What is the tangential component of any vector?

Question

There's a statement I discovered in the book I am reading which says

Kinetic energy changes only when speed changes and that happens when the resultant force has a tangential component.

Does that mean kinetic energy changes only when there is a resultant force component that is parallel to the velocity component? Or is something lacking in my understanding of the word tangential?

Answer 1

Yes. If the velocity vector is $\mathbf{v}$ the kinetic energy is $K=(1/2)m|\mathbf{v}|^2$ - it only depends on the magnitude of the vector, not is direction. So a force that is tangential to the velocity will change its direction but not magnitude, leaving kinetic energy unchanged. A force with a component along $v$ will change the magnitude and hence the kinetic energy.

Another way of seeing this is to write the energy as $K'=(1/2)m(\mathbf{v}\cdot\mathbf{v})$ and take the time derivative: $$K'=(1/2)m \left(\frac{d\mathbf{v}}{dt}\cdot \mathbf{v}+ \mathbf{v}\cdot \frac{d\mathbf{v}}{dt}\right) = m \left(\frac{d\mathbf{v}}{dt}\cdot \mathbf{v}\right).$$ But $\mathbf{F}=m\mathbf{a}=m\frac{d\mathbf{v}}{dt}$, so $$K'=\mathbf{F}\cdot \mathbf{v}.$$

That means that if the scalar product is zero (tangential force), there is no change. It is only the component of the force parallel to the velocity that does change it.

Answer 2

• A force causes an acceleration (Newton’s second law $\mathbf{F}=m\mathbf{a}$).
• Next, acceleration is by definition the rate of change of velocity $\mathbf{a}:=\frac{d\mathbf{v}}{dt}$.
• Speed is by definition the magnitude of velocity $v=|\mathbf{v}|=\sqrt{\mathbf{v}\cdot\mathbf{v}}$.

So, the first bullet point tells us that if there is a non-zero force then there is a non-zero acceleration, and so by the second bullet point, there will be a change in the velocity. But, does this mean the speed changes? No, not necessarily. To figure out what causes a change in the speed, and how this is related to velocity and acceleration, let’s do some basic calculus: \begin{align} \frac{dv}{dt}&=\frac{1}{2\sqrt{\mathbf{v}\cdot\mathbf{v}}}\cdot 2\mathbf{v}\cdot\frac{d\mathbf{v}}{dt}=\frac{1}{v}(\mathbf{v}\cdot \mathbf{a}). \end{align} So, if we want the speed to change, i.e if we want $\frac{dv}{dt}$ to be non-zero, then we need the dot product $\mathbf{v}\cdot\mathbf{a}$ to be non-zero, i.e we need the acceleration and velocity vectors of the particle to be non-zero, and we also need their directions to be such that the component of the acceleration along the direction of the velocity is non-zero.

It is tradition to call the direction of the velocity vector $\mathbf{v}$ the tangential direction (because the velocity is by definition the tangent vector to the curve traced out by the particle). So, putting all this together, a non-zero force always causes a change in velocity, but in order for it to change the speed, we need the direction of the force to have a non-zero tangential component (i.e a component along the velocity), hence the answer to your question

Does that mean Kinetic energy will change only when there is a resultant force component that is parallel to the velocity component?

is yes.

Answer 3

That is correct. The work done by some force over a given distance is $$W=\vec F\cdot \vec s$$ where $\vec s$ is a vector describing the displacement over which the force happend and $\cdot$ denotes the dot product. (See links below). A velocity is just the rate of displacement so this formula is closely connected as I will show.

\begin{align} \frac{d}{dt}\left(\tfrac 1 2 mv^2\right)=\vec F\cdot \vec v \end{align}

Kinetic energy, $T$, can only change when some external force does work on the body. This means that $$\Delta T=\Delta W$$ or $$\Delta T=\vec F\cdot\Delta \vec s$$

Dividing by $\Delta t$ and taking a limit gives: $$\frac {dT}{dt}=\vec F\cdot \vec v$$

So now we can investigate how the kinetic energy changes just by looking at the alignment between force and velocity.

The dot product has three cases:

1. The dot product is positive. This means that the force and displacement are (at least somewhat) aligned. The kinetic energy will increase and the body will accelerate
2. The dot product is zero. This means the force is at $90^\circ$ with respect to the velocity. The kinetic energy will stay constant. This happens for example in circular motion and it holds approximately true for planets in circular orbits.
3. The dot product is negative. This means the force points (at least somewhat) in the opposite direction of the velocity. The body will slow down.

https://www.khanacademy.org/math/multivariable-calculus/thinking-about-multivariable-function/x786f2022:vectors-and-matrices/a/dot-products-mvc

https://betterexplained.com/articles/vector-calculus-understanding-the-dot-product/

Answer 4

What is the tangential component of any vector?

a force component that is parallel to the velocity component? Or is something lacking in my understanding of the word tangential?

Yes, the tangential component of a vector is the component of the vector that is in the direction of motion, that is, the scalar projection of the vector onto the velocity vector. Here the word "tangent" refers to being tangent to the object's path.