Where is the center or mass of a wrench?

Center-of-Mass

A mechanical system moves as if all of its mass were concentrated at this point

A very special point used to describe the overall motion of a system

ProjectilesMotion can be described in two ways:

Rotation about a special point Parabolic curve traced by that same point

Where is the center of mass of a ball?

Where is the center or mass of a barbell?

Binary Stars

CoM The weighted average position of the system’s mass

xCM m1x1 m2x2 m3x3 ... mnxn

m1 m2 m3 ... mn

xCM miximi

yCM miyiM

zCM miz iM

Capital M is the sum of the masses.

In 3-D, you have to calculate it 3 times ; once along the x-axis, then y , then Z. This gives you the (x,y,z coordinates of the center of mass)

Complex, asymetric bodies of uniform density can be simplified by breaking them into easily identified, symmetric pieces and considering all of the mass of that piece to be at it’s geometric center.

L/2

L/2L

Answer= (0.5ML, 0.3ML)

Complex, asymetric bodies of uniform density can be simplified by breaking them into easily identified, symmetric pieces and considering all of the mass of that piece to be at it’s geometric center.

L/2

L/2L

CMx = M·0+3M·(L/2) + M·(L)

5M

CMy = M·0+3M·(L/2) + M·(0)

5M

Answer= (0.5ML, 0.3ML)

1. Chop it into manageable pieces.

2. Locate their CMs by symmetry

3. Fix an origin

4. Find CM of whole system by finding the x and y distance to each little CM and plugging into the CM formula.

We’ve now created a 5-body problem that is easier to handle than the original whole shape by taking advantage of symmetries.

1. Chop it into manageable pieces.

2. Locate their CMs by symmetry

3. Fix an origin

4. Find CM of whole system by finding the x and y distance to each little CM and plugging into the CM formula.

We’ve now created a 5-body problem that is easier to handle than the original whole shape by taking advantage of symmetries.

Answer: x = - 0.25 m, y = 0M\Therefor the missing piece of Q is Q /9 since Area = (2/6)2.

(-1.5,1.5)

(-1.5,-1.5)

(1.5,2)

(-1.5, -2)

(0.5,0)

¼ Q(-1.5) + ¼Q (-1.5) + 1/6 Q(1.5) + 1/6Q (1.5) + 1/18Q(0.5)__________________________________________

Q 8/9

Call the total mass before the cut out “Q”

CoM

The CoM can also be defined by its position vector, which starts at the origin and goes to the CM point.

rCM mixi ˆ i miyi ˆ j miz i

ˆ k M

rCM miri

M

rCM xCMˆ i yCM

ˆ j zCMˆ k

RCM

Example

Consider the following masses and their coordinates which make up a "discrete mass" or “rigid body”.

What are the coordinates for the center of mass of this system?

M

zmr

M

ymr

M

xmr

N

i iicm

N

i iicm

N

i iicm zyx

111

16

)10)(1()7)(10()2)(5(

16

)17)(1()2)(10()0)(5(

16

)10)(1()4)(10()3)(5(

1

1

1

M

zmr

M

ymr

M

xmr

N

i iicm

N

i iicm

N

i iicm

z

y

x

-0.94

0.19

4.4

kjircm ˆ4.4ˆ19.0ˆ94.0

Locating the CoM of an object without calculation

Throw it, watch for the point it

spins about.

Hang it from two different points and find the intersection

The CoM of a symmetric object lies on an axis of symmetry and on any plane of symmetry

Balance it on your finger

The CoM is a.k.a. the Center of Gravity

Each element of mass making up an object is acted upon by gravity.

The CoG is another special point where all gravitational force acts (Mg).

This is the same location as the CoM if g is constant over the object.

If an object is pivoted at is CoG, it balances in any orientation

Where is the Center of Gravity (CG)?

CG is at the midpoint for uniform objects.

CG is the balance point.

CG will be directly below a single point of suspension.

CG may exist where these is no actual material.

Old Physics formulas applied to the CoM

momentum

M

vm

t

rv iiCMCM

MvCM mivi pi ptot

Take the time derivative of the position of the center of mass The total linear

momentum of the system equals the total mass

multiplied by the velocity of the CoM.

Fireworks”: What does the CoM do?

See p. 193

Take another time derivativeAcceleration of the

Center of MassM

am

t

va iiCMCM

MaCM mia i FiThe CoM moves like an

imaginary particle of mass M under the

influence of the resultant external force on the

system.t

pMaF tot

CMext

Where is the CG?

In the head

In the air in the center

Where is the CG?

In the center air closer to the diamond

In the hole!

Toppling

If the CG of an object is above the area of support, the object will remain upright.

If the CG extends outside the area of support, the object will topple.

Stability

An object with a low CG is usually more stable than an object with a high CG.

Equilibrium

Unstable equilibrium An object balanced so that any displacement lowers its center

of gravity.

Stable equilibrium Any displacement raises its CG

Neutral equilibrium CG is neither raised or lowered with displacement

CoM

For odd-shaped, extended objects with continuous mass distribution?

xCM ximiMWhy is this only approximate?

We must let the number of mass elements approach infinity so this is not an approximation.

Replace the sum with an integral and the element

Δm with differential element dm:

M

mxx ii

mCM

0lim