Center of Mass
Torque
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.
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).
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.
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.
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.
Locating Center of Mass
• Center of mass can be “outside” an object.
7-9 CM for the Human Body
The x’s in the small diagram mark the CM of
the listed body segments.
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.
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.
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)
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.
CM of Two Particles
For two masses on a frictionless bar, where
is the center of mass?
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.
General Formulas for CM
M is total mass of system
M  m1  m2  m3  ....
xCM
x1m1  x2 m2  x3 m 3 ....


M
xm
i
M
i
Example
• m1=5 kg is located at x=10m from origin
• m2=10 kg is located at x=16m from origin
xCM
xCM
m1 x1  m2 x2

M
(5kg )(10)  (10kg )(16) 50  160 210



 14m
(5kg  10kg )
15
15
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.
xCM
m1 x1  m2 x2 (2kg )(0)  (3kg )(10m) 30



 6m
M
2kg  3kg
5

0
2kg
10m
3kg
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.
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.
Lever Arm
• To get the maximum
torque, the force
should be applied in a
direction that creates
the greatest lever
arm.
8-4 Torque
Here, the lever arm for FA 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.
Rate the forces using scale 1-3 according to their
effectiveness in turning the nut?
(1-most effective, 3- least effective).
8-4 Torque
A longer lever
arm is very
helpful in
rotating objects.
a
b
c
d
Torque
τ= Frsinθ
r – distance from the axis of rotation to the
point of the application of the force
The symbol for torque is Greek letter tau
SI unit of torque is N•m Newton-meter

Bar With Pivot
The bar is balanced if the net torque =0

20 N
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.
  0
(20 N )(1m)  F (2m)
F  10 N
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.
Positive torque
Negative torque
Balanced Torques
• Balance is achieved
if the torque that
tends to produce
clockwise rotation
by the boy equals
the torque that
tends to produce
counterclockwise
rotation by the girl.
Balanced Torques
20 N
10 N
1m
0.5 m
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.
F1r1  ( F2 r2 )  0
F1r1  F2 r2
Balanced Torques
20 N
10 N
1m
0.5 m
Left torque = 10 N x 1 m = 10 N m
Right torque = 20 N x 0.5 m = - 10 N m
Torquetotal  10 N  10 N  0
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.
What mass must be added to balance
the scale?
• Find the force that balances the torque
4m
2m
2kg
5m
1kg
?kg
Example
• Find the force that balances the torque
2m
10N
5m
?N
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?
F1  Weightcat
F2  500 N
F1r1  F2 r2
 mg  (4kg )(10m / s / s )  40 N
(40 N )(2m)  (500 N )r2
r2  80 / 500  0.16m
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?
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