Year+13+Translational+Motion

= = Center of Mass

This element of Mechanics includes Centre of mass (one and two dimensions), conservation of momentum and impulse (two dimensions only)


 * **__Center of Mass: __**

When calculating trajectories and collisions, it’s convenient to treat extended bodies, such as boxes and balls, as point masses. That way, we don’t need to worry about the shape of an object, but can still take into account its mass and trajectory. We can treat objects, and even systems, as point masses, even if they have very strange shapes or are rotating in complex ways. We can make this simplification because **there is always a point in the object or system that has the same trajectory as the object or system as a whole would have if all its mass were concentrated in that point**. That point is called the object’s or system’s center of mass.

Consider the trajectory of a diver jumping into the water. **The diver’s trajectory can be broken down into the translational movement of his center of mass, and the rotation of the rest of his body about that center of mass**. A human being’s center of mass is located somewhere around the pelvic area. We see here that, though the diver’s head and feet and arms can rotate and move gracefully in space, the center of mass in his pelvic area follows the inevitable parabolic trajectory of a body moving under the influence of gravity. If we wanted to represent the diver as a point mass, this is the point we would choose.



The center of mass or mass center is the mean location of all the mass in a system. In the case of a rigid body, the position of the center of mass is fixed in relation to the body. In the case of a **loose distribution of masses in free space**, such as shot from a shotgun or the planets of the solar system, the **position of the center of mass is a point in space among them that may not correspond to the position of any individual mass**. The center of mass is the point where all of the mass of the object is concentrated. **When an object is __supported at its center of mass there is no net torque acting on the body__ and it will remain in static equilibrium**. An easy way to determine the location of the center of mass of a rigid pole is to support the pole horizontally on one finger from each hand. Gently slide your fingers together. When your fingers meet, you will be at the center of mass at which time you can easily hold up the pole with only one finger as long as it can withstand the entire weight of the pole. Try it with a bat or a broom. If the object is uniform, for example a meter stick, the center of mass will be at the exact geometric center; if the object is irregular in shape the center of mass will be closer to the heavier end.



However, as long as a "plumb line" dropped from the center of mass falls within the area of an object's base of support, an object will not topple - for example:
 * the leaning tower of Pisa ,
 * a truck parked on a hillside,
 * a race car moving through a banked curve,
 * an acrobatic troop's pyramid act in the circus,
 * a professional unicyclist sensing how to lean as he accommodates the various dynamically-changing weight vectors in the system to keep his center of mass above the wheel's base of support, or simply
 * a person bending over to pick up an object from the floor.

Balancing Torques: m1gx1= m2gx2 <span style="font-family: 'Palatino Linotype','Book Antiqua',Palatino,serif;">m1x1= m2x2 <span style="font-family: 'Palatino Linotype','Book Antiqua',Palatino,serif;">m1x1= m2(d-x1)

<span style="font-family: 'Palatino Linotype','Book Antiqua',Palatino,serif;">POINTS OF THE DAY: COM can be for one or many objects. <span style="font-family: 'Palatino Linotype','Book Antiqua',Palatino,serif;"> COM is closer towards the heavier ‘end’.

<span style="display: block; font-family: 'Palatino Linotype','Book Antiqua',Palatino,serif; font-size: 120%; text-align: center;">To your right is an example of a page of this pdf.

<span style="display: block; font-family: 'Palatino Linotype','Book Antiqua',Palatino,serif; font-size: 120%; text-align: center;">This is a great tutorial on how to make animations believable in relation to physics principles.

<span style="display: block; font-family: 'Palatino Linotype','Book Antiqua',Palatino,serif; font-size: 120%; text-align: center;">Very interesting and very helpful for COM ideas and why things fall over etc. or look "wrong".

<span style="display: block; font-family: 'Palatino Linotype','Book Antiqua',Palatino,serif; font-size: 120%; text-align: center;">It's a big file but it's worth the wait, I promise.