used to measure torque, which is the rotational force that causes an object to rotate about an axis, pivot, or fulcrum. It is the product of the applied force and the perpendicular distance between the axis of rotation and the line of action of the force.
Non-Conservative Force: A non-conservative force is a type of force that does not satisfy the work-energy theorem. Unlike conservative forces, the work done by a non-conservative force depends on the path taken by the object, rather than just the initial and final positions. This means the work done by a non-conservative force cannot be expressed solely in terms of the object's position.
Particle: A particle is a discrete, indivisible unit of matter or energy that exhibits fundamental properties of a physical system. Particles are the basic building blocks that make up all matter and energy in the universe.
Potential Energy: Potential energy is the stored energy possessed by an object due to its position or state, which can be converted into kinetic energy or other forms of energy when the object is moved or transformed. This term is central to understanding various physical phenomena and the conservation of energy.
Projectile motion: Projectile motion is the motion of an object thrown or projected into the air, subject to only the acceleration due to gravity. It involves two components of motion: horizontal and vertical.
Projectile Motion: Projectile motion is the motion of an object that is launched into the air and moves solely under the influence of gravity and without any additional force acting on it. It is a type of motion that follows a curved trajectory, with the object's position and velocity changing over time in a predictable manner.
Roller Coaster: A roller coaster is an amusement park ride that features a track with steep inclines, sharp turns, and sudden drops, creating an exhilarating and thrilling experience for riders. Roller coasters are closely tied to the concepts of work, energy, and conservation of energy, making them an important topic in the study of college physics.
Sled: A sled is a flat or curved vehicle used for transportation over snow or ice, often pulled by animals or propelled by the user. It is a fundamental device in the context of the Work-Energy Theorem, as the motion of a sled can be used to demonstrate the principles of work, energy, and their relationship.
Velocity: Velocity is a vector quantity that describes the rate of change of an object's position with respect to time. It includes both the speed and the direction of an object's motion, making it a more complete description of an object's movement compared to just speed alone.
Work: Work is a physical quantity that describes the energy transferred by a force acting on an object as the object is displaced. It is the product of the force applied and the displacement of the object in the direction of the force.
Work-energy theorem: The work-energy theorem states that the net work done on an object is equal to its change in kinetic energy. Mathematically, it is expressed as $W_{net} = \Delta KE$.
Work-Energy Theorem: The work-energy theorem is a fundamental principle in physics that states the change in the kinetic energy of an object is equal to the net work done on that object. It establishes a direct relationship between the work performed on an object and the resulting change in its kinetic energy, providing a powerful tool for analyzing and solving problems involving energy transformations.
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