Mechanical Power

Mechanical power is a concept that has a much different meaning in physics than it does in daily life. How do you use the term 'power' in real life? For example, we call people who are engaged in bodybuilding sports or who show resilience in the face of a difficult situation powerful people.

But what does 'power' mean in physics? In physics, being powerful means doing work as quickly as possible. For example, Mark carries a box of books from the ground floor of the school to the 2^nd floor in 1 minute. If Kevin does the same work in 2 minutes, Mark displays more mechanical power than Kevin.

Mechanical Power Definition

Let's look into the definition of power.

It is helpful to keep track of how fast the energy of a system changes with time. A system can receive energy and can transfer it to other systems as well. Energy can be converted from one type of energy to another. For example, kinetic energy can be converted into heat. Power is a time-dependent physical quantity. For example, your power also changes if you take a different amount of time to get the work done.

Mechanical Power Formula

Work is the change of energy in a system when a force acts on it. If a force does an amount of work \(W\) in a time interval, we can calculate the average power due to the force as

\[\begin{align}P_{\text{avg}}&=\frac W{\Delta t},\\P_{\text{avg}}&=\frac{\Delta E}{\Delta t}.\end{align}\]

On the other hand, the instantaneous power is the instantaneous time rate of doing work. We can calculate it from

\[P=\frac{\text dW}{\text dt}.\]

Instantaneous power is useful when we have a work function dependent on time, and we want to know the power at a specific instant. Then we take the time derivative of the work function and plug the instant time into the derived function.

The rate at which a force does work on a particle (or particle-like item) may be expressed in terms of the force and the particle's velocity. For a particle traveling in a straight path and subjected to a constant force \(\vec{F}\) directed at an angle \(\theta\) to that path, we can write the power equation as:

$$\begin{align*}P&=\frac{\text{d}W}{\text{d}t},\\P&=\frac{F\cos{\theta}\text{d}x}{\text{d}t},\\P&=F\cos{\theta}\left(\frac{\text{d}x}{\text{d}t}\right),\\P&=Fv\cos{\theta}.\end{align*}$$

We must notice that the component of the force that acts along the direction of displacement is the one responsible for doing work and moving the object. If we reorganize the equation according to the dot product, we end up with: \(P=\vec{F}\cdot\vec{v}.\)

Mechanical Power Example

Let's look at examples of mechanical power.

Mechanical Power Output

We can study the power of a mechanical system by splitting it into input power and output power. The output power will always be equal to or smaller than the input power, as, in real life, machines use energy to do work. The input power refers to how much energy a system can receive, while the output power refers to how much energy the system can use to do work. Let's say that the input force \(F_{\text A}\) acts on a system that moves with velocity \(v_{\text A},\) and the output force \(F_{\text B}\) acts on a system that moves with velocity \(v_{\text B}\). If the system does not lose any mechanical power, then the input and output power are equal:

$$F_{\mathrm A}v_{\mathrm A}=F_{\mathrm B}v_{\mathrm B.}$$

This case allows us to create an expression for the mechanical advantage \(a\), which is another way of measuring the output energy in terms of the input energy or the efficiency \(e\):

$$\begin{align*}a&=\frac{F_{\mathrm B}}{F_{\mathrm A}},\\a&=\frac{v_{\mathrm B}}{v_{\mathrm A}},\\e&=\frac{\text{output energy}}{\text{input energy}}\times100\%.\end{align*}$$

Even if energy cannot be destroyed, it can be converted to a different type of energy. As a consequence, the efficiency of the device is slower, as the output is less than the input. For example, a light bulb's input power is provided by electrical energy, while its output power is in the form of light and heat.

Difference Between Mechanical and Electrical Power

While mechanical power refers to the rate at which work can be done, electrical power is the rate at which an electric circuit transfers electrical energy. An electric motor's input power is provided by electrical power, while the output power is mechanical so that it causes the car to move. The equation for electrical power is given by

$$P=IV,$$

where the electrical current \(I\) is expressed in amperes \(\left(\text A\right)\) and the applied voltage is expressed in volts \(\left(\text V\right)\).

We have previously discussed mechanical power for translational motion. Motors undergo rotational motion so that we can express the rotational form of mechanical power. It depends on the rotational analogous to the force and velocity, which are the torque and the angular velocity:

$$P=\tau \omega.$$