Unit #4 Momentum

Momentum – Center of Mass

http://www.aplusphysics.com/courses/honors/momentum/honors_center_of_mass.html

Unit #4 Momentum

Objectives and Learning Targets

Define and calculate the momentum of an object.

Determine the impulse given to an object.

Use impulse to solve a variety of problems.

Interpret and use force vs. time graphs.

Apply conservation of momentum to solve a variety of problems.

Distinguish between elastic and inelastic collisions.

Calculate the center of mass for a system of point particles.

Unit #4 Momentum

Defining Center of Mass

• The motion of real objects is considerably more complex than that of simple theoretical particles. However, we can treat an entire object as if its entire mass were contained at a single point, known as the object's center of mass (CM). Mathematically, the center of mass of an object is the weighted average of the location of mass in an object.

• We can find the center of mass of a system of particles by taking the sum of the mass of the particles, multiplied by their positions, and dividing that by the total mass of the object. Looking at this in two dimensions, the center of mass in the x- and y-directions would be:

Unit #4 Momentum

Defining Center of Mass• No matter how complex an object may be, we can calculate its center of mass and then

treat the object as a point particle with total mass M. This allows us to apply our basic physics principles to complex objects without adding unnecessary mathematical complexity to our analyses!

Unit #4 Momentum

Sample Problem #1Center of Mass 1D

• Find the center of mass of an object modeled as two separate masses on the x-axis. The first mass is 2 kg at an x-coordinate of 2 and the second mass is 6 kg at an x-coordinate of 8.

Answer:

Unit #4 Momentum

Sample Problem #2Center of Mass in Multiple Dimensions

• We can use the same strategy for finding the center of mass of a multi-dimensional object.

• Question: Find the coordinates of the center of mass for the system shown below.

Unit #4 Momentum

Sample Problem #2Center of Mass in Multiple Dimensions

• Question: Find the coordinates of the center of mass for the system shown below.

• Answer:

Unit #4 Momentum

Center of Mass vs. Center of Gravity

Center of Gravity• Note that center of mass is not the same as center of gravity. In a uniform gravitational

field, they are the same, but center of gravity refers to the location at which the force of gravity acts upon an object as if it were a point particle with all its mass focused at that point, a subtle but important difference.