Power in Particle Systems

Related to the concepts of work and energy is the concept of power. At its core, power is the rate at which work is being done, which because work is equal to the change in energy in a system, will also be equal to the rate at which energy is changing within our system.

Power at any instant is defined as the derivative of work with respect to time. If we look at the average power over a set period, we can simply measure the work done and divide that by the time. Work is defined as the force times the distance traveled, and distance over time is the velocity of a object, giving us many possible options for relating power, work, force, distance, time, and velocity.

\[P=\frac{dW}{dt}\]
\[P_{ave}=\frac{W}{t}=\frac{F*d}{t}=F*v\]

The common units of power are watts for metric, where one watt is defined as a joule per second, or a newton meter per second, and horsepower in the US Customary system where one horsepower is defined as 550 foot pounds per second. Maximum power ratings are often a primary specification for motors and engines as gear trains can easily change the torque provided by a motor, but the overall power will not be altered by gearing.

Two cars with differing power.
Assuming the two cars above have the same mass, it would take the same amount of work to get them up to a set speed (such as 60 miles per hour). However, the more powerful car would be able to get to this speed in a much shorter time period.

Worked Problems:

Question 1:

If a car delivers an average 100 hp to the road and weighs a total of 1.2 tons, how long will it take to go from 0-60 mph?

Problem 1 Diagram

Solution:



PDF Solution

Question 2:

Your car broke down and now needs to be repaired. How much power is required for a lift to raise your 1.2 ton car 6 ft off the ground in 15 seconds?

Problem 2 Diagram

Solution:



PDF Solution

Question 3:

The drag force of air on a car is equal to...
\[F_{d}=\frac{1}{2}\rho v^2c_{d}A\] where 𝜌 is the density of the air, 𝑣 is the velocity, 𝑐d is the drag coefficient, and 𝐴 is the frontal area. If a Mazda RX7 has a drag coefficient of .29, a frontal area of 5.95 square feet, and a max power output of 146 hp, and the density of air is .002326 slug/ft3 what is the theoretical top speed of the Mazda assuming it only has to fight wind resistance?

Problem 3 Diagram

Solution:



PDF Solution

Question 4:

You ask your little cousin to move a 1 kilogram box up a hill with a coefficient of kinetic friction of 0.2. Rather than carrying the box, he over-thinks things and drags the box up the hill with a rope. Determine the average power exerted by your little cousin if he applies a 10 newton force over a 3 meter distance with an incline of 30 degrees.

Problem 4 Diagram

Solution:



PDF Solution

Transcript