TEACHER NOTES

ISSUES RELATED TO THE PROBLEM

There is often a lot of interest in how the motor in a car works. This problem is oriented towards the physics of the piston-cylinder arrangement. It suggests the first step in finding the forces and torques in the engine, this step being an analysis of the engine's motion. The first goals, then, are the acceleration of the piston, as well as the angular velocity and acceleration of the connecting rod. The accelerations of the centers of mass of the connecting rod and the crank are also important; they may be investigated in an extension to this problem.

The kinematics of bodies which move in a plane is not too difficult. In this case it is possible to use the geometry of a simple triangle which represents the instantaneous configuration of the device. Two of the sides of the triangle have constant length. The independent variable is q, the angle the crank makes with the positive x direction. Differentiation of the trigonometric description of the triangle produces the needed expressions.

One difficulty that the students face is the management of the unknown quantities in the problem. It requires some definite advance planning. If the student is using a CAS, then this issue is less important.

Prerequisites

The student should be aware of the definitions of position, velocity and acceleration.

The student should have some familiarity with geometry, and be able to differentiate using the chain rule. The student must also have enough familiarity with reciprocating engines to be able to visualize the motion. This may be a problem for some students. Many science books contain good 'cut-away' drawings of the engine. It might be a good idea to have the student sketch the basic configuration for several different crank angles, including 0, 90, 180 and 270 degrees.

Time allotment - time management

If you are using a computer algebra system

The problem is not very hard. The key is to identify the necessary geometric relationships (see POSSIBLE SOLUTIONS) and then to key them into the system. The plots may then be obtained posthaste. It would probably be a 20 minute problem. There would be some time to explore the suggested extensions.

If you are not using a computer algebra solution

This problem is a lengthy one. It takes students some time just to grasp what needs to be done. The work of grinding out the derivatives and seeing how the pieces fit together to give the solution is also long. This problem might take more than 50 minutes unless hints were given. If you give hints and stop at finding the velocity and acceleration of the piston, students might finish in less than 50 minutes.

The use of a graphing calculator is strongly recommended.

Expectations

Trouble may result from the following:
(1) Not knowing how to use the chain rule in this situation.
(2) Defeat due to complexity of expressions. If the problem is done by hand, and the plots done with a graphing calculator, then develop the expressions in the following order:
f,
df/dt
d2f/dt2
x,
dx/dt,
d2x/dt2
Do not substitute these expressions into each other. The expressions just get too messy! If you proceed in this order and enter the functions into the graphing calculator correctly, then things should work out pretty well.

Future payoffs

This problem is intended to help students begin to recognize the links between elementary calculus and kinematics. It connects these ideas to the real world.

Extensions

There is an obvious extension to the problem: the dynamic analysis of the engine. To do the above extension, one needs the velocity and acceleration of the mass centers of the crank and the connecting rod, in addition to those of the piston, which has just been computed.