• If you are citizen of an European Union member nation, you may not use this service unless you are at least 16 years old.

  • You already know Dokkio is an AI-powered assistant to organize & manage your digital files & messages. Very soon, Dokkio will support Outlook as well as One Drive. Check it out today!

View
 

Power - Introduction

Page history last edited by Mark Eichenlaub 6 years, 1 month ago

Prerequisite:

Kinetic energy and the work-energy theorem  

 

Anything that's moving has kinetic energy, and the formula for its kinetic energy is 

 

 

What if the speed is changing? If the speed is going up, the kinetic energy is increasing. Energy is coming in. Where is this energy coming from, and how fast is it flowing in to the object? If the object is slowing down, its kinetic energy is decreasing. Energy is flowing out. Where is that energy going?

 

Energy is transferred by forces doing work,

 

 

but this formula doesn't quite answer our questions about energy flows. It tells us that if, for example, a pitcher applies a force of 50N to a baseball over a distance of 2m, the pitcher put in a total of 100J of energy. It tells us where the energy of the ball came from (the pitcher's hand, since that's what applied for force, F), and it tell us how much total energy went into the ball. It doesn't tell us how quickly energy was flowing into the ball.

 

To think about how quickly energy flows, we introduce the rate of change of energy, which is called the power, P. 

 

Let's define the power received by an object as the derivative of its kinetic energy.

 

 

Power is how fast the kinetic energy changes. For example, you have probably heard of the horsepower ("horsepower" is a specific unit of power, the way "inch" is a specific unit of length) of a car. The higher the horsepower of a car engine, the more kinetic energy the engine can put into the car per unit time, and the faster the car accelerates. 

 

In physics, we more commonly use the unit "watts" (symbol: W) for power. (This unit is named after James Watt, an inventor whose work on steam engines helped usher in the industrial revolution).

 

Problem:

Remembering the definition of power in terms of things you already know, can you guess the definition of a watt in terms of other SI units? What are the dimensions of watts?

 

Solution:

Our definition of power was "rate of change of kinetic energy". Kinetic energy is measured in joules, so that's a good starting point. But we need a rate of change. The rate of change of position (units of meters) is velocity (units of meters/sec). The rate of change of momentum (units of kg*m/s) is force (units of kg*m/s^2). When we take a rate of change in the SI unit system, we divide the units by seconds, so the units of power are

 

 

If we break the units of joules down even further, we find

 

This last expression lets us find the dimensions of power:

 

 

 

-include plots of kinetic energy vs time-

 

-give a very simple example-

 

-give an example to find the dimensions of power-

 

-give an example reasoning from power's dimensions-

 

Because the change in an object's kinetic energy is the work done on it, we can also think of power as the rate that work is done.

 

P = rate of doing work

 

-this is not exactly dW/dt, since if we write W = F Delta x, you then have the problem of why you don't differentiate F with respect to t to find the power; you only differentiate \Delta x-

 

-give an heuristic justification for P = Fv-

 

-give an example problem using P = Fv-

 

-give an example using both P = Fv and P = d/dt(1/2 mv^2)-

 

Even though a car with higher power output can have greater acceleration, power and acceleration are not exactly the same thing. First, we might notice that they have different dimensions.

 

[a] = L/T^2

[P] = M L^2/T^3

 

The extra dimension of mass in the power shouldn't affect much; we'll assume the mass of any object we're looking at is constant, so it's just an overall constant factor that doesn't affect the functional form. But the extra dimensional of length and time in the power indicate that it should behave differently from acceleration, even though more power intuitively means more acceleration, because you need high power to increase something kinetic energy a lot.

 

- examine a constant-acceleration scenario

- examine a linearly-decaying acceleration scenario

 

-make the point that the higher the velocity, the harder it is to accelerate; power depends on velocity and acceleration-

-make a connection to the work-energy theorem and introduce P = Fv, then break this equation down-

 

 

* in fact, there are two horsepowers....

Comments (0)

You don't have permission to comment on this page.