Deal with combining rigid motions. Given two rigid motions M and N, we can combine the two rigid motions by first applying it and then applying X to the result. The rigid motion defined by combining M and N (M goes first, N goes second) is called the product of M and N. In each case, state whether the rigid motion M is proper or improper. (a) M is the product of a proper rigid motion and an improper rigid motion. (b) M is the product of an improper rigid motion and an improper rigid motion. (c) M is the product of a reflection and a rotation. (d) M is the product of two reflections.
The wheels on a moving bicycle have both translational (or linear) and rotational motions. What is meant by the phrase "a rigid body, such as a bicycle wheel, is in equilibrium"? (a) The body cannot have translational or rotational motion of any kind. (b) The body can have translational motion, but it cannot have rotational motion. (c) The body cannot have translational motion, but it can have rotational motion. (d) The body can have translational and rotational motions, as long as its translational acceleration and angular acceleration are zero.
Deal with combining rigid motions. Given two rigid motions M and N, we can combine the two rigid motions by first applying it and then applying X to the result. The rigid motion defined by combining M and N (M goes first, N goes second) is called the product of M and N. Suppose that a rigid motion it is the product of a reflection with axis $$ l_1 $$ and a reflection with axis $$ l_2 $$ , where $$ l_1 $$ and $$ l_2 $$ intersect at a point C. Explain why M must be a rotation with center C.
You have a pendulum pulled to the side, a heavy beach ball pushed partly under water, and a medicine ball lifted some height above the ground. Each object is then released. (a) Draw a picture of the motion of each object and a motion diagram for each. (b) Indicate the equilibrium position. (c) Draw two or more force diagrams at key points in the motion. (d) Use (a) through (c) to reason whether the motions of these objects can be considered vibrational motions.
For one day, write down a brief description of the different kinds of motions you see. Work with a group to generate a common list of the different kinds of motions observed. What reference points did you use to detect the motion? How did these reference points help you determine motion? Compare your list with those from other groups in your class.
At a homeowners' association meeting, a board member can vote yes, vote no, or abstain on a motion. There are three motions on which a board member must vote. (a) Determine the number of sample points in the sample space. (b) Construct a tree diagram and determine the sample space. If all votes are equally likely, determine the probability that a board member votes (c) No, yes, no in that order. (d) Yes on exactly two of the motions. (e) Yes on at least one motion.
A music department consists of a band director and a music teacher. Decisions on motions are made by voting. If both members vote in favor of a motion, it passes. If both members vote against a motion, it fails. In the event of a tie vote, the principal of the school votes to break the tie. For this voting scheme, determine the Banzhaf power index for each department member and for the principal.
You are an observer on the ground. (a) Draw two motion diagrams representing the motions of two runners moving at the same constant speeds in opposite directions toward you.Runner 1, coming from the east, reaches you in 5 s, and runner 2 reaches you in 3 s. (b) Draw a motion of diagram for the second runner as seen by the first runner.