Work, Energy, and Power - Calculus Based Classical Mechanics
Kinetic energy is energy of motion. It is a scalar quantity calculated by:
1 2
𝐾= 2
𝑚𝑣
Potential energy is a scalar quantity takes many forms, but here we will discuss gravitational
potential energy and spring potential energy. Gravitational potential energy is given by:
𝑈𝑔 = 𝑚𝑔ℎ
Where “h” is defined as any height above/below a surface, and g is the acceleration due to
gravity. Although two people may disagree on where they define “h”, the change in potential
energy of a falling/rising object will always be the same as the change in height is always the
same.
Spring potential energy is given by:
1 2
𝑈𝑠 = 2
𝑘𝑥
Where “k” is the spring constant and “x” is the displacement of the spring at a given point.
● Potential energy is NOT the energy of something to “potentially happen”
● Potential energy IS the energy of relative motion in a system.
Where “v” is the magnitude of the velocity vector. Direction does not affect kinetic energy as it
is a scalar quantity.
Work is a scalar quantity defined by force applied to an object across a certain distance. No
distance = no work done, even if you applied a large amount of force. For a constant force, it can
be calculated by:
𝑊= 𝐹 ·𝑑
The “F dot d” notation means the dot product between force and distance. This simply means
that work is the component of force parallel to the displacement of the object multiplied by the
distance it moves. Or:
𝑊 = 𝐹𝑑𝑐𝑜𝑠θ
where theta is the angle between the force and the displacement
Kinetic energy is energy of motion. It is a scalar quantity calculated by:
1 2
𝐾= 2
𝑚𝑣
Potential energy is a scalar quantity takes many forms, but here we will discuss gravitational
potential energy and spring potential energy. Gravitational potential energy is given by:
𝑈𝑔 = 𝑚𝑔ℎ
Where “h” is defined as any height above/below a surface, and g is the acceleration due to
gravity. Although two people may disagree on where they define “h”, the change in potential
energy of a falling/rising object will always be the same as the change in height is always the
same.
Spring potential energy is given by:
1 2
𝑈𝑠 = 2
𝑘𝑥
Where “k” is the spring constant and “x” is the displacement of the spring at a given point.
● Potential energy is NOT the energy of something to “potentially happen”
● Potential energy IS the energy of relative motion in a system.
Where “v” is the magnitude of the velocity vector. Direction does not affect kinetic energy as it
is a scalar quantity.
Work is a scalar quantity defined by force applied to an object across a certain distance. No
distance = no work done, even if you applied a large amount of force. For a constant force, it can
be calculated by:
𝑊= 𝐹 ·𝑑
The “F dot d” notation means the dot product between force and distance. This simply means
that work is the component of force parallel to the displacement of the object multiplied by the
distance it moves. Or:
𝑊 = 𝐹𝑑𝑐𝑜𝑠θ
where theta is the angle between the force and the displacement
