Translational Kinetic Energy

Review of Key Terms

Before diving into translational kinetic energy, let us review some key terms, starting with the term scalar.

Energy is a scalar quantity.

Energy exists in many forms; however, all energy can be classified as kinetic or potential. Therefore, when calculating energy, the mathematical formula required will depend on the type of energy in question. However, for the purpose of this article, we will only focus on kinetic energy and translational motion.

The SI unit of energy is the joule, denoted by \( \mathrm{J}. \)

Translational Kinetic Energy Definition

Having reviewed energy and kinetic energy, let us now define and discuss translational kinetic energy.

Translational Kinetic Energy

Translational kinetic energy is a scalar quantity, which means it has no direction. It only has magnitude.

Translational Kinetic Energy Formula

The mathematical formula corresponding to the definition of translational kinetic energy is

\[K_{\mathrm{T}} = \frac{1}{2}mv^2\]

where \(m \) is the mass measured in \( \mathrm{kg} \) and \( v \) is the measured velocity in \( \mathrm{\frac{m}{s}}. \) Note that \(v^2 = |\vec{v}|^2.\) However, since translational kinetic energy is scalar, vector notation can be suppressed.

Translational Kinetic Energy vs. Rotational Kinetic Energy

Before comparing translational and rotational kinetic energy, we must first define rotational kinetic energy and discuss its corresponding formula.

Rotational Kinetic Energy

Rotational kinetic energy is also a scalar quantity, which means it has no direction. Unlike translational kinetic energy, however, its magnitude is determined by the angular speed of an object.

The corresponding mathematical formula to the definition of rotational kinetic energy is

\[K_{\mathrm{rot}} = \frac{1}{2}I\omega^2\]

where \( I \) is the moment of inertia measured in \( \mathrm{{kg}\,{m^2}} \) and \( \omega \) is the angular velocity measured in \( \mathrm{\frac{rad}{s}}. \)

Relationship between Rotational Kinetic Energy and Translational Kinetic Energy

Rotational and translational kinetic energies describe different types of motion, yet their formulas are very similar in form. Why? This is the result of the relationship between linear and rotational motion as they are equivalent counterparts of one another. Linear velocity, \( v \), and angular velocity, \( \omega \), relate to one another by the formulas

\[ v=\omega{r} \]

and

\[\omega = \frac{v}{r}.\]

However, the relationship between mass and inertia is not as straightforward. The rotational analog of mass is called the moment of inertia. An object's moment of inertia describes how mass is distributed relative to the axis of rotation. Meanwhile, mass itself describes the amount of matter within an object. Hence, mass is the measure of an object's resistance to changing its motion. Therefore, the more inertia an object has, the more mass it has.

Total Kinetic Energy of a System

Although systems can contain only translational or rotational kinetic energy, both types can be present in a single system. For example, cars move with translational kinetic energy, while their tires move with translational and rotational kinetic energy. Therefore, the total kinetic energy of the system is the sum of rotational kinetic energy and translational kinetic energy. Its corresponding formula is

\[K_{\mathrm{total}} = K_{\mathrm{T}} + K_{\mathrm{rot}}.\]

Reference Frames and Translational Kinetic Energy

Translational kinetic energy may be measured differently depending on an observer’s reference frame.

Reference frames are classified as inertial or non-inertial. Inertial reference frames are frames where Newton's first law applies, objects at rest remain at rest, and objects in motion remain in motion. These types of frames can be stationary or move with a constant velocity. A non-inertial reference frame is an accelerating frame, undergoing either linear acceleration or angular acceleration around an axis.

This concept can be hard to understand, so let us complete a thought experiment. Imagine you are riding a train, which moves with constant velocity. You notice a child throwing a ball up into the air. As the train barrels past the next platform, people on the platform see this as well. However, although you each witness the ball thrown in the air at the same time, you have two different reference frames. Your frame of reference is moving while the frame of reference of people on the platform is stationary. Consequently, you witness the ball moving vertically up and down because of gravity. The people on the platform, however, see a parabola where the horizontal part of the ball's velocity is equal to the train's velocity.

Fig. 2: The motion of an object is viewed differently depending on an observer's frame of reference.

Translational Kinetic Energy Examples

To solve translational kinetic energy problems, one can apply the formula for translational kinetic energy to different problems. As we have defined translational kinetic energy and rotational energy and discussed their relationship, let us work through some examples to gain a better understanding of the concepts. Note that before solving a problem, we must always remember these simple steps:

1. Read the problem and identify all variables given within the problem.
2. Determine what the problem is asking and what formulas are needed.
3. Apply the necessary formulas and solve the problem.
4. Draw a picture if necessary to provide a visual aid

Examples

Let us apply our new knowledge of translational kinetic energy to the following two examples.

Now let's complete a slightly more difficult example.