Centre of MassCreated by J HarrisJharris.school@gmail.com

Centre of Mass

Find the mass of a random 2D shape:Cut of a random 2D shape out of paper / cardPut a pin through your card into the wall, at an edge of your card. Hang a plumb bob from the pin.Trace the line the string from the plumb bob makes Change the position of the pin, and trace the new line the string makesCan you find the centre of mass?

Find the mass of a random 2D shape:Start with any arbitrary shape

Find the mass of a random 2D shape:An arbitrary shapeThe line of a plumb bob from the edge of the shapeThe line of a plumb bob from a different edge gives the location of the centre of mass

*Physics 1 (Garcia) SJSULocating Center of GravityBalance an object to find center of gravityCenterofGravity

Centre of Mass:The centre of mass (CoM) is an imaginary point that can lie either inside (e.g. a shot put) or outside a body (hula hoop). A systems mass responds to external forces and torques, as if its entire mass were concentrated about that point.

Where is the COM? The position of centre of mass depends on the shape of bodyThis is how the high jumper can have his CoM pass under the bar but he could still clear the bar.

CoM depends on shape

Where is the COM?

*Physics 1 (Garcia) SJSUDemo: Balanced BirdWhere is the birds center of gravity?

Try this one at home!

*Physics 1 (Garcia) SJSUCheck YourselfThree trucksare parked ona slope. Whichtruck(s) tipover?

CGCGCG

Pictures of motion of COM

Centre of Mass & BalanceWhat 3 mechanical principles enable a body to become more stable?The lower the C.O.M. the more stable the bodyThe wider the base of support the more stable the bodyThe closer the line of gravity to the centre of the base of support, the more stable the body

*Physics 1 (Garcia) SJSUDemo: Balance the CanIf a small amount of water is added to an empty soda can then the can may be balanced on its bottom edge. xCGPEPSI

*Physics 1 (Garcia) SJSUStabilityObject is stable if CG is above the base.CGCGWeightWeightBASEAxisAxisBASESTABLEUNSTABLE

What is the difference between COG & COM?The term Centre of Mass is often confused with the Centre of Gravity.The two terms are so similar that they can be used interchangeably.The COG of an object coincides with its COM if the object is in a completely uniform gravitational field.If it is not in a uniform gravitational field, then the COG and the COM will be at too different locations.

Finding CoM in 2D

M = m1 + m2The Centre of MassMm1g x d1 = m2g x d2

Centre of Mass (1D)CoMM = m1 + m2M xcm = m1 x1 + m2 x2

See pic on pg 35 of rutter

Why study Center of Mass?(a) A ball tossed into the air follows a parabolic path. (b) The center of mass (black dot) of a baseball bat flipped into the air follows a parabolic path, but all other points of the bat follow more complicated curved paths.

The motion of the Centre of Mass is a simple parabola.(just like a point particle)The motion of the entire object is complicated.This motion resolves to motion of the CMThe Centre of Massmotion of points around the CM

Types of CollisionsMomentum is conserved in any collisionInelastic collisions: rubber ball and hard ballKinetic energy is not conservedPerfectly inelastic collisions occur when the objects stick togetherElastic collisions: billiard ballboth momentum and kinetic energy are conservedActual collisionsMost collisions fall between elastic and perfectly inelastic collisions

Conservation of MomentumIn an isolated and closed system, the total momentum of the system remains constant in time. Isolated system: no external forcesClosed system: no mass enters or leavesThe linear momentum of each colliding body may changeThe total momentum P of the system cannot change.

Motion of the Centre of MassThe complex motion of many separate masses can often be made clear by only considering the single motion of the CoM. Even though the separate masses may collide and interact, the velocity of the CoM vcm,does not change, assuming no external forces are present.

Motion of the Centre of MassThe velocity of the centre of mass can be found by adding up all the individual momentums. Then to find the velocity use p=mv. So divide by the total mass will give the average / overall velocity called the velocity of the centre of mass.

http://www.youtube.com/watch?v=gPDjaMsG_rg

Centre of Mass (3D)For a collection of masses in 3D

Xcm = 16/15 = 1.07 mycm = 20/15 = 1.33 m

****See video. **********