General Physics I with Calculus

Assignment #8 Due: 11:00pm on Saturday, July 20, 2013 You will receive no credit for items you complete after the assignment is due. Grading Policy Center of Mass and External Forces Learning Goal: Understand that, for many purposes, a system can be treated as a point-like particle with its mass concentrated at the center of mass. A complex system of objects, both point-like and extended ones, can often be treated as a point particle , located at the system's center of mass . Such an approach can greatly simplify problem solving. Before you use the center of mass approach, you should first understand the following terms: System: Any collection of objects that are of interest to you in a particular situation. In many problems, you have a certain freedom in choosing your system. Making a wise choice for the system is often the first step in solving the problem efficiently. Center of mass: The point that represents the "average" position of the entire mass of a system. The postion of the center of mass can be expressed in terms of the position vectors of the particles as . The x coordinate of the center of mass can be expressed in terms of the x coordinates of the particles as . Similarly, the y coordinate of the center of mass can be expressed. Internal force: Any force that results from an interaction between the objects inside your system. As we will show, the internal forces do not affect the motion of the system's center of mass. External force: Any force acting on an object inside your system that results from an interaction with an object outside your system. Consider a system of two blocks that have masses and . Assume that the blocks are point-like particles and are located along the x axis at the coordinates and as shown . In this problem, the blocks can only move along the x axis. Part A r  cm r  i = r  cm ∑ m i r  i ∑ m i x cm ( r x ) i = x cm ∑ ( m i r x ) i ∑ m i m 1 m 2 x 1 x 2 Find the x coordinate of the center of mass of the system. Express your answer in terms of , , , and . ANSWER: Correct Part B If , then the center of mass is located: ANSWER: Correct Part C If , then the center of mass is located: ANSWER: Correct Part D Recall that the blocks can only move along the x axis. The x components of their velocities at a certain moment are and . Find the component of the velocity of the center of mass at that moment. Keep in mind that, in general: . x cm m 1 m 2 x 1 x 2 = x cm + m 1 x 1 m 2 x 2 + m 1 m 2  m 2 m 1 to the left of at a distance much greater than to the left of at a distance much less than to the right of at a distance much less than to the right of at a distance much greater than to the right of at a distance much less than to the left of at a distance much less than m 1 - x 2 x 1 m 1 - x 2 x 1 m 1 - x 2 x 1 m 2 - x 2 x 1 m 2 - x 2 x 1 m 2 - x 2 x 1 = m 2 m 1 at at half-way between and the answer depends on and m 1 m 2 m 1 m 2 x 1 x 2 v 1x v 2x x ( v cm ) x = dx / dt v x

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