Mechanical Energy

Systems and Energy Conservation

• ^^Open systems^^ are physical systems that exchange energy, matter, or both with their surroundings. Examples of open systems include a car engine, a living organism, or a water heater. In open systems, energy and matter can flow in or out of the system, and the system is not isolated from its surroundings.
• ^^Closed systems^^ are physical systems that ^^do not^^ exchange energy, matter, or both with their surroundings. Examples of closed systems include a closed container of gas, a roller coaster, or a pendulum. In closed systems, energy and matter cannot flow in or out of the system, and the system is isolated from its surroundings.
• The principle of conservation of energy states that ^^energy cannot be created or destroyed^^, only transformed from one form to another. This means that the total amount of energy in a closed system is constant, and any change in the energy of one part of the system must be compensated by a corresponding change in the energy of another part of the system.

Work

• Work is defined as the transfer of energy through the application of a force over a distance.
  • The unit of work is the joule (J), which is defined as the amount of work done when a force of 1 newton (N) is applied over a distance of 1 meter (m).
  • We can calculate the work done by a force by using the following formula:
  • W = Fdcos(theta)
  • where W is the work done, F is the force applied, d is the distance over which the force is applied, and theta is the angle between the force and the displacement.
• When the angle between the force and the displacement is 90 degrees, the work done is equal to zero. This is because the force and the displacement are perpendicular to each other, so there is no transfer of energy.
• When the angle between the force and the displacement is less than 90 degrees, the work done is positive. This is because the force and the displacement are in the same direction, so there is a transfer of energy.
• When the angle between the force and the displacement is greater than 90 degrees, the work done is negative. This is because the force and the displacement are in opposite directions, so there is a transfer of energy in the opposite direction.
• The Work-Kinetic Energy Theorem states that the change in kinetic energy of an object is equal to the work done on the object. This can be expressed mathematically as:
  • ΔKE = W
  • where ΔKE is the change in kinetic energy, and W is the work done on the object. The work done on an object is equal to the force applied to the object multiplied by the displacement of the object in the direction of the force.
• To calculate the kinetic energy of an object, we can use the following formula:
  • KE = 1/2mv^2
  • where KE is the kinetic energy, m is the mass of the object, and v is the velocity of the object. The kinetic energy of an object is equal to its mass multiplied by its velocity squared, divided by 2.

Power

• Power is a measure of the rate at which work is done or energy is transferred. It is defined as the amount of work done or energy transferred per unit of time.
  • The unit of power is the watt (W), which is defined as 1 joule per second (J/s).
  • We can calculate the power of a force by using the following formula:
  • P = W / t
  • where P is the power, W is the amount of work done by the force, and t is the time interval over which the work is done.
• We can also calculate the power of an object by using the following formula:
  • P = F*v
  • where P is the power, F is the force acting on the object, and v is the velocity of the object.
• The rate of transfer of energy (power) can be demonstrated through experiments involving closed systems. A closed system is a system in which no matter or energy can enter or leave. Examples of closed systems include a sealed container with a gas, a sealed container with a liquid, and a sealed container with a solid.
  • In a closed system, the total energy of the system is conserved. This means that the total energy of the system cannot change unless it is acted upon by an external force.

Impulse and Momentum

• Impulse is defined as the force applied to an object over a given time interval. It is a measure of the change in momentum of an object.

• The unit of impulse is the newton-second (N*s).

• We can calculate the impulse applied to an object by using the following formula:

  • I = F*t
  • where I is the impulse, F is the force applied, and t is the time interval over which the force is applied.

• Momentum is defined as the product of an object's mass and its velocity. It is a measure of the motion of an object.

• The unit of momentum is the kilogram-meter per second (kg*m/s).

• We can calculate the momentum of an object by using the following formula:

  • p = m*v
  • where p is the momentum, m is the mass of the object, and v is the velocity of the object.

• The principle of conservation of momentum states that the total momentum of a closed system is conserved. This means that the total momentum of the system cannot change unless it is acted upon by an external force.

• In a collision, the brief application of a force creates an impulse. The impulse changes the momentum of the colliding objects.

• We can calculate the change in momentum of an object by using the following formula:

  • Δp = pf - pi
  • where Δp is the change in momentum, pf is the final momentum of the object, and pi is the initial momentum of the object.

Types of Collisions:

• Elastic Collisions:

  • In elastic collisions, the kinetic energy of the colliding objects is conserved. This means that the kinetic energy of the colliding objects is unchanged before and after the collision.
  • To calculate the final velocities of the colliding objects in an elastic collision, we can use the following formulas:
  • v1f = (m1v1i + m2v2i - m2*(v1i - v2i)) / (m1 + m2)
  • v2f = (m1v1i + m2v2i - m1*(v1i - v2i)) / (m1 + m2)
  • where v1f and v2f are the final velocities of the colliding objects, m1 and m2 are the masses of the colliding objects, v1i and v2i are the initial velocities of the colliding objects, and v1i - v2i is the relative velocity of the colliding objects.

• Inelastic Collisions:

  • In inelastic collisions, the kinetic energy of the colliding objects is not conserved. This means that some of the kinetic energy of the colliding objects is converted into other forms of energy, such as thermal energy or sound energy. As a result, the kinetic energy of the colliding objects decreases after the collision.
  • To calculate the final velocities of the colliding objects in an inelastic collision, we can use the following formulas:
  • v1f = (m1v1i + m2v2i) / (m1 + m2)
  • v2f = (m1v1i + m2v2i) / (m1 + m2)
  • where v1f and v2f are the final velocities of the colliding objects, m1 and m2 are the masses of the colliding objects, and v1i and v2i are the initial velocities of the colliding objects.