Center of Mass - PowerPoint Presentation

Slide1

Center of MassTorqueSlide2

Center of Mass

When analyzing the motion of an extended object, we treat the entire object as if its mass were contained in a single point, known as the object’s

center of mass (CM).

Mathematically, the CM of an object is the weighted average of the location of the mass in an object.Slide3

7-8 Center of Mass

In (a), the diver’s motion is pure

translation

; in (b) it is

translation

plus

rotation.There is one point that moves in the same path a

particle would take if subjected to the same force as the diver. This point is the center of mass (CM).Slide4

7-8 Center of Mass

The

general

motion of an object can be considered as the

sum

of the

translational motion of the CM, plus rotational, vibrational

, or other forms of motion about the CM.Slide5

7-8 Center of Mass

The

center of gravity

is the point where the gravitational force can be considered to act. It is the same as the

center of mass

as long as the gravitational force does not

vary among different parts of the object.Slide6

7-8 Center of Mass

The center of gravity can be found

experimentally

by

suspending

an object from different points. The CM need not be

within

the actual object – a doughnut’s CM is in the center of the hole.Slide7

Locating Center of Mass

Center of mass can be “outside” an object.Slide8

7-9 CM for the Human Body

The x’s in the small diagram mark the CM of the listed

body

segments.Slide9

7-9 CM for the Human Body

The

location

of the center of mass of the leg (circled) will depend on the

position

of the leg.Slide10

7-9 CM for the Human Body

High jumpers have developed a technique where their CM actually passes

under

the bar as they go over it. This allows them to clear

higher

bars.Slide11

7-10 Center of Mass and Translational Motion

The

total momentum

of a system of particles is equal to the product of the

total mass

and the

velocity of the center of mass.The sum of all the forces acting on a system is equal to the total mass of the system multiplied by the acceleration

of the center of mass:

(7-11)Slide12

7-10 Center of Mass and Translational Motion

This is particularly useful in the analysis of

separations

and

explosions

; the center of mass (which may not correspond to the position of any particle) continues to move according to the net force.Slide13

CM of Two Particles

For two masses on a frictionless bar, where is the center of mass?Slide14

7-8 Center of Mass

For two particles, the

center of mass

lies closer to the one with the most mass:

where

M is the total

mass.Slide15

General Formulas for CM

M is total mass of systemSlide16

Example

m1=5 kg is located at x=10m from origin

m

2

=10 kg is located at x=16m from originSlide17

Do Now

Find the center of mass of a system

two masses

if a 2 kg mass is located at x=0 and a 3 kg mass is located at x=10m.

2kg

3kgSlide18

Torque

To make an object

start

rotating

,

a force is needed.If you push at the edge of the door with a force perpendicular to the door, the door rotates around the axes that passes through the hinge.The ability of a force to rotate an object around some axis is measured by a quantity called a torque.Slide19

8-4 Torque

The

position

and

direction

of the force

that creates rotation is very important.The perpendicular distance from the axis of rotation to the line along which the force acts is called the lever arm.Slide20

Lever Arm

To get the maximum torque, the force should be applied in a direction that creates the greatest

lever arm.

Slide21

8-4 Torque

Here, the lever arm for F

A

is the distance from the

knob

to the

hinge; the lever arm for FD is zero; and the lever arm for FC is as shown.Slide22

Rate the forces using scale 1-3 according to their

effectiveness in turning the nut?

(1-most effective, 3- least effective).Slide23

8-4 Torque

A

longer

lever arm is very helpful in rotating objects.

Slide24
Slide25
Slide26
Slide27

Torque

τ

=

Frsin

θ

r – distance from the axis of rotation to the point of the application of the forceThe symbol for torque is Greek letter tauSI unit of torque is N•m Newton-meterSlide28

Bar With Pivot

The bar is balanced if the net torque =0

Pivot

Weight of bar =20N Acts as if it was at the center of mass.

What force should be applied at x to balance the torque? The length of the bar = 2m.Slide29
Slide30

The Sign of a Torque

Torque is a vector.

If

a positive torque is applied, the object will start rotating in a counter-clockwise

direction.

Negative torque produces clockwise rotation.

Negative torquePositive torqueSlide31

Balanced Torques

Balance is achieved if the torque that tends to produce

c

lockwise rotation by the boy equals the torque that tends to produce counterclockwise rotation by the girl.Slide32

Balanced Torques

10 N

20 N

1 m

0.5 mSlide33

Balanced Torques

If the total torque

on a motionless object is zero, the object will be balanced and not start rotating.

Thus the sum of all torques on an object at equilibrium must be zero.Slide34

Balanced Torques

10 N

20 N

1 m

0.5 m

Left torque = 10 N x 1 m =

10 N

m

Right torque = 20 N x 0.5 m =

- 10

N mSlide35

Rotational Equilibrium

When an object is in rotational equilibrium,

the total torque applied to it is zero.

Rotational equilibrium is often used to determine unknown forces.Slide36

What mass must be added to balance the scale?

Find the force that balances the torque

?kg

2kg

5m

4

m1kg2mSlide37

Example

Find the force that balances the torque

?

N

10N

5m

2mSlide38

Example

A boy and his cat sit on a

seesaw. The

cat has a mass of 4 kg and sits 2 m from the center of rotation.

If

the boy has a mass of 50 kg, where should he sit so that the see-saw will balance?Slide39

Questions:

1) What is torque?

2)How do we calculate torque?

3) What are the units of measurement of torque?

4) What is rotational equilibrium?Slide40

Practice Problem

What is the torque on a bolt applied with a wrench that has a lever arm of 45 cm with a force of 10 N?

Torque = F x r = (10 N)(0.45m) = 4.5 N m