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.

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Power is the rate of change of ___ with respect to time.

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True/False: The energy input and output can be provided by different types of energy, like mechanical energy, electrical energy, and heat.

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True/False: The equation for electrical power is the same as mechanical power.

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If the power of an engine is \(2\,\text W\) for \(2\, \text{seconds}\), how much work does it do?

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A horizontal force of \(F= 10\, \text N\) on a frictionless horizontal path moves the object \(20\, \text m\) for \(5\, \text{seconds}\). According to this, what is the power consumed by the force \(F\)?

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An object can only be lifted to a height of \(10\, \text m\) in \(5\, \text{seconds}\) by consuming \(300\, \text W\) of power.Since the velocity of the object is constant throughout the motion, what is the mass of the object, in \(\text{kg}\)? (Assume \(g=10\,\frac{\text m}{\text{s}^2}\))

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The efficiency of an engine is given by 

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The mechanical power is also related to the change in energy by

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An automobile engine applies a force of \(2000\, \text N\) to the car, allowing it to travel \(500\,\text m\) in \(5\, \text {seconds}\). A constant friction force of \(500\, \text N\) acts on the car along the road. What's the power of the car?

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A worker lifts a \(15\, \text{kg}\) load from the 1st floor to the 5th floor of an apartment building, each floor of which has a height of \(2.5\, \text m\), in \(1\) minute. What is the power of the worker? (Assume \(g=10\,\frac{\text m}{\text s}^2\))

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Helping Ms. Kerrie to carry the \(12\, \text{kg}\) shopping bags in her hand, Mark, with a mass of \(48\, \text{kg}\), takes the bags from the entrance of the apartment to the house on the 2nd floor, which is \(8\, \text m\) tall, at a constant speed, in \(15\, \text{seconds}\). What is Mark's power? (Assume \(g=10\;\mathrm N/\mathrm{kg}\))

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Power is the rate of change of ___ with respect to time.

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  • Mo

True/False: The energy input and output can be provided by different types of energy, like mechanical energy, electrical energy, and heat.

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  • Immunology
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  • Mo

True/False: The equation for electrical power is the same as mechanical power.

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  • + Add tag
  • Immunology
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  • Mo

If the power of an engine is \(2\,\text W\) for \(2\, \text{seconds}\), how much work does it do?

Show Answer
  • + Add tag
  • Immunology
  • Cell Biology
  • Mo

A horizontal force of \(F= 10\, \text N\) on a frictionless horizontal path moves the object \(20\, \text m\) for \(5\, \text{seconds}\). According to this, what is the power consumed by the force \(F\)?

Show Answer
  • + Add tag
  • Immunology
  • Cell Biology
  • Mo

An object can only be lifted to a height of \(10\, \text m\) in \(5\, \text{seconds}\) by consuming \(300\, \text W\) of power.Since the velocity of the object is constant throughout the motion, what is the mass of the object, in \(\text{kg}\)? (Assume \(g=10\,\frac{\text m}{\text{s}^2}\))

Show Answer
  • + Add tag
  • Immunology
  • Cell Biology
  • Mo

The efficiency of an engine is given by 

Show Answer
  • + Add tag
  • Immunology
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  • Mo

The mechanical power is also related to the change in energy by

Show Answer
  • + Add tag
  • Immunology
  • Cell Biology
  • Mo

An automobile engine applies a force of \(2000\, \text N\) to the car, allowing it to travel \(500\,\text m\) in \(5\, \text {seconds}\). A constant friction force of \(500\, \text N\) acts on the car along the road. What's the power of the car?

Show Answer
  • + Add tag
  • Immunology
  • Cell Biology
  • Mo

A worker lifts a \(15\, \text{kg}\) load from the 1st floor to the 5th floor of an apartment building, each floor of which has a height of \(2.5\, \text m\), in \(1\) minute. What is the power of the worker? (Assume \(g=10\,\frac{\text m}{\text s}^2\))

Show Answer
  • + Add tag
  • Immunology
  • Cell Biology
  • Mo

Helping Ms. Kerrie to carry the \(12\, \text{kg}\) shopping bags in her hand, Mark, with a mass of \(48\, \text{kg}\), takes the bags from the entrance of the apartment to the house on the 2nd floor, which is \(8\, \text m\) tall, at a constant speed, in \(15\, \text{seconds}\). What is Mark's power? (Assume \(g=10\;\mathrm N/\mathrm{kg}\))

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    Mechanical Power An image of a man lifting weight in the gym StudySmarter

    Fig. 1 - The concept of power is used differently in daily life and physics. In physics, power is the rate at which a force does work.

    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 2nd 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.

    Power is the rate at which a force does work.

    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 SI unit of power is the joule per second \(\mathrm{J\, s^{-1}}\), which is also called the watt \(W\) after James Watt.

    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.

    A block is moving on a frictionless floor with the effect of force \(F = 20\,\text{N}\) with a speed of \(v=5\,\frac{\text{m}}{\text{s}}\) instantaneously as shown in figure 2. What is the power due to the force acting on that block at that instant?

    Mechanical Power The block is moving on a frictionless floor under the effect of a force. The force has both vertical and horizontal components StudySmarterFig. 2 - The block is moving on a frictionless floor under the effect of a force. The force has both vertical and horizontal components.

    Answer:

    To find instantaneous power, we need the magnitude of the force acting on the object, and the instant speed. The force acts on the box at \(60^\circ\). Because the vertical component of the force does not do work, we need the horizontal component to find the instantaneous power. We can calculate it using the equation \(P=Fv\cos{\theta}\). We know that the force is \(F=20\,\text{N}\) and the velocity is \(v=5\,\frac{\text{m}}{\text{s}}\). If we insert these known values into the formula, we can calculate the instantaneous power:

    \begin{align*}P&=\left(20\,\text{N}\right)\left( 5\,\frac{\text{m}}{\text{s}}\right) \cos{60^\circ},\\P&=\left(20\,\text{N}\right)\left( 5\,\frac{\text{m}}{\text{s}}\right)\left(\frac12\right),\\P&=50\,\text{W}.\end{align*}

    Since the rate of transfer of energy is different than zero, the velocity will increase.

    Let's look at one more example where two forces are acting.

    A block is moving on a frictionless floor with a force of \(F_1=20\,\text{N}\) at an angle of \(60^\circ\) with the floor, and the second force of \(F_2=10\,\text{N}\) pulling directly to the left as shown in figure 3. The block is moving at a speed of \(5\, \frac{\text{m}}{\text{s}}\) instantaneously.

    What is the net power due to the forces acting on that block at that instant?

    Mechanical Power An image of a block moving on a frictionless floor under the effect of two forces at opposite directions StudySmarterFig. 3 - A block is moving on a frictionless floor. Two forces act on the object in opposite directions.

    Answer:

    Let's calculate the instantaneous power of the individual forces.

    First, calculate the instantaneous power \(P_1\) due to \(F_1\):

    \begin{align*}P_1&=F_1 v\cos{60^\circ},\\P_1&=\left(20\,\text{N}\right) \left(5\,\frac{\text{m}}{\text{s}}\right)\left(\frac12\right),\\P_1&=50\,\text{W}.\end{align*}

    Then, calculate the instantaneous power \(P_2\) due to \(F_2\):

    \begin{align*}P_2&=F_2 v\cos{180^\circ},\\P_2&=\left(10\,\text{N}\right) \left(5\,\frac{\text{m}}{\text{s}}\right)\left(-1\right),\\P_2&=-50\,\text{W}.\end{align*}

    To find the net power, we can add up \(P_1\) and \(P_2\):

    \begin{align*}P_{\text{net}}&=P_1+P_2,\\P_{\text{net}}&=50\,\text{W}-50\,\text{W},\\P_{\text{net}}&=0.\end{align*}

    Since the net power is zero, that means the rate of transfer of kinetic energy is zero as well. So, the speed of the block will remain the same.

    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.

    Determine the efficiency of a light bulb that releases \(60\,\text{kJ}\), while its input energy is \(1550\,\text{kJ}\).

    Answer:

    The efficiency of the light bulb is given by

    $$\begin{align*}e&=\frac{60\,\text{kJ}}{1550\,\text{kJ}}\times 100\%,\\e&=3.87\%.\end{align*}$$

    The lightbulb is very inefficient.

    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:

    $$P=\tau \omega.$$

    Mechanical Power - Key takeaways

    • The power due to the force is defined as the rate at which a force does work.
    • If a force does an amount of work W in a time interval, then average power can be calculated from \(P=\frac{W}{\Delta t}\).
    • The instantaneous power is the instantaneous time rate of doing work, \(P=\frac{\text dW}{\text dt}\).
    • The rate at which a force does work on a particle (or particle-like item) may alternatively be expressed in terms of the force and the particle's velocity: \(P=\vec{F}\cdot\vec{v}\).

    References

    1. Fig. 1 - The concept of power is differently used in daily life and physics. In physics, power is the rate of doing work (https://pixabay.com/es/photos/hombre-persona-poder-fuerza-fuerte-1282232/), by Pexels (https://pixabay.com/es/users/pexels-2286921/), licensed by Pixabay (https://pixabay.com/es/service/license/)
    2. Fig. 2 - The block is moving on a frictionless floor under the effect of a force. The force has both vertical and horizontal components, StudySmarter Originals
    3. Fig. 3 - A block is moving on a frictionless floor. Two forces act on the object in opposite directions, StudySmarter Originals
    Frequently Asked Questions about Mechanical Power

    What is meant by mechanical power?

    Mechanical power is the rate that a force does work on an object. It is the amount of energy transferred into a system over a period of time.

    How do you calculate mechanical power?

    You calculate mechanical power by doing the work divided by the amount of time that the work was being done.

    What are the types and uses of mechanical power?

    Some types of mechanical power are water turbines, electric motors, hydraulic presses, steam turbines, and wind turbines. The power that these machines produce is used to power our transportation, electricity, and other necessities.

    What is the 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.

    What is the formula for mechanical power?

    Mechanical power equals the change in energy divided by the change in time. It can also be expressed as the work divided by the change in time.

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    Power is the rate of change of ___ with respect to time.

    True/False: The energy input and output can be provided by different types of energy, like mechanical energy, electrical energy, and heat.

    True/False: The equation for electrical power is the same as mechanical power.

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