Understanding Work and Kinetic Energy

Work Done

• Work is done when a force causes a displacement of an object.
• Work is a scalar quantity.
• The amount of work done is calculated as the product of the force component along the direction of displacement and the magnitude of the displacement.
• If the force and displacement are in the same direction, the work done is positive.
• If the force and displacement are in opposite directions, the work done is negative.
• If the force is perpendicular to the displacement, the work done is zero.
• Formula: W = F ⋅ d ⋅ cos(θ), where W is work, F is the force, d is the displacement, and θ is the angle between the force and displacement vectors.
• The unit of work is the joule (J), which is equal to one newton-meter (N⋅m).
• Work done is the energy transferred to or from an object by means of a force acting on the object.
• Work done can change an object's kinetic energy.
• The net work done on an object equals the change in its kinetic energy (work-energy theorem).

Kinetic Energy

• Kinetic energy is the energy possessed by an object due to its motion.
• It is a scalar quantity.
• Kinetic energy depends on the mass of the object and its speed.
• Formula: KE = 1/2 ⋅ m ⋅ v², where KE is kinetic energy, m is mass, and v is speed.
• The unit of kinetic energy is the joule (J).
• Kinetic energy is always positive, as mass and speed squared are always positive.
• An object at rest has zero kinetic energy.
• Kinetic energy is directly proportional to the square of the speed.
• If the speed of an object doubles, its kinetic energy quadruples.
• The work-energy theorem states that the net work done on an object is equal to the change in its kinetic energy.
• W_net = ΔKE = KE_final - KE_initial

Gravitational Potential Energy

• Gravitational potential energy (GPE) is the energy an object possesses due to its position in a gravitational field.
• It is a scalar quantity.
• GPE depends on the object's mass, the acceleration due to gravity, and its height above a reference point.
• Formula: GPE = m ⋅ g ⋅ h, where GPE is gravitational potential energy, m is mass, g is the acceleration due to gravity (approximately 9.8 m/s² on Earth's surface), and h is the height above a reference point.
• The unit of gravitational potential energy is the joule (J).
• The choice of the reference point (h = 0) is arbitrary; only changes in GPE are physically significant.
• When an object is lifted, work is done against gravity, and its GPE increases.
• When an object falls, gravity does work on it, and its GPE decreases.
• Changes in GPE are independent of the path taken, only depending on the initial and final heights.

Conversion of Energy

• Energy can be converted from one form to another.
• A common example is the conversion between kinetic energy and gravitational potential energy.
• When an object is lifted, work is done to increase its gravitational potential energy: W = ΔGPE
• As an object falls, its gravitational potential energy is converted into kinetic energy.
• In an ideal system without non-conservative forces like friction, the total mechanical energy (KE + GPE) is conserved.
• Total mechanical energy (E) is the sum of kinetic energy (KE) and potential energy (PE): E = KE + PE
• In a closed system, the total energy remains constant, although it may change forms.
• If only conservative forces (e.g., gravity) are acting, the total mechanical energy is conserved: KE_initial + GPE_initial = KE_final + GPE_final
• When non-conservative forces like friction are present, some mechanical energy is converted into thermal energy, and the total mechanical energy decreases.
• The work done by non-conservative forces is equal to the change in total mechanical energy: W_nc = ΔE = (KE_final + GPE_final) - (KE_initial + GPE_initial)
• An example of energy conversion is a pendulum, where potential energy is converted to kinetic energy at the bottom of the swing, and kinetic energy back to potential energy at the top of the swing.
• Another example is a roller coaster, where potential energy at the top of a hill is converted to kinetic energy as it goes down, and kinetic energy is converted back to potential energy as it goes up another hill (ignoring friction).