Centre of Mass Created by J Harris [email protected].

Centre of MassCreated by J Harris

[email protected]

Centre of Mass

Find the mass of a random 2D shape:

• Cut of a random 2D shape out of paper / card

• Put 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 makes• Can 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 shape

The line of a plumb bob from the edge of the shape

The line of a plumb bob from a different edge gives the location of the centre of mass

Apr 18, 2023 Physics 1 (Garcia) SJSU

Locating Center of Gravity

Balance an object to find center of gravity

Centerof

Gravity

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 system’s 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 body

• This 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?

Apr 18, 2023 Physics 1 (Garcia) SJSU

Demo: Balanced Bird

Where is the bird’s center of gravity?

Try this one at home!

Apr 18, 2023 Physics 1 (Garcia) SJSU

Check Yourself

Three trucks

are parked on

a slope. Which

truck(s) tip

over?CG

CG

CG

BASEBASE

Pictures of motion of COM

Centre of Mass & Balance

• What 3 mechanical principles enable a

body to become more stable?

The lower the C.O.M. the more stable the body

The wider the base of support the more stable the body

The closer the line of gravity to the centre of the base of support, the more stable the body

Apr 18, 2023 Physics 1 (Garcia) SJSU

Demo: 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.

x CG

PEPSI

Apr 18, 2023 Physics 1 (Garcia) SJSU

Stability

Object is stable if CG is above the base.

CG CG

Wei

ght

Wei

ght

BASEAxis

AxisBASE

STABLE

UNSTABLE

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

m1 m2

m1

m2

m1 m2

m 1

m 2

m1 m2

m1g m2g

d1 d2

M = m1 + m2

The Centre of Mass

CMM

m1g x d1 = m2g x d2

Centre of Mass (1D)

0x1 x2

xcm

m1CoM m2

M = m1 + m2

M xcm = m1 x1 + m2 x2

Mxmxm

x 2211cm

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 CM

The Centre of Mass

•motion of points around the CM

Types of Collisions

• Momentum is conserved in any collision• Inelastic collisions: rubber ball and hard

ball– Kinetic energy is not conserved– Perfectly inelastic collisions occur when the

objects stick together

• Elastic collisions: billiard ball– both momentum and kinetic energy are conserved

• Actual collisions– Most collisions fall between elastic and perfectly

inelastic collisions

Conservation of Momentum

• In an isolated and closed system, the total momentum of the system remains constant in time. – Isolated system: no external forces– Closed system: no mass enters or

leaves– The linear momentum of each

colliding body may change– The total momentum P of the

system cannot change..

Motion of the Centre of Mass

• The 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 Mass

The 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.

m

mv

m

pvcm

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

Centre of Mass (3D)

iicm rmM1

r

For a collection of masses in 3D

cmr1

r3r2

y

0

2

21

1

3 kg

8 kg

4 kg

iicm rmM1

r

iicm ymM1

y

)1x42x80x3(15

1ycm

Xcm = 16/15 = 1.07 m

ycm = 20/15 = 1.33 m

15 kg

rCM

1.071.

33

iicm xmM

1x

)2x41x80x3(15

1xcm