Student Resources
Integrals: Kinetic energy
This activity is meant as a group discussion, a group quiz, or a take home assessment with opportunity for revision based on feedback
Activity
Your overall goal in this activity is to calculate the kinetic energy of a uniformly dense rod that is rotating about one of its ends at a speed of one rotation per minute. Kinetic energy captures the energy of an object in motion. If an object has mass \(m\), and the entire object is moving at the same constant speed, \(v\), then the kinetic energy of the object is calculated as
\(\displaystyle KE=\frac{1}{2}m\cdot v^2.\)
Task One:
Use the provided random number generator to set the length and mass of the rod in the following GeoGebra applet.
Suppose that your randomly generated rod rotates counter-clockwise about one of its ends, as depicted in the following applet. (See “Applet Details” below for instructions on manipulating this applet.)
Give the integral that calculates the exact kinetic energy of your randomly generated rod, and use the Fundamental Theorem of Calculus to evaluate the integral. Be sure to specify what each symbol in the integral formula means in this context.
Task Two: Give the integral that calculates the exact kinetic energy of a rotating rod with mass \(M\) and length \(L\), and use the Fundamental Theorem of Calculus to evaluate the integral. Be sure to specify what each symbol in the integral formula means in this context.
Task Three: If the rod length is halved and the rotational speed doubles, how does this affect the kinetic energy? Be sure to specify what each symbol in your integral formula(s) means in this context.
Applet Details:
In this applet, you first input the randomly generated values of \(L\) and \(M\) for the "Rod Length" and "Rod Mass".
The "t=" slider shows the rod's position at time \(t\)-seconds as it rotates around during the minute-long rotation.
The "Segment Location=" and “Segment Length” sliders allow you to choose a size and location for a sample segment of the rod. On the left screen, the applet will simulate that segment of the rod moving a distance \(D\) on the time interval \([0,t]\). The distance on the left screen (visually represented by the length of the dashed green line) is calculated to be the same distance as the dashed green arc length shown on the right screen. Notice that altering the “Sample Radius” within the segment does indeed alter the arc length.
Given this sample information, the applet will calculate an approximate kinetic energy for the sample, as if the sample segment was entirely moving at the same speed.