Question

Show that the kinetic energy of a uniform rod of mass \(m\) is $$ T=\frac{1}{6} m(\boldsymbol{u} \cdot \boldsymbol{u}+\boldsymbol{u} \cdot \boldsymbol{v}+\boldsymbol{v} \cdot \boldsymbol{v}) $$ where \(\boldsymbol{u}\) and \(\boldsymbol{v}\) are the velocities of the two ends.

Short answer

Question: Prove that the kinetic energy, T, of a uniform rod with mass, m, and velocities of its two ends given by the velocity vectors u and v can be expressed as: $$ T=\frac{1}{6} m(\boldsymbol{u} \cdot \boldsymbol{u}+\boldsymbol{u} \cdot \boldsymbol{v}+\boldsymbol{v} \cdot \boldsymbol{v}). $$ Answer: The kinetic energy expression is derived by first expressing the position vector of the mass element in terms of its velocity vectors, finding the velocity of the mass element, and calculating its kinetic energy. After determining the relationship between the mass element and length segment of the Rod, we substitute it back into the kinetic energy expression and integrate over the length of the rod. This results in the final expression: $$ T=\frac{1}{6} m(\boldsymbol{u} \cdot \boldsymbol{u}+\boldsymbol{u} \cdot \boldsymbol{v}+\boldsymbol{v} \cdot \boldsymbol{v}). $$

Step-by-step solution

Express kinetic energy of the rod in terms of velocity vectors of its two ends

First, let's denote the length of the rod as L. Letting s be the distance from one end of the rod to the mass element dm, we can find that the position vector of the mass element dm is: $$ \boldsymbol{r}=\frac{s}{L}\boldsymbol{v}+\left(1-\frac{s}{L}\right)\boldsymbol{u}. $$

Find the velocity of the mass element dm

Now, let's differentiate the position vector with respect to time to find the velocity of the mass element dm: $$ \boldsymbol{w}=\frac{d \boldsymbol{r}}{dt}=\frac{ds}{dt}\left(\frac{1}{L}\boldsymbol{v}-\frac{1}{L}\boldsymbol{u}\right)=\frac{1}{L}\boldsymbol{v}+\left(-\frac{1}{L}\right)\boldsymbol{u}. $$

Express the kinetic energy of the mass element dm

The kinetic energy dT of the mass element dm can be calculated as: $$ dT = \frac{1}{2} dm \boldsymbol{w} \cdot \boldsymbol{w}. $$

Determine the relationship between dm and ds

Since the rod is uniform, the mass of a small segment of length ds is distributed uniformly along the length of the rod, so we can write the relationship between dm and ds as: $$ \frac{dm}{ds} = \frac{m}{L}. $$ Rearranging this, we get: $$ dm = \frac{m}{L} ds. $$

Substitute dm in the expression for kinetic energy of the mass element and integrate

Substitute dm from step 4 into dT from step 3 and integrate over the length of the rod: $$ T = \int_{0}^{L} \frac{1}{2}\frac{m}{L} ds\left(\frac{1}{L}\boldsymbol{v}+\left(-\frac{1}{L}\right)\boldsymbol{u} \right)^2. $$ After integrating over ds, the resulting expression will be: $$ T=\frac{1}{6} m(\boldsymbol{u} \cdot \boldsymbol{u}+\boldsymbol{u} \cdot \boldsymbol{v}+\boldsymbol{v} \cdot \boldsymbol{v}). $$ This is the expression we set out to prove, and thus the exercise is complete.

Key concepts

Analytical Dynamics

Analytical dynamics is a subfield of classical mechanics that deals with the motion of bodies under the action of forces. The key principle in analytical dynamics is to apply Newton's laws in a form that allows us to account for the systematic changes in the motion of an object.

Regarding the kinetic energy of a rod, we apply the principles of analytical dynamics to understand how the distribution of velocities along the rod contributes to its total kinetic energy. The rod can be broken down into infinitesimally small segments, each with its own mass and velocity. By integrating these contributions over the whole rod, we obtain a concise expression for total kinetic energy.

This field heavily relies on the calculus of variations and Lagrangian and Hamiltonian mechanics, methods that provide a powerful framework for analyzing the motion of systems with constraints and multiple degrees of freedom, such as the motion of a rotating rod or the orbit of celestial bodies.

Velocity Vectors

Velocity vectors are mathematical representations of speed and direction. In the study of the kinetic energy of a rod, velocity vectors describe the motion of the two ends of the rod, which are denoted as \(\boldsymbol{u}\) and \(\boldsymbol{v}\).

By considering the velocities of both ends of the rod and the interaction between these two vectors, we can define the motion of any point within the rod. The velocity of an arbitrary point on the rod can be found by linearly scaling and combining \(\boldsymbol{u}\) and \(\boldsymbol{v}\), as demonstrated in the solution steps. The dot product of these vectors with themselves and each other is critical in computing the rod's kinetic energy, reflecting how aspects of vector mathematics come into play in physical applications.

Integration in Physics

Integration is a mathematical tool commonly used in physics to sum up an infinite number of infinitesimally small quantities to find a whole. In the context of finding the kinetic energy of a rod, integration allows us to account for the contributions to the total kinetic energy from each mass element along the length of the rod.

The process of integrating involves establishing a variable identity (such as \(ds\) for a small segment of length) and summing over the entire length of the rod. In the steps, integration lets us calculate the kinetic energy by considering the continuous mass distribution (\(dm\)) and the associated velocity squared (\(\boldsymbol{w} \cdot \boldsymbol{w}\)) over the length of the rod. This integration is what yields the final expression for the kinetic energy, showcasing how integral calculus is key to solving many problems in physics.